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1.14.7: Boundary

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.07%3A_Boundary

The term ‘boundary’ is encountered in several contexts in thermodynamics. Indeed the definition becomes more complicated as we concentrate on the thermodynamic properties of systems.As a starting point, a boundary separates system and surroundings. A boundary is an infinitely thin surface separating system and surroundings such that the properties of system and surroundings change abruptly at the boundary. In these terms a reaction vessel is part of the surroundings. We support this careful distinction by observing that if chemical reaction inside the system is exothermic, the liberated heat warms the reaction vessel. In this case, the boundary encloses the system. No molecules can either enter or leave the system. However heat is allowed to cross the boundary. Thus the whole universe is divided into system and surroundings, the only role of the boundary is to facilitate communication between system and surroundings. In these terms chemical substances, heat, and electric charge may cross a boundary between a system and surroundings.In some cases the container (e.g. reaction vessel) may be considered part of the system. In many cases it is correct to do so and so the boundary is again a hypothetical surface separating ‘reaction solution + reaction vessel’ and the surroundings.In general terms, it is important to define the boundary for a given system. Another term for boundary is ‘envelope’ which indicates something which can be quite dynamic in terms of shape and volume rather than, for example, a glass vessel. Moreover the boundary may be selectively permeable to one or more chemical substances rather like the envelope of unit cells in living systems.The term ‘boundary’ in the context of surface chemistry means a boundary phase (or, capillary phase). In such a phase there is a concentration gradient of one or more chemical substances across the boundary phase between system and surroundings. Indeed surface chemistry can be described as the chemistry of boundaries.In summary we repeat the point that in a thermodynamic analysis of experimental results, a first requirement is that the system, boundary and surroundings are carefully defined. For the most part we assume that the boundary is an infinitely thin envelope separating system and surroundings.Footnotes K. S. Pitzer, Thermodynamics, McGraw-Hill, New York,1995, 3rd. edn., page 6. G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, New York, 1923, page 10. D. H. Everett, Chemical Thermodynamics, Longmans, London, 1959, page 8. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979,page 1. E. B. Smith, Basic Chemical Thermodynamics, Clarendon Press, Oxford, 3rd. edn.,1982, page 2. S. E. Wood and R. Battino, Thermodynamics of Chemical Systems, Cambridge University Press, Cambridge, 1990, page 3. J. N. Bronsted, Physical Chemistry, Heinemann, London, 1937, page 359.This page titled 1.14.7: Boundary is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.8: Calculus

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.08%3A_Calculus

Consider a variable u defined by the independent variables \(x\) and \(y\). \[\text { We write } u=u[x, y]\]Equation (b) is the general exact differential of equation (a). \[\mathrm{du}=\left(\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \, \mathrm{dx}+\left(\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)_{\mathrm{x}} \, \mathrm{dy}\]In other words the change in u is related to the differential dependence of \(\mathrm{u}\) on \(x\) at constant \(y\) and the differential dependence of \(\mathrm{u}\) on \(y\) at constant \(x\). For the case where u does not change, \[\left(\frac{\partial u}{\partial x}\right)_{y}=-\left(\frac{\partial u}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial x}\right)_{u}=0 \text { and }\left(\frac{\partial y}{\partial x}\right)_{u}=-\left(\frac{\partial u}{\partial x}\right)_{y} \,\left(\frac{\partial y}{\partial u}\right)_{x}\]A variable \(z\) is defined by the independent variables \(x\) and \(y\). \[z=z[x, y]\]Equation (e) is the general differential of equation (d). d z=\left(\frac{\partial z}{\partial x}\right)_{y} \, d x+\left(\frac{\partial z}{\partial y}\right)_{x} \, d y\]We direct attention to the dependence of \(z\) on \(x\) along a pathway for which \(\mathrm{u}\) is constant. \[\text { Then }\left(\frac{\partial z}{\partial x}\right)_{u}=\left(\frac{\partial z}{\partial x}\right)_{y}+\left(\frac{\partial z}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial x}\right)_{u}\]The latter equation contains the differential dependence of \(y\) on \(x\) at constant \(\mathrm{u}\). The latter dependence can be reformulated using equation (c). Therefore \[\left(\frac{\partial z}{\partial x}\right)_{u}=\left(\frac{\partial z}{\partial x}\right)_{y}-\left(\frac{\partial u}{\partial x}\right)_{y} \,\left(\frac{\partial y}{\partial u}\right)_{x} \,\left(\frac{\partial z}{\partial y}\right)_{x}\]The key point to emerge from this exercise centres is the way in which the condition on the partial differential \((\partial z / \partial x)\) can be changed from ‘at constant \(y\)’ to ‘at constant \(\mathrm{u}\)’.Another important operation concerns a variable \(\mathrm{q}\). \[\text { Thus, }\left(\frac{\partial x}{\partial y}\right)_{z}=\left(\frac{\partial x}{\partial q}\right)_{z} \,\left(\frac{\partial q}{\partial y}\right)_{z}\]For composite functions such as \(z=z[\mathrm{u}, \mathrm{v}]\), where \(z=z[x, y]\), and \(\mathrm{u}=\mathrm{u}[\mathrm{x}, \mathrm{y}]\), further important equations are found. \[\text { Thus }\left(\frac{\partial z}{\partial u}\right)_{v}=\left(\frac{\partial z}{\partial x}\right)_{y} \,\left(\frac{\partial x}{\partial u}\right)_{v}+\left(\frac{\partial z}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial u}\right)_{v}\]Equation (i) is an example of the well-known chain rule, a similar equation holding for \((\partial z / \partial v)_{u}\). This rule allows the total change of independent variables from \(z=z[\mathrm{u}, \mathrm{v}]\) to \(z=z[x, y]\). \[\text { Also }\left(\frac{\partial z}{\partial x}\right)_{y}=\left(\frac{\partial z}{\partial x}\right)_{y, v}+\left(\frac{\partial z}{\partial v}\right)_{y, x} \,\left(\frac{\partial v}{\partial x}\right)_{y}\]The latter equation is useful for introducing an extra constraint on a given differential.Footnote H. B. Callen, Thermodynamics and an Introduction to Thermostatics, Wiley, New York, 2dn. Edn.,1985, Appendix A.This page titled 1.14.8: Calculus is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.9: Clausius - Clapeyron Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.09%3A_Clausius_-_Clapeyron_Equation

A give closed system contains chemical substance j present in both liquid and gas phases. The system is at equilibrium. In terms of the Phase Rule, the following parameters are defined; \(\mathrm{P} = 2\), \(\mathrm{C} = 1\) and hence \(\mathrm{F} = 1\). Hence, if the temperature is fixed by the observer, the equilibrium pressure \(\mathrm{p}^{\mathrm{eq}}\) is defined. The equilibrium can be described in terms of an equality of chemical potentials of pure liquid and pure gas.\[\mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \label{a}\]Both chemical potentials in Equation \ref{a} are functions of both \(\mathrm{T}\) and \(\mathrm{p}\). In general terms,\[\mathrm{d} \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\left(\frac{\partial \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp} \label{b}\]or\[ \mathrm{d} \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \label{c}\]Similarly\[\mathrm{d} \mu_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})=-\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \label{d}\]The condition in Equation \ref{a} applies at all \(\mathrm{T}\) and \(\mathrm{p}\).\[\begin{aligned} &\text { Then, }-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \\ &=-\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \end{aligned}\]or,\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{e q}=\frac{\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})-\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}\]\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{\mathrm{eq}}=\frac{\Delta_{\mathrm{vap}} \mathrm{S}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}\]But at equilibrium, \[\Delta_{\text {vap }} \mathrm{G}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})=\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})-\mathrm{T} \, \Delta_{\text {vap }} \mathrm{S}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})=0\]\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{\text {eq }}=\frac{\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{T} \, \Delta_{\mathrm{vap}} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}\]The latter is the Clausius-Clapeyron Equation. In a modern development, equation (i) was exactly integrated. Equation(i) does not have the form of an exact differential in the independent variables \(\mathrm{p}\) and \(\mathrm{T}\). The corresponding integrating factor is \(\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}\).Thus\[\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*} \, \mathrm{dp}-\mathrm{T}^{-2} \, \Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*} \, \mathrm{dT}=0\]or,\[\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*} \, \mathrm{dp}+\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*} \, \mathrm{dT}^{-1}=0\]The latter equation is an exact differential as a consequence of equation (\(\ell\)).\[\left(\frac{\partial\left(\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}\right)}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{p}}=\left(\frac{\partial \Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]A mathematical solution is known for differential equations having the form of equation (k). A comprehensive set of equations have been derived describing first order transitions for pure substances and hence the phase equilibrium curves For liquid-vapour equilibria, both \(\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p}) \text { and } \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p}) \text { are }>0\). Therefore the equilibrium vapor pressure of a liquid increases with increase in temperature. A useful approximation assumes that gas \(j\) is a perfect gas; i.e. \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp})=\mathrm{R} \, \mathrm{T}\) and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})>>\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{l} ; \mathrm{T} ; \mathrm{p})\).\[\left(\frac{\mathrm{d} \ln (\mathrm{p})}{\mathrm{dT}}\right)^{\text {eq }}=\frac{\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{R} \, \mathrm{T}^{2}}\]\[\left(\frac{\mathrm{d} \ln (\mathrm{p})}{\mathrm{d}\left(\mathrm{T}^{-1}\right)}\right)^{\mathrm{eq}}=-\frac{\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{R}}\]Within the limits of the approximations outlined above, \(\ln \left(p^{e q}\right)\) is a linear function of \(\mathrm{T}^{-1}\).Exactly integrated equations have also been established for other first-order transitions (\(\mathrm{p}{\mathrm{eq}}\), \(\mathrm{T}{\mathrm{eq}}\)) curves of pure substances.Footnotes \(\frac{\left[\mathrm{N} \mathrm{m}^{-2}\right]}{[\mathrm{K}]}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}=\frac{\left[\mathrm{J} \mathrm{m}^{-3}\right]}{[\mathrm{K}]}=\frac{\left[\mathrm{N} \mathrm{m}^{-2}\right]}{[\mathrm{K}]}\) We have derived the equation for vapor-liquid equilibrium which is the generally quoted form. An equivalent form expresses \(\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{e q}\) for the equilibrium for chemical substance \(j\) in two phases \(\alpha\) and \(\beta\). C. Mosselman, W. H. van Vugt and H. Vos, J. Chem. Eng. Data 1982,27,246. From \(\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}\)\(\begin{aligned} &\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{T} \, \left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{V} \\ &\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{T} \, \left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{T}}+\mathrm{V} \\ &\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{T}^{-1} \, \left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{T}}+\mathrm{V} \\ &\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial\left(\mathrm{T}^{-1} \, \mathrm{V}\right)}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{T}} \end{aligned}\) L. Q. Lobo and A. G. M. Ferreira, J. Chem. Thermodyn., 2001,33,1597.This page titled 1.14.9: Clausius - Clapeyron Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.1: "Excess" Thermodynamic Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.14%3A_Excess_and_Extra_Thermodynamics/1.14.1%3A_%22Excess%22_Thermodynamic_Properties

Description of the thermodynamic properties of both solutions and liquid mixtures in terms of excess properties has considerable merit. However care has to be exercised in defining excess properties. It is not sufficient, for example, to argue that all properties of a given binary liquid mixture would be linear functions of mole fraction composition in the event that the properties of this mixture are “ideal”. There is of course no rule that states one is forbidden from doing this. But it is not allowed to identify immediately deviations “ideal” with those features responsible for deviation from thermodynamically defined ideal based on, for example, Gibbs energies of mixing. Similarly it is not good practice to use an arbitrary definition of the properties of ideal solutions and to account for deviations from ideal directly in terms of fraction responsible for deviations in the thermodynamic properties of the solution from ideal.Care has to be exercised in anticipating patterns in the properties of solutions which can be claimed as ideal from a thermodynamic viewpoint. We illustrate the point with reference to expansivities in the context of binary liquid mixtures. A given liquid mixture is prepared at defined \(\mathrm{T}\) and \(\mathrm{p}\) using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{2}\) moles of liquid 2. Then, \[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix})\]At fixed pressure (and at ‘\(\mathrm{A} = 0\)’), \[[\partial \mathrm{V}(\mathrm{mix}) / \partial \mathrm{T}]_{\mathrm{p}}=\mathrm{n}_{1} \,\left[\partial \mathrm{V}_{1}(\mathrm{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{n}_{2} \,\left[\partial \mathrm{V}_{2}(\mathrm{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}}\]We divide through by \(\mathrm{V}(\mathrm{mix})\) and incorporate new terms. \[\begin{aligned} & {[1 / V(\text { mix })][\partial \mathrm{V}(\text { mix }) / \partial \mathrm{T}]_{\mathrm{p}} } \\ =& {\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{mix}) / \mathrm{V}(\mathrm{mix})\right] \,\left[1 / \mathrm{V}_{1}(\mathrm{mix})\right]\left[\partial \mathrm{V}_{1}(\mathrm{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}} } \\ &+\left[\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) / \mathrm{V}(\mathrm{mix})\right]\left[1 / \mathrm{V}_{2}(\mathrm{mix})\right]\left[\partial \mathrm{V}_{2}(\mathrm{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}\]The term \(\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix}) / \mathrm{V}(\operatorname{mix})\right]\) is an effective volume fraction for substance 1, \(\phi_{1}\); similarly for \(\phi_{2}\). We define effective component expansibilities, \(\alpha_{1}\) and \(\alpha_{2}\). Thus, \[\alpha_{1}=\left[1 / V_{1}(\mathrm{mix})\right] \,\left[\partial \mathrm{V}_{1}(\mathrm{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}}\]and \[\alpha_{2}=\left[1 / V_{2}(\operatorname{mix})\right] \,\left[\partial V_{2}(\operatorname{mix}) / \partial T\right]_{p}\]Hence (using equation (c) \[\alpha_{p}(\operatorname{mix})=\phi_{1}(\operatorname{mix}) \, \alpha_{1}(\operatorname{mix})+\phi_{2}(\operatorname{mix}) \, \alpha_{2}(\operatorname{mix})\]For an ideal mixture, \[\mathrm{V}(\text { id:mix })=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Since \(\phi_{1}(\mathrm{id})\) equals \(\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{V}(\mathrm{id}: \operatorname{mix})\), the volume fraction for the ideal mixture [similarly for \(\phi_{2} \text { (id) }\)], then \[\alpha_{\mathrm{p}}(\mathrm{id}: \operatorname{mix})=\phi_{1}(\mathrm{id}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\phi_{2}(\mathrm{id}) \, \alpha_{\mathrm{p} 2}^{*}(\ell)\]An excess (isobaric) thermal expansivity is given by equation (i). \[\alpha_{p}^{E}(\operatorname{mix})=\alpha_{p}(\operatorname{mix})-\alpha_{p}(\text { id : mix })\]Equation (h) confirms that \[\alpha_{p}(\text { id : mix }) \neq \mathrm{x}_{1} \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \alpha_{\mathrm{p} 2}^{*}(\ell)\]In other words the definition of an ideal property using the otherwise conventional form in equation (j) is invalid.Footnotes M. I. Davis and G. Douheret, Thermochim. Acta,1991,190,267. H. L. Friedman, J.Chem.Phys.,1960,32,1351. R. W. Missen, Ind. Eng. Chem., Fundam.,1969,8,81.This page titled 1.14.1: "Excess" Thermodynamic Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.10: Extrathermodynamics - Solvent Effects in Chemical Kinetics

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.14%3A_Excess_and_Extra_Thermodynamics/1.14.10%3A_Extrathermodynamics_-_Solvent_Effects_in_Chemical_Kinetics

An enormous chemical literature describes the effects of solvents on rates of chemical reactions. C. K. Ingold in his classic monograph actually uses the phrase 'solvent polarity' when commenting on the relative rates of reactions through a series of solvents of diminishing "polarity". One of the reactions discussed by Ingold concerns the solvolysis of 2-chloro-2-methyl -propane, \(\left(\mathrm{CH}_{3}\right)_{3}\mathrm{CCl}\). In 1948, Winstein and Grunwald used this reaction as a basis for a quantitative treatment of solvent polarities leading to the definition of solvent Y-value. The basis of their analysis can be understood using an extrathermodynamic analysis. We use a description based on substituent zone \(\mathrm{R}\) and reaction zone \(\mathrm{X}\) for a solute molecule \(\mathrm{RX}\). Here the interaction between the two zones is solvent dependent.The starting point is kinetic data describing an (assumed) unimolecular first order solvolysis of a solute \(\mathrm{RX}\). The chemical reaction proceeds through a transition state \(\mathrm{RX}^{neq}\). For a given solvent medium \(\mathrm{M}\) at defined \(\mathrm{T}\) and \(\mathrm{p}\), transition state theory describes the standard activation Gibbs energy as follows. \[\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX} ; \mathrm{M})=\mu^{0}\left(\mathrm{RX} \mathrm{X}^{\neq} ; \mathrm{M}\right)-\mu^{0}(\mathrm{RX} ; \mathrm{M})\]The basic postulate states that the reference chemical potential of solute \(\mathrm{RX}\) in solution, \(\mu^{0}(\mathrm{RX} ; \mathrm{sln})\) at defined \(\mathrm{T}\) and \(\mathrm{p}\) is given by the sum of contributions from the substituent zone, \(\mu^{0} (\mathrm{R})\) and reaction zone , \(\mu^{0} (\mathrm{X})\) together with terms describing the interaction of \(\mathrm{R}\) and \(\mathrm{X}\) with the solvent, \(\mathrm{I}(\mathrm{R}, \mathrm{M})\) and \(\mathrm{I}(\mathrm{X}, \mathrm{M})\) and the effect of solvent on this interaction \(\mathrm{II}(\mathrm{R}, \mathrm{X}, \mathrm{M})\). \[\begin{aligned} &\mu^{0}(\mathrm{RX} ; \text { in medium } \mathrm{M})= \\ &\qquad \mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}, \mathrm{M})+\mathrm{I}(\mathrm{X}, \mathrm{M})+\mathrm{II}(\mathrm{R}, \mathrm{X}, \mathrm{M}) \end{aligned}\]Thus the solvent \(\mathrm{M}\) contributes to the interaction between \(\mathrm{R}\) and \(\mathrm{X}\). A key postulate is advanced at this stage which states that the interactions terms can be factorised. \[\begin{aligned} &\mu^{0}(\mathrm{X} ; \text { in medium } \mathrm{M})= \\ &\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}) \, \mathrm{I}(\mathrm{M})+\mathrm{I}(\mathrm{X}) \, \mathrm{I}(\mathrm{M})+\mathrm{II}(\mathrm{R}) \, \mathrm{II}(\mathrm{X}) \, \mathrm{II}(\mathrm{M}) \end{aligned}\]A similar equation is set down for the transition state. \[\begin{aligned} &\mu^{0}\left(\mathrm{RX}^{\neq} ; \text {in medium } \mathrm{M}\right)= \\ &\mu^{0}\left(\mathrm{R}^{\neq}\right)+\mu^{0}\left(\mathrm{X}^{\neq}\right)+\mathrm{I}\left(\mathrm{R}^{\neq}\right) \, \mathrm{I}(\mathrm{M})+\mathrm{I}\left(\mathrm{X}^{\neq}\right) \, \mathrm{I}(\mathrm{M})+\mathrm{II}\left(\mathrm{R}^{\neq}\right) \, \mathrm{II}\left(\mathrm{X}^{\neq}\right) \, \mathrm{II}(\mathrm{M}) \end{aligned}\]Equations (c) and (d) are combined with equation (a). For reaction in solvent medium \(\mathrm{M}\), \[\begin{aligned} &\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX}, \mathrm{M})=\\ &\left.\left[\mu^{0}\left(R^{\neq}\right)-\mu^{0}(R)\right]+\left[\mu^{0}\left(X^{\neq}\right)-\mu^{0} X\right)\right]\\ &+\mathrm{I}(\mathrm{M}) \,\left[\mathrm{I}\left(\mathrm{R}^{\neq}\right)-\mathrm{I}(\mathrm{R})\right]+\mathrm{I}(\mathrm{M}) \,\left[\mathrm{I}\left(\mathrm{X}^{\neq}\right)-\mathrm{I}(\mathrm{X})\right]\\ &+\mathrm{II}(\mathrm{M}) \,\left[\mathrm{II}\left(\mathrm{R}^{*}\right) \, \mathrm{II}\left(\mathrm{X}^{\neq}\right)-\mathrm{II}(\mathrm{R}) \, \mathrm{II}(\mathrm{X})\right] \end{aligned}\]A second postulate state that \(\mathrm{II}(\mathrm{M})\) and \(\mathrm{I}(\mathrm{M})\) are simply related; i.e. equation (f). \[\mathrm{II}(\mathrm{M})=\alpha \, \mathrm{I}(\mathrm{M})\]If \(\Delta_{m} \Delta^{\neq} G^{0}(\mathrm{RX})\) describes the effect of solvent \(\mathrm{M}\) on \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX})\), \(\Delta_{\mathrm{m}} \Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX})\) is given by the product of a (solvent operator) and a (substrate operator). By definition, \[\Delta_{\mathrm{m}} \Delta^{\pm} \mathrm{G}^{0}(\mathrm{RX})=\Delta_{\mathrm{m}} \mathrm{Y} \,(\text { substrate operator })\]Originally the substrate operator was set to unity for \(\left(\mathrm{CH}_{3}\right)_{3}\mathrm{CCl}\), and \(\mathrm{Y}\) was set to zero for an 80:20 ethanol + water mixture. The outcome was a set of \(\mathrm{Y}\)-values for many solvents, particularly alcohol + water mixtures at \(298.5 \mathrm{~K}\) and ambient pressureFootnotes C. K. Ingold, Structure and Mechanism in Organic Chemistry, G. Bell, London, 1953; see page 347. E. Grunwald and S. Winstein, J. Am. Chem. Soc., 1948,70, 841; 846. J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, New York, 1963; Dover Publications , New York,1989. \(20 \mathrm{~cm}^{3}\) of ethanol(\(\ell\)) was poured from a volumetric flask containing \(1 \mathrm{~dm}^{3}\) of ethanol(\(\ell\)). The liquid in the flask was then ‘topped up’ with water(\(\ell\)). The mixture is a good solvent for both apolar and polar solutes. Unfortunately the exact composition of the mixture is unknown. As rarely stated, the volume of water required is slightly larger than \(20 \mathrm{~cm}^{3}\). Professor Ross E Robertson (University of Calgary) viewed with interest that so much information in the chemical literature describes rates of chemical reactions where the solvent is vodka.This page titled 1.14.10: Extrathermodynamics - Solvent Effects in Chemical Kinetics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.10: Gibbs - Helmholtz Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.10%3A_Gibbs_-_Helmholtz_Equation

The Gibbs energy and enthalpy of a closed system are related; \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]The two properties \(\mathrm{G}\) and \(\mathrm{H}\) are also related by the Gibbs - Helmholtz equation through the dependence of \(\mathrm{G}\) on temperature at fixed pressure. We envisage a situation in which a closed system at equilibrium having Gibbs energy \(\mathrm{G}\) is displaced to a neighbouring equilibrium state by a change in temperature at constant pressure. We are interested in the partial derivative, \(\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}\). In general terms we consider the isobaric differential dependence of \((\mathrm{G} / \mathrm{T})\) on temperature. \[\frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{p}-\frac{\mathrm{G}}{\mathrm{T}^{2}}\]\[\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=\mathrm{T} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{G}\]But \[\mathrm{S}=-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]For an equilibrium change, equations (b) and (c) yield equation (e). \[\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=-(\mathrm{G}+\mathrm{T} \, \mathrm{S})\]But \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\). Then, \[\mathrm{H}=-\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}\]For an equilibrium change, \[\Delta \mathrm{H}(\mathrm{A}=0)=-\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p} ; \mathrm{A}=0}\]or, \[\Delta \mathrm{H}(\mathrm{A}=0)=\frac{\mathrm{d}}{\mathrm{dT}^{-1}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p} ; \mathrm{A}=0}\]In a similar manner we obtain the Gibbs -Helmholtz equation for a system perturbed at constant composition. \[\Delta \mathrm{H}(\text { fixed } \xi)=\frac{\mathrm{d}}{\mathrm{dT}^{-1}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}, \bar{\xi},}\]Equation (f) is the starting point for the development of another important equation. Thus, \[\mathrm{H}=-\mathrm{T}^{2} \,\left[-\frac{\mathrm{G}}{\mathrm{T}^{2}}+\frac{1}{\mathrm{~T}} \, \frac{\mathrm{dG}}{\mathrm{dT}}\right]\]Hence, \[\mathrm{H}=\mathrm{G}-\mathrm{T} \,\left[\frac{\mathrm{dG}}{\mathrm{dT}}\right]\]Equation (k) is differentiated with respect to temperature at constant pressure and at ‘\(\mathrm{A}=0\)’. \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{p, A=0}-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}\]Hence, \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}, \mathrm{A}=0}\]But \[\left(\frac{\partial^{2} G}{\partial T^{2}}\right)_{p, A=0}=\frac{\partial}{\partial T}\left(\frac{\partial G}{\partial T}\right)=-\left(\frac{\partial S}{\partial T}\right)_{p, A=0}\]Also the equilibrium isobaric heat capacity, \[C_{p}(A=0)=\left(\frac{\partial H}{\partial T}\right)_{p, A=0}\]Equations (m), (n) and (o) yield equation (p). \[\left(\frac{\partial S}{\partial T}\right)_{p, A=0}=\frac{C_{p}(A=0)}{T}\]Equation (p) relates the isobaric equilibrium dependence of entropy of a closed system on temperature to the isobaric heat capacity. Also starting from, \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\), then \[(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{T}}=(\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}}+\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}\]Using a Maxwell Equation, \[(\partial H / \partial p)_{T}=\mathrm{V}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\]Similarly, \[(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\]And \[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}=-\mathrm{p}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \,(\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}}\]Footnote There are many thermodynamic equations which are of the GibbsHelmholtz type. As a common feature they conform to the following calculus property.Given \[\mathrm{f}=\mathrm{f}(\mathrm{x}, \mathrm{y})\]Then \[\left(\frac{\partial(f / x)}{\partial(1 / x)}\right)_{y}=-x^{2} \,\left(\frac{\partial(f / x)}{\partial x}\right)_{y}=f-x \,\left(\frac{\partial f}{\partial x}\right)_{y}\]Similarly, \[\left(\frac{\partial(f / y)}{\partial(1 / y)}\right)_{x}=-y^{2} \,\left(\frac{\partial(f / x)}{\partial y}\right)_{x}=f-y \,\left(\frac{\partial f}{\partial y}\right)_{x}\]Normally \(\mathrm{f}\) stands for a thermodynamic potential and \(x\) and \(y\ for its natural variables. Thus a total of 8 equations of the Gibbs - Helmholtz type holding for closed systems can be constructed from \(\mathrm{U}=\mathrm{U}(\mathrm{S}, \mathrm{V}), \mathrm{F}=\mathrm{F}(\mathrm{T}, \mathrm{V}), \mathrm{H}=\mathrm{H}(\mathrm{S}, \mathrm{p}) \text { and } \mathrm{G}=\mathrm{G}(\mathrm{T}, \mathrm{p})\).This page titled 1.14.10: Gibbs - Helmholtz Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.11: Guggenheim-Scatchard Equation / Redlich-Kister Equation

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For binary liquid mixtures at fixed \(\mathrm{T}\) and \(\mathrm{p}\), an important task is to fit the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\) to an equation in order to calculate the derivative \({\mathrm{dG}_{\mathrm{m}}}^{\mathrm{E}} / \mathrm{dx}_{2}\) at required mole fractions. The Guggenheim-Scatchard (commonly called the Redlich-Kister ) equation is one such equation. This equation has the following general form.\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \left(1-\mathrm{x}_{2}\right) \sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{k}} \mathrm{A}_{\mathrm{i}} \left(1-2 \mathrm{x}_{2}\right)^{\mathrm{i}-1} \label{a}\]\(\mathrm{A}_{\mathrm{i}}\) are coefficients obtained from a least squares analysis of the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\). The equation clearly satisfies the condition that \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is zero at \(x_{2} = 0\) and at \(x_{2} = 1\). In fact the first term in the \(\mathrm{G} - \mathrm{~S}\) equation has the following form.\[X_{m}^{E}=x_{2} \left(1-x_{2}\right) A_{1}\label{b}\]According to Equation \ref{b} \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) is an extremum at \(x_{2} = 0.5\), the plot being symmetric about the line from \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) to ‘\(x_{2} = 0.5\)’. In fact for most systems the \(\mathrm{A}_{1}\) term is dominant. For the derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{2}\), we write Equation \ref{a} in the following general form.\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\left(\mathrm{x}_{2}-\mathrm{x}_{2}^{2}\right) \mathrm{Q}\label{c}\]Then\[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{x}_{2}=\mathrm{x}_{2} \left(1-\mathrm{x}_{2}\right) \mathrm{dQ} / \mathrm{dx}_{2}+\left(1-2 \mathrm{x}_{2}\right) \mathrm{Q}\label{d}\]where\[\mathrm{dQ} / \mathrm{dx}_{2}=-2 \sum_{\mathrm{i}=2}^{\mathrm{i}=\mathrm{k}}(\mathrm{i}-1) \mathrm{A}_{\mathrm{i}} \left(1-2 \mathrm{x}_{2}\right)^{\mathrm{i}-2}\label{e}\]Equation \ref{a} fits the dependence with a set of contributing curves which all pass through points, \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}=0\) at \(x_{1} = 0\) and \(x_{1} =1\). The usual procedure involves fitting the recorded dependence using increasing number of terms in the series, testing the statistical significance of including a further term. Although Equation \ref{a} has been applied to many systems and although the equation is easy to incorporate into computer programs using packaged least square and graphical routines, the equation suffers from the following disadvantage. As one incorporates a further term in the series, (e.g. \(\mathrm{A}_{j}\)) estimates of all the previously calculated parameters (i.e. \(\mathrm{A}_{2}\), \(\mathrm{A}_{3}\), ... \(\mathrm{A}_{j-1}\)) change. For this reason orthogonal polynomials have been increasingly favoured especially where the appropriate computer software is available. The only slight reservation is that derivation of explicit equations for the required derivative \({\mathrm{dX}_{\mathrm{m}}}^{\mathrm{E}}\) is not straightforward. The problem becomes rather more formidable when the second and higher derivatives are required. The derivative \(\mathrm{d}^{2}{\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) is sometimes required by calculations concerning the properties of binary liquid mixtures.The derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\) and \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) are combined (see Topic EZ20) to yield an equation for \(\ln\left(\mathrm{f}_{1}\right)\).\[\ln \left(f_{1}\right)=\frac{G_{m}^{E}}{R T}+\frac{\left(1-x_{1}\right)}{R T} \frac{d G_{m}^{E}}{d x_{1}}\label{f}\]A similar equation leads to estimates of \(\ln\left(\mathrm{f}_{2}\right)\). Hence the dependences are obtained of both \(\ln\left(\mathrm{f}_{1}\right)\) and \(\ln\left(\mathrm{f}_{2}\right)\) on mixture composition. It is of interest to explore the case where the coefficients \(\mathrm{A}_{2}, \mathrm{~A}_{3} \ldots\) in Equation \ref{a} are zero. Then\[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \left(1-\mathrm{x}_{2}\right) \mathrm{A}_{1}\label{g}\]and\[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{2}=\left(1-2 \mathrm{x}_{2}\right) \mathrm{A}_{1}\label{h}\]With reference to the Gibbs energies,\[\ln \left(\mathrm{f}_{2}\right)=(1 / \mathrm{R} \mathrm{T}) \left[\mathrm{x}_{2} \left(1-\mathrm{x}_{2}\right)+\left(1-\mathrm{x}_{2}\right) \left(1-2 \mathrm{x}_{2}\right)\right] \mathrm{A}_{1}^{\mathrm{G}} \label{i}\]\[\ln \left(f_2\right)=\left(A_1^G / R T\right) \left[1-2 x_2 + x_2^{2} \right] \label{j}\]or,\[\ln \left(f_{2}\right)=\left(A_{1}^{\mathrm{G}} / \mathrm{R} \mathrm{T}\right) \left[1-\mathrm{x}_{2}\right]^{2}\label{k}\]In fact the equation reported by Jost et al. has this form.Rather than using the Redlich-Kister equation, recently attention has been directed to the Wilson equation written in Equation \ref{l} for a two-component liquid.\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{R} \mathrm{T}=-\mathrm{x}_{1} \ln \left(\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}\right)-\mathrm{x}_{2} \ln \left(\mathrm{x}_{2}+\Lambda_{21} \mathrm{x}_{1}\right)\label{l}\]Then , for example,\[\ln \left(f_{1}\right)=-\ln \left(x_{1}+\Lambda_{12} x_{2}\right)+x_{2} \left(\frac{\Lambda_{12}}{x_{1}+\Lambda_{12} x_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} x_{1}+x_{2}}\right)\label{m}\]The Wilson equation forms the basis for two further developments, described as the NRTL (non-random, two-liquid) equation and the UNIQUAC equation. E. A. Guggenheim, Trans. Faraday Soc.,1937,33,151; equation 4.1. G. Scatchard, Chem. Rev.,1949,44,7;see page 9. O. Redlich and A. Kister, Ind. Eng. Chem.,1948,40,345; equation 8. F. Jost, H. Leiter and M. J. Schwuger, Colloid Polymer Sci., 1988, 266, 554. G. M. Wilson, J. Am. Chem. Soc.,1964,86,127. See also From Equation \ref{l},\[\begin{aligned} \frac{1}{\mathrm{R} \mathrm{T}} \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=&-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}\right)-\frac{\mathrm{x}_{1} \left(1-\Lambda_{12}\right)}{\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}} \\ &+\ln \left(\Lambda_{21} \mathrm{x}_{1}+\mathrm{x}_{2}\right)-\frac{\mathrm{x}_{2} \left(\Lambda_{21}-1\right)}{\Lambda_{21} \mathrm{x}_{1}+\mathrm{x}_{2}} \end{aligned}\]Then using Equation \ref{f} with \(1− x_{1} = x_{2}\),\[\begin{aligned} \ln \left(f_{1}\right)=&-x_{1} \ln \left(x_{1}+\Lambda_{12} x_{2}\right)-x_{2} \ln \left(\Lambda_{21} x_{1}+x_{2}\right) \\ &-x_{2} \ln \left(x_{1}+\Lambda_{12} x_{2}\right)-\frac{x_{1} x_{2} \left(1-\Lambda_{12}\right)}{x_{1}+\Lambda_{12} x_{2}} \\ &+x_{2} \ln \left(\Lambda_{21} x_{1}+x_{2}\right)+\frac{\left(x_{2}\right)^{2} \left(1-\Lambda_{21}\right)}{\Lambda_{21} x_{1}+x_{2}} \end{aligned}\]Or,\[\begin{aligned} \ln \left(f_{1}\right) &=-\left(x_{1}+x_{2}\right) \ln \left(x_{1}+\Lambda_{12} x_{2}\right) \\ &+x_{2} \left[\frac{\Lambda_{12} x_{1}-x_{1}}{x_{1}+\Lambda_{12} x_{2}}-\frac{\Lambda_{21} x_{2}-x_{2}}{\Lambda_{21} x_{1}+x_{2}}\right] \end{aligned}\]But\[\Lambda_{12} \mathrm{x}_{1}-\mathrm{x}_{1}=\Lambda_{12} \left(1-\mathrm{x}_{2}\right)-\mathrm{x}_{1}=\Lambda_{12}-\left(\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}\right)\]Hence,\[\begin{aligned} \ln \left(\mathrm{f}_{1}\right) &=-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}\right) \\ &+\mathrm{x}_{2} \left[\frac{\Lambda_{12}-\left(\mathrm{x}_{1}-\Lambda_{12} \mathrm{x}_{2}\right)}{\mathrm{x}_{1}+\Lambda_{12} \mathrm{x}_{2}}-\frac{\Lambda_{21}-\left(\Lambda_{21} \mathrm{x}_{1}+\mathrm{x}_{2}\right)}{\Lambda_{21} \mathrm{x}_{1}+\mathrm{x}_{2}}\right] \end{aligned}\]Or,\[\ln \left(f_{1}\right)=-\ln \left(x_{1}+\Lambda_{12} x_{2}\right)+x_{2} \left[\frac{\Lambda_{12}}{x_{1}+\Lambda_{12} x_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} x_{1}+x_{2}}\right]\] D. Abrams and J. M. Prausnitz, AIChE J.,1975,21,116. R. C. Reid, J. M. Prausnitz and E. B. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 4th edn.,1987, chapter 8. J. M. Prausnitz, R. N. Lichtenthaler and E. G. de Azevedo, Molecular Themodyanamics of Fluid Phase Equilibria, Prentice –Hall, Upper Saddle River, N.J., 3rd edn.,1999,chapter 6.This page titled 1.14.11: Guggenheim-Scatchard Equation / Redlich-Kister Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.12: Legendre Transformations

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Many important thermodynamic equations are closely related. These relationships are often highlighted by the mathematical technique, Legendre Transformations. With reference to thermodynamics, Callen discusses application of Legendre Transformations. The essential features of Legendre Transformations can be understood in the following terms.A primary variable \(\mathrm{Q}\) is defined by two dependent variables \(x\) and \(y\). Thus \[Q=Q[x, y]\]Then \[\mathrm{dQ}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \, \mathrm{dx}+\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{y}}\right)_{\mathrm{x}} \, \mathrm{dy}\]By definition, \[\mathrm{u}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \quad \text { and } \quad \mathrm{v}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{y}}\right)_{\mathrm{x}}\]Then from equation (b), \[\mathrm{dQ}=\mathrm{u} \, \mathrm{dx}+\mathrm{v} \, \mathrm{dy}\]A new variable \(\mathrm{Z}\) is defined by equation (e). \[\mathrm{Z}=\mathrm{Z}[\mathrm{u}, \mathrm{y}] \text { where } \mathrm{Z}=\mathrm{Q}-\mathrm{u} \, \mathrm{x}\]Then, \[\mathrm{dZ}=\mathrm{dQ}-\mathrm{x} \, \mathrm{du}-\mathrm{u} \, \mathrm{dx}\]Hence using equation (d), \[d Z=u \, d x+v \, d y-x \, d u-u \, d x\]Or, \[\mathrm{d} Z=-\mathrm{x} \, \mathrm{du}+\mathrm{v} \, \mathrm{dy}\]Hence, \[x=-\left(\frac{\partial Z}{\partial u}\right)_{y} \quad \text { and } \quad v=\left(\frac{\partial Z}{\partial y}\right)_{u}\]Comparison of equations (a) and (e) reveals the transformation, \(\mathrm{Q}[\mathrm{x}, \mathrm{y}] \rightarrow \mathrm{Z}[\mathrm{u}, \mathrm{y}]\). We now explore thermodynamic transformations. The following Master Equation relates the change in thermodynamic energy \(\mathrm{U}\) with the changes in entropy \(\mathrm{S}\) at temperature \(\mathrm{T}\), volume \(\mathrm{V}\) at pressure \(\mathrm{p}\) and composition \(\xi\) at affinity \(\mathrm{A}\); \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi]\). \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi\]By definition, enthalpy \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\); \[\mathrm{dH}=-\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]Using equation (j), \[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]Then \(\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\) Or, \[\mathrm{H}=\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]\]The transformation is- \(\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi] \rightarrow \mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]\) By definition, Gibbs energy \[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}\]Or using equation (k), \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]Then \[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}\]Hence from equation (l) \[\mathrm{dG}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}\]Or, \(\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\) And, \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\]The transformation is \(\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi] \rightarrow \mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\) Similarly, \(\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi] \rightarrow \mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\) Ledgendre transformations can be examined in the broad context of chemistry and biochemistry. Their importance lies in establishing the general mathematical structure of thermodynamics.Footnotes A. M. Legendre; an eighteenth century mathematician. C. Paus at http://web.mit.edu /8.21/www/ notes/notes/ node7.html B. Callen, Thermodynamics, Wiley, New York,1961. E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York , 1997. R. A. Alberty, Chem. Revs.,1994,94,1457. D. Kondepudi and I. Prigogine, Modern Thermodynamics, Wiley, New York,1998.This page titled 1.14.12: Legendre Transformations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.13: Closed System

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A closed system is effectively a closed reaction vessel. As chemists we are interested in changes in chemical composition of the closed system. The condition “closed” means that while observing the processes taking place inside the system, we do not add more chemical substances to the system from the surroundings or remove chemical substance from the system into the surroundings. Actually the thermodynamic treatment of closed systems is simpler than for other systems. The system and surroundings interact by virtue of, for example, heat passing between system and surroundings.This page titled 1.14.13: Closed System is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.14: Cohesive Energy Density

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The molar enthalpy of vaporisation \(\Delta_{\text {vap }} \mathrm{H}^{*}\) is the change in enthalpy for one mole of chemical substance \(j\) on going from the liquid to the (perfect) gaseous state. The properties of a given liquid-\(j\) are determined by \(j-j\) intermolecular forces. By definition, there are no intermolecular forces in a perfect gas. Hence \(\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)\) offers an insight into the strength of intermolecular forces in the liquid state. We have to be careful not to use the word ‘energy’. By definition enthalpy \(\mathrm{H}\) equals (\(\mathrm{U} + \mathrm{p} \, \mathrm{V}\)). For phase I at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), \[\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{I})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})\]\[\text { Similarly for phase II, } \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{II})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})\]\[\text { Hence, } \quad U_{j}^{*}(\text { II })-U_{j}^{*}(I)=H_{j}^{*}(\text { II })-H_{j}^{*}(I)-p \,\left[V_{j}^{*}(\text { II })-V_{j}^{*}(I)\right]\]If phase II is a perfect gas and phase I is the corresponding liquid, \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})>>\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})\); for one mole of chemical substance \(j\), \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{R} \, \mathrm{T}\). \[\text { Consequently } \Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}\]\(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) is the molar thermodynamic energy of vaporisation for liquid \(j\) at temperature \(\mathrm{T}\). Having calculated \(\Delta_{\text {vap }} \mathrm{H}_{j}^{*}(\mathrm{~T})\) from experimental data we obtain \(\Delta_{\text {vap }} U_{j}^{*}(T)\), a measure of the strength of inter-molecular interactions in the liquid.The differential quantity \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) defines the internal pressure \(\pi_{int}(j)\) of chemical substance \(j\). For liquid \(j\), \[\pi_{\mathrm{int}}^{*}(\ell ; \mathrm{j})=\left[\partial \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]_{\mathrm{T}}\]The internal pressure for liquids, of the order \(10^{8} \mathrm{~Pa}\), is an indicator of the strength of intermolecular forces. The structure of the terms in equation (e) prompts a slight rewrite using properties that are either readily measured or calculated, namely \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) and the molar volume of the liquid at temperature \(\mathrm{T}\), \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T})\). The result is the cohesive energy density, c.e.d., a measure of the cohesion within a liquid. \[\text { By definition, c.e.d. }=\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T}) / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]Intuitively, \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\) is a measure of cohesive interactions in the liquid whereas volume is a measure of the repulsive interactions, keeping the molecules in the liquid apart. At constant \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})\), c.e.d decreases with increase in molar volume; c.f. repulsion. But at constant \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T})\), c.e.d. increases with increase in \(\Delta_{\text {vap }} \mathrm{U}_{j}^{*}(\mathrm{~T})\), the attractive part. \[\text { If the vapour is a perfect gas, c.e.d. }=\left[\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)-\mathrm{R} \, \mathrm{T}\right] / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]If the molar mass of the liquid \(j\) equals \(\mathrm{M}_{j}\) and the density equals \(\rho_{\mathrm{j}}^{*}(\ell)\) \[\text { c.e.d. }=\left[\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\ell)-\mathrm{R} \, \mathrm{T}\right] \, \rho_{\mathrm{j}}^{*}(\ell) / \mathrm{M}_{\mathrm{j}}\]\(\mathrm{M}_{j}\) is expressed in \(\mathrm{kg mol}^{-1}\) and \(\rho_{\mathrm{j}}^{*}(\ell)\) in \(\mathrm{kg m}^{-3}\), consistent with c.e.d. being expressed in (\(\mathrm{J mol}^{–1} \mathrm{~m}^{–3}\)). At \(298.2 \mathrm{~K}\), \(\mathrm{R} \, \mathrm{T}=2.48 \mathrm{~kJ} \mathrm{~mol}^{-1}\). The ratio of internal pressure \(\pi_{\text {int }}(\mathrm{j})\) to c.e.d. defines a property \(\mathrm{n}\) using equation (i). \[\mathrm{n}=\left[\partial \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]_{\mathrm{T}} /\left[\Delta_{\mathrm{vap}} \mathrm{U}_{\mathrm{j}}^{*}(\ell) / \mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]\]The dimensionless ratio \(\mathrm{n}\) has been used to comment on the strength of intermolecular forces in a liquid.In the context of the properties of liquid mixtures, using the definition of enthalpy \(\mathrm{H}(=\mathrm{U}+\mathrm{p} \, \mathrm{V})\) we can write the following equation for a given phase I containing \(\mathrm{n}_{1}\) moles of substance 1 and \(\mathrm{n}_{2}\) moles of substance 2. \[\mathrm{U}\left(\mathrm{I}, \mathrm{n}_{1}+\mathrm{n}_{2}\right)=\mathrm{H}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2}\right)-\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2}\right)\]We assert that phase I is an ideal binary liquid mixture. Then, \[\begin{aligned} &\mathrm{U}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{mix} ; \mathrm{id}\right)= \\ &\quad \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell)-\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \text { mix } ; \mathrm{id}\right) \end{aligned}\]We assert that phase II is a perfect gas comprising \(\mathrm{n}_{1}\) moles of substance 1 and \(\mathrm{n}_{2}\) moles of substance 2. Then \[\begin{aligned} \mathrm{U}\left(\mathrm{II} ; \mathrm{n}_{1}+\right.&\left.\mathrm{n}_{2} ; \mathrm{pfg}\right)=\\ & \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\mathrm{pfg})+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\mathrm{pfg})-\mathrm{p} \, \mathrm{V}\left(\mathrm{II} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{pfg}\right) \end{aligned}\]\[\text { For a perfect gas, } \mathrm{p} \, \mathrm{V}\left(\mathrm{II} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \mathrm{pfg}\right)=\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \, \mathrm{R} \, \mathrm{T}\]We express the thermodynamic energy of vaporisation for (\(\mathrm{n}_{1} + \mathrm{n}_{2}\)) moles passing from phase I to phase II. \[\begin{aligned} &\Delta_{\text {vap }} \mathrm{U}\left(\mathrm{id}, \mathrm{n}_{1}+\mathrm{n}_{2}\right)= \\ &\begin{aligned} \mathrm{n}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}+\mathrm{n}_{2} \, \Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}-\left(\mathrm{n}_{1}\right.&\left.+\mathrm{n}_{2}\right) \, \mathrm{R} \, \mathrm{T} \\ &+\mathrm{p} \, \mathrm{V}\left(\mathrm{I} ; \mathrm{n}_{1}+\mathrm{n}_{2} ; \text { mix; id }\right) \end{aligned} \end{aligned}\]Therefore for one mole, \[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}+\mathrm{x}_{2} \, \Delta_{\text {vap }} \mathrm{H}_{2}^{*}-\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\mathrm{I} ; \mathrm{mix} ; \mathrm{id})\]Suppose however that the thermodynamic properties of the liquid mixture are not ideal. We rewrite equation (k) in the following form (for one mole of mixture) where \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\) and \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) are the excess molar enthalpies and excess molar volumes of mixing. \[\begin{aligned} &\mathrm{U}_{\mathrm{m}}(\mathrm{I}, \operatorname{mix})= \\ &\quad\left[\mathrm{x}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{H}_{2}^{*}(\ell)+\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{p} \,\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\right] \end{aligned}\]\[\operatorname{Or} \mathrm{U}_{\mathrm{m}}(\mathrm{I}, \operatorname{mix})=\left[\mathrm{x}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{H}_{2}^{*}(\ell)+\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\operatorname{mix})\]Therefore the molar thermodynamic energy of vaporisation on going from the real mixture to the perfect gas in given by equation (r). \[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}}=\left[\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \, \Delta_{\text {vap }} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right]-\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\]The cohesive energy density, c.e.d., for a real binary liquid mixture is given by equation (s). \[\begin{aligned} \text { c.e.d. }=\left\{\left[\mathrm{x}_{1} \, \Delta_{\mathrm{vap}} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \,\right.\right.&\left.\left.\Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right] / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\right\} \\ &-\left\{\mathrm{R} \, \mathrm{T} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\right\}+\mathrm{p} \end{aligned}\]The c.e.d. for a given binary mixture is given by the molar enthalpies of vaporisation of the pure components, the excess molar enthalpy of mixing and the molar volume of the mixture. For the corresponding ideal binary mixture, c.e.d.(id) is given by equation (t). \[\text { c.e.d.(id) } \begin{aligned} =\left\{\left[\mathrm{x}_{1} \, \Delta_{\text {vap }} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \,\right.\right.&\left.\left.\Delta_{\text {vap }} \mathrm{H}_{2}^{*}(\mathrm{~T})\right] / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right\} \\ &-\left\{\mathrm{R} \, \mathrm{T} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right\}+\mathrm{p} \end{aligned}\]The difference between \(\Delta_{\text {vap }} \mathrm{U}_{\mathrm{m}} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and \(\Delta_{\mathrm{vap}} \mathrm{U}_{\mathrm{m}} / \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\) is the excess cohesive energy density, \((\text { c.e.d. })^{\mathrm{E}}\). The sign of \((\text { c.e.d. })^{\mathrm{E}}\) is controlled to a significant extent by the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) and the excess molar enthalpy \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\). In fact equations (s) and (t) lead to m equation (u). \[(\text { c.e.d. })^{\mathrm{E}}=\frac{-\left[\mathrm{x}_{1} \, \Delta_{\mathrm{vap}} \mathrm{H}_{1}^{*}(\mathrm{~T})+\mathrm{x}_{2} \, \Delta_{\mathrm{vap}} \mathrm{H}_{2}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}\right]}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix}) \, \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}-\frac{\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}\]Recently the cohesive energy density of a liquid has been described as a ‘solvation pressure’ acting on, for example, ethanol in ethanol + water and ethanol + trichloromethane liquid mixtures.Footnotes M. R. J. Dack, Aust. J. Chem.,1976,27,779. \(\text { c.e.d. }=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]\); the unit of pressure. \(\mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) A variety of units are used for cohesive energy densities. Despite the fact that there are good grounds for using the unit \(\mathrm{J m}^{-3}\), the commonly used unit is calories per \(\mathrm{cm}^{3}\), \mathrm{cal cm}^{-3}\). \[\begin{aligned} &\text { For liquids } 298.15 \mathrm{~K} \text { and ambient pressure. }\\ &\begin{array}{lc} \text { Liquid } & \text { c.e.d./ } \mathrm{cal} \mathrm{cm}^{-3} \\ \text { water } & 547 \\ \text { methanol } & 204 \\ \text { benzene } & 85 \\ \text { tetrachloromethane } & 74 \end{array} \end{aligned}\] A. F. M. Barton, J.Chem.Educ.,1971, 48,156. For comments on the role of cohesive energy densities of solvents and rates of disproportionation, see A.P. Stefani, J. Am. Chem. Soc., 1968,90,1694. For comments on cohesive energy densities of binary aqueous mixtures see, N.W. A.van Uden, H. Hubel, D. A. Faux, A. C. Tanczos, B. Howlin and D. J. Dunstan, J. Phys.: Condens. Mater,2003,15,1577.This page titled 1.14.14: Cohesive Energy Density is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.15: Degree of Dissociation

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A given aqueous solution is prepared using \(\mathrm{n}_{1}^{0}\) moles of water and \(\mathrm{n}_{\mathrm{A}}^{0}\) moles of a weak acid \(\mathrm{HA}\). The composition of the solution at equilibrium (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is described as follows.If the volume of the system is \(1 \mathrm{~dm}^{3}\) then, \(\mathrm{c}_{\mathrm{A}}^{0} \,(1-\alpha) \quad \alpha \, \mathrm{c}_{\mathrm{A}}^{0} \quad \alpha \, \mathrm{c}_{\mathrm{A}}^{0} \quad \mathrm{~mol} \mathrm{dm}^{-3}\)By definition, the degree of dissociation, \(\alpha=\xi^{\mathrm{eq}} / \mathrm{n}_{\mathrm{A}}^{0}\); \(\alpha\) is an intensive variable describing the ‘degree’ of dissociation. If the total volume of the solution is \(\mathrm{V}\), the concentration \(\mathrm{c}_{\mathrm{A}}^{0}=\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\). If the thermodynamic properties of the solution are ideal, the composition of the solution can be described by an equilibrium acid dissociation constant \(\mathrm{K}_{\mathrm{A}}\). \[\mathrm{K}_{\mathrm{A}}=\alpha^{2} \, \mathrm{c}_{\mathrm{A}}^{0} /(1-\alpha)\]If \[1-\alpha \cong 1, \alpha^{2}=\mathrm{K}_{\mathrm{A}} / \mathrm{c}_{\mathrm{A}}^{0}\]If the acid is dibasic, the analysis is a little more complicated.By definition \(\mathrm{c}_{\mathrm{A}}^{0}=\mathrm{n}_{\mathrm{A}}^{0} / \mathrm{V}\) where \(\mathrm{V}\) is the volume of solution expressed in \mathrm{dm}^{3}\). Also by definition \(\alpha_{1}=\xi_{1} / \mathrm{n}_{\mathrm{A}}^{0}\) and \(\alpha_{2}=\xi_{2} / \mathrm{n}_{\mathrm{A}}^{0}\)Hence from equation (d)\(\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}-\alpha_{2}\right]\mathrm{~mol}\)Total amount of \(\mathrm{H} in the system \[=2 \,\left(\mathrm{n}_{\mathrm{A}}^{0}-\xi_{1}\right)+\xi_{1}+\xi_{2}+\xi_{1}-\xi_{2}=2 \, \mathrm{n}_{\mathrm{A}}^{0}\]Total amount of \(\mathrm{A}\) in the system \[=\mathrm{n}_{\mathrm{A}}^{0}-\xi_{1}+\xi_{1}-\xi_{2}+\xi_{2}=\mathrm{n}_{\mathrm{A}}^{0}\]If the thermodynamic properties of the solution are ideal, \[\mathrm{K}_{1}=\mathrm{c}_{\mathrm{A}}^{0} \,\left[\alpha_{1}+\alpha_{2}\right] \,\left[\alpha_{1}-\alpha_{2}\right] /\left[1-\alpha_{1}\right]\]If \[\mathrm{K}_{2}=0, \alpha_{2}=0, \mathrm{~K}_{1}=\mathrm{c}_{\mathrm{A}}^{0} \, \alpha_{1}^{2} /\left(1-\alpha_{1}\right)\]But \[\mathrm{K}_{2}=\left(\alpha_{1}+\alpha_{2}\right) \, \alpha_{2} \, \mathrm{c}_{\mathrm{A}}^{0} /\left(\alpha_{1}-\alpha_{2}\right)\]This page titled 1.14.15: Degree of Dissociation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.16: Energy and First Law of Thermodynamics

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A central axiom of chemical thermodynamics is that a given system has a property called energy. In fact the First Law of Thermodynamics centres on the concept of energy. In its broadest terms, the law requires that the energy of the universe is constant. This is a rather overwhelming statement. A more attractive statement is that the thermodynamic energy \(\mathrm{U}\) of a typical chemistry laboratory is constant.\[\mathrm{U} = \text{ constant} \label{a}\]The latter is the principle of conservation of energy; energy can be neither created nor destroyed. A chemist ‘watches’ energy “move” between system and surroundings. As a consequence of Equation \ref{a} we state that,\[\Delta \mathrm{U}(\text { system })=-\Delta \mathrm{U}(\text { surroundings }) \label{b}\]We cannot know the actual energy \(\mathrm{U}\) of a closed system although we agree that it is an extensive property of a system. In describing the energy changes we need a convention. We use the acquisitive convention, describing all changes in terms of how the system is affected. Thus \(\Delta \mathrm{U} < 0\), means that the energy of the system falls whereas \(\Delta \mathrm{U} > 0\) means that the energy increases. In the context of chemistry, chemists agree that the energy of a given closed system can be increased in two ways:\[\Delta \mathrm{U}=\mathrm{q}+\mathrm{w} \label{c}\]As it stands the symbols \(\mathrm{U}\), \(\mathrm{q}\) and \(\mathrm{w}\) seem rather uninformative. It is the task of chemists to flesh out the meaning of these terms. If only ‘\(\mathrm{p}-\mathrm{V}\)’ work is involved,\[\mathrm{w}=-\mathrm{p} \, \mathrm{dV} \label{d} \]The point of Equation \ref{c} is to separate the work term from the heat term. The significance for chemists is that \(\mathrm{q}\) links to the Second Law of Thermodynamics. Thus chemists know that heat flows spontaneously from high to low temperatures. This concept of ‘spontaneous change’ is picked up with enormous impact in the second law. Peter Atkins (Galileo’s Finger, Oxford University press, 2003, page 107) speculates that the total energy of the universe ‘may be exactly zero’. In principle it is possible to calculate the total energy of a given system using a scale in conjunction with Einstein’s famous equation, \(\mathrm{E}=\mathrm{m} \, \mathrm{c}^{2}\). However the mass corresponding to \(1 \mathrm{~kJ}\) is only about \(10^{-14} \mathrm{~kg}\).This page titled 1.14.16: Energy and First Law of Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.17: Energy and Entropy

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.17%3A_Energy_and_Entropy

This Topic takes a rather different approach from the other Topics in this Notebook. Across the galaxy of terms used in thermodynamics, two terms stand out, namely Energy and Entropy. With respect to a given closed system, both terms describe extensive properties, using the letter \(\mathrm{U}\) to identify energy and the letter \(\mathrm{S}\) to identify entropy.The term ‘energy’ is used quite generally in everyday life. One dictionary describes energy as ‘the power and ability to be physically active’. Perhaps we might not be too happy at the use of the term ‘power’ in this context on the grounds that this term is normally linked to the rate at which energy is supplied. Indeed in every day life we refer to powerful engines in, for example, fast sports cars. Nevertheless in a thermodynamic context the concepts of energy and energy change are part of the language of chemistry; e.g. bond energy, energy of activation, radiant energy… The First Law of Thermodynamics formalises the concept of energy change using the following ostensibly simple equation. \[\Delta \mathrm{U}=\mathrm{q}+\mathrm{w}\]Here \(\mathrm{U}\) is the thermodynamic energy, a function of state; \(\Delta \mathrm{U}\) describes the increase in thermodynamic energy of a closed system when heat \(\mathrm{q}\) flows from the surroundings into a given system and work w is done by the surroundings on that system.The distinction between heat and work is crucial. There are many ways in which the surroundings can do work on a system. Caldin lists many examples in which work is given by the product of Intensity and Capacity Factors; e.g. intensity factor pressure, \(\mathrm{p}\) and capacity factor volume, \(\mathrm{V}\) such that \(\mathrm{w}=\mathrm{p} \, \mathrm{dV}\).In the context of energy, chemical thermodynamics quite generally describes quite modest changes in energy. Even in the case of an explosion involving, for example, ignition of a mixture of hydrogen and oxygen gases, the energy change turns out to involve transitions between electronic energy levels in atoms and molecules. Much more dramatic are nuclear reactions which involve the conversion of mass, \(\mathrm{m}\) into energy \(\mathrm{E}\) as described by Einstein’s famous equation. \[\mathrm{E}=\mathrm{m} \, \mathrm{c}^{2}\]Here \(\mathrm{c}\) is the speed of light, \(3.00 \times 10^{8} \mathrm{~m s}^{-1}\). In nuclear fission the nucleus of an atom breaks into two smaller nuclei of similar mass. Thus \(\text{uranium}^{235}\) nuclei bombarded by neutrons split into barium-142 and krypton-92 nuclei. Einstein’s equation shows that \(1.0 \mathrm{~g}\) of this uranium isotope undergoes fission with the release of \(7.5 \times 10^{10} \mathrm{~J}\), an awesome amount of energy.Here we return to the domain of chemical properties and chemical reactions where nuclei of atoms are not destroyed. Our interest centres on the thermodynamic variable, entropy \(\mathrm{S}\), an extensive function of state. However in every day conversation and in articles in newspapers and magazines the term ‘entropy’ is rarely used suggesting that it is not important. This conclusion is incorrect and the message quite misleading.In these Topics we describe the Second Law using an equation based on the formulation given by Clausius as follows. \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi\]Here a positive \(\mathrm{q}\) describes heat passing from the surroundings into a closed system; \(\mathrm{A}\) is the affinity for spontaneous change, the change being described by the property \(\xi\). Following De Donder as discussed by Prigogine and Defay the Second Law is simply stated as follows. \[A \, d \xi \geq 0\]For a system moving between equilibrium states (i.e. the system and surroundings are at all stages at equilibrium where \(\mathrm{A}\) is zero), \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}\]Hence \(\mathrm{q}\), measured using a calorimeter, is a direct measure of the change in entropy accompanying a change where the system is always in equilibrium with the surroundings. In fact this statement provides a useful answer to the question ‘what is entropy?’. There is therefore a fundamental link between the two quantities \(\mathrm{dS}\) and heat \(\mathrm{q}\). Indeed we understand immediately the importance of calorimeters in thermodynamics. At the same time we understand the importance of chemical kinetics because this subject is built around equation (d) which in the basis of the Law of Mass Action.In summary we see how the two foundation stones of thermodynamics, namely energy and entropy, are formalised in two laws for which there are no exceptions. So we can end the Topics here. But chemists do not although there are new hazards.It follows from equation (d) that for a process where \(\mathrm{q} < 0\), the entropy of the system decreases. [It is interesting to note that the unit of entropy \(\mathm{J K}^{-1}\) is the same as that for heat capacity.] At this point we review the arguments advanced by Lewis and Randall.An important reference system in thermodynamics is the perfect gas. No such gas is actually known in the real world but the concept is very valuable. The properties of a perfect gas conform to the two laws [7b].We envisage a closed system, volume \(\mathrm{V}\) containing n moles of a perfect gas. The first condition states that the thermodynamic energy \(\mathrm{U}\) is only a function of temperature. Thus, \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=0\]The second condition states that the following equation relates the pressure, volume and temperature of \(\mathrm{n}\) moles of a perfect gas. \[\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T}\]Thus for one mole of a perfect gas, having molar volume \(\mathrm{V}_{\mathrm{m}}\), \[\mathrm{p} \, \mathrm{V}_{\mathrm{m}}=\mathrm{R} \, \mathrm{T}\]A key equation (Topic 2500) relates the change in thermodynamic energy \(\mathrm{U}\) to the changes in entropy, volume and composition. Thus \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi\]For an equilibrium transformation, the affinity of spontaneous change is zero. Hence for an equilibrium process, \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}\]For 1 mole of an ideal gas, \[\mathrm{dU}_{\mathrm{m}}=\mathrm{T} \, \mathrm{dS}_{\mathrm{m}}-\mathrm{p} \, \mathrm{dV}_{\mathrm{m}}\]Or, \[\mathrm{dS}_{\mathrm{m}}=\frac{1}{\mathrm{~T}} \, \mathrm{dU}_{\mathrm{m}}+\frac{\mathrm{P}}{\mathrm{T}} \, \mathrm{dV} \mathrm{m}_{\mathrm{m}}\]Molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vm}}\) is related to \(\mathrm{dU}_{\mathrm{m}}\) by equation (m). \[\mathrm{C}_{\mathrm{V}_{\mathrm{m}}}=\left(\partial \mathrm{U}_{\mathrm{m}} / \partial \mathrm{T}\right)_{\mathrm{V}(\mathrm{m})}\]Then, \[\mathrm{dS}_{\mathrm{m}}=\frac{\mathrm{C}_{\mathrm{Vm}^{\mathrm{m}}}}{\mathrm{T}} \, \mathrm{dT}+\frac{\mathrm{p}}{\mathrm{T}} \, \mathrm{dV} \mathrm{V}_{\mathrm{m}}\]We define the molar entropy of an ideal gas using equation (o). \[\mathrm{S}_{\mathrm{m}}=\mathrm{S}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{V}_{\mathrm{m}}\right]\]The total differential of equation (o) takes the following form. \[\mathrm{dS}_{\mathrm{m}}=\left(\frac{\partial \mathrm{S}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})} \mathrm{dT}+\left(\frac{\partial \mathrm{S}_{\mathrm{m}}}{\partial \mathrm{V}_{\mathrm{m}}}\right)_{\mathrm{T}} \mathrm{dV} \mathrm{V}_{\mathrm{m}}\]Comparison of equations (n) and (p) reveals the following relation. \[\left(\frac{\partial S_{m}}{\partial V_{m}}\right)_{T}=\frac{p}{T}\]Hence using equation h), \[\left(\frac{\partial S_{m}}{\partial V_{m}}\right)_{T}=\frac{R}{V_{m}}\]Thus at constant temperature, \[\mathrm{dS}_{\mathrm{m}}=\mathrm{T} \, \mathrm{d} \ln \left(\mathrm{V}_{\mathrm{m}}\right)\]Hence the change in entropy for the isothermal expansion of an ideal gas between states where the volumes are \(\mathrm{V}_{\mathrm{m}}(\mathrm{B})\) and \(\mathrm{V}_{\mathrm{m}}(\mathrm{A})\) is given by equation (t). \[S_{m}(B)-S_{m}(A)=R \, \ln \left[\frac{V_{m}(B)}{V_{m}(A)}\right]\]We turn now to a consideration of changes in entropy from a statistical point of view.A given experiment uses two glass flasks of equal volumes connected by a glass tube which includes a tap, all at the same temperature \(\mathrm{T}\). The system contains \(\mathrm{N}\) gas molecules; e.g. oxygen. The gas molecules pass freely between the two flasks through the open tap. On examining the contents of the two flasks we would not be surprised to discover that there are equal numbers of the gas molecules in the two flasks. The probability of this results from experiment A is expressed by stating that \(\mathrm{P}_{\mathrm{y}}^{\mathrm{A}}\) is unity.We return to the two flasks and close the tap. The probability that all the oxygen molecules are to be found in one flask is \({(1/2)}^{\mathrm{N}}\); i.e. a very low probability. If the total system contained only 20 molecules this probability signals a chance of 1 in \(2^{20}\). Thus the probability \(\mathrm{P}_{\mathrm{y}}^{\mathrm{B}}\) for experiment B is very small; effectively zero.An interesting exercise characterises these probabilities by a property \(\sigma\). Then, \[\sigma=\frac{R}{N} \, \ln \left(P_{Y}\right)\]Note that the auxiliary property \(\sigma\) is generally negative because statistical probabilities vary between zero and unity. For the two experiments, \[\sigma_{B}-\sigma_{A}=\frac{R}{N} \, \ln \left(\frac{P_{Y}^{B}}{P_{Y}^{A}}\right)\]Hence, \[\sigma_{B}-\sigma_{A}=\frac{R}{N} \, \ln \left(\frac{(1 / 2)^{N}}{1}\right)\]Or, \[\sigma_{\mathrm{B}}-\sigma_{\mathrm{A}}=-\mathrm{R} \, \ln\]We can express this result in general terms describing the expansions of one mole of gas from volume \(\mathrm{V}_{\mathrm{A}}\) to \(\mathrm{V}_{\mathrm{B}}\). Then, \[\sigma_{B}-\sigma_{A}=R \, \ln \left[V_{m}(B) / V_{m}(A)\right]\]At this point comparison between equations (t) and (y) is rewarding. Thus we may write the following equation. \[\mathrm{S}_{\mathrm{m}}(\mathrm{B})-\mathrm{S}_{\mathrm{m}}(\mathrm{A})=\frac{\mathrm{R}}{\mathrm{N}} \,\left[\ln \left(\mathrm{P}_{\mathrm{Y}}^{\mathrm{B}}\right)-\ln \left(\mathrm{P}_{\mathrm{Y}}^{\wedge}\right)\right]\]In other words the difference between the entropies in the ideal gas state is related to a probability. Thus we might conclude that \(\mathrm{S}_{\mathrm{m}}(\mathrm{B})\) is larger that \(\mathrm{S}_{\mathrm{m}}(\mathrm{A})\) because there are more ways of arranging molecules in system B than in system A. The state with the more ordered arrangement is the state with the lower entropy. It is a small step (but a very dangerous step) to draw comparison between entropy and (if there is such a word) the muddled-up-ness of a given system. But these are treacherous waters and outside the province of the classic thermodynamics which form the basis of the Topics. Indeed strong feelings are aroused. McGlashan, for example, takes to task chemists who assume that an increase in entropy implies an increase of disorder or of randomness or of ‘mixed-upness’. We leave the debate here except to note that both authors of these Topics favour the view advanced by McGlashan although this view would not a win a popularity contest. But ‘popularity’ is not an acceptable criterion in thermodynamics.Footnotes Cambridge International Dictionary of English, Cambridge University Press, Cambridge, 1995. P. W. Atkins, Concepts in Physical Chemistry, Oxford University Press, Oxford, 1995. E. F. Caldin, An Introduction to Chemical Thermodynamics, Oxford University Press, Oxford, 1958. S. Glasstone, Sourcebook of Atomic Energy, MacMillan, London, 1954. P. W. Atkins and L. Jones, Chemistry: Molecules, Matter and Change, W H Freeman, New York, 3rd edition, 1997, p.875. Th. de Donder, Bull. Acad. Roy. Belg. (Cl.Sc), 1922, 7, 197, 205. I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1954, (a) chapter 3; (b) chapter 4. G. N. Lewis and M. Randall, Thermodynamics , McGraw-Hill, New York, 1923, chapter VI. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979,pages 112-113; M.L. McGlashan, J. Chem. Educ.,1966,43,226.This page titled 1.14.17: Energy and Entropy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.18: Electrochemical Units

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.18%3A_Electrochemical_Units

Electric CurrentThe SI base electrical unit is the AMPERE which is that constant electric current which if maintained in two straight parallel conductors of infinite length and of negligible circular cross section and placed a metre apart in a vacuum would produce between these conductors a force equal to \(2 \times 10^{-7}\) newton per metre length. It is interesting to note that definition of the Ampere involves a derived SI unit, the newton. Except in certain specialised applications, electric currents of the order ‘amperes’ are rare. Starter motors in cars require for a short time a current of several amperes.When a current of one ampere passes through a wire about \(6.2 \times 10^{18}\) electrons pass a given point in one second.The coulomb (symbol \(\mathrm{C}\)) is the electric charge which passes through an electrical conductor when an electric current of one \(\mathrm{A}\) flows for one second. Thus \[[\mathrm{C}]=[\mathrm{As}]\]In order to pass an electric current thorough an electrical conductor a difference in electric potential must exist across the electrical conductor. If the energy expended by a flow of one ampere for one second equals one Joule the electric potential difference across the electrical conductor is one volt.If the electric potential difference across an electrical conductor is one volt when the electrical current is one ampere, the electrical resistance is one ohm, symbol \(\Omega\). The inverse of electrical resistance , the conductance, is measured using the unit siemens, symbol [S].This famous phenomenological law describes the ability of a system to conduct electrical charge. This law describes the relationship between three properties of an electrical conductor; e.g. a salt solution. The three properties areOhm’s Law is a phenomenological law in that it describes the phenomenon of electrical conductivity. Unlike the laws of thermodynamics, there are exceptions to Ohm’s law for very high voltages and high frequency alternating electric currents. The (electrical) conductance \(\mathrm{G}\) of a system is given by the inverse of its resistance \(\mathrm{R}\). The conductivity \(\kappa\) (\(\equiv \sigma\)) of a system is given by equation (c). \[\mathrm{j}=\kappa \, \mathrm{E}\]Here \(\mathrm{j}\) is the electric current density and \(\mathrm{E}\) is the electric field strength. Chemists prefer to think in terms the charge carrying properties of a given system; i.e. the conductance \(\mathrm{G}\) (\(=1 / \mathrm{R}\)) using the unit siemens, symbol S.An interesting contrast often emerges between chemists and physicists, the latter seem to emphasise the property of ‘resistance’ whereas chemists are more interested in how systems transport electrical charge. Certainly the classic subject in chemistry concerns the electrical conductivities of salt solutions where the charge carriers are ions. The subject is complicated by the fact that there are two types of charge carriers in a given solution, cations and anions. Moreover the subject is further complicated by the fact that these charge carriers move in opposite directions with different velocities. In a solution containing a single salt the fraction of electric current carried by the cations and anions are called transport numbers; i.e. \(\mathrm{t}_{+}\) and \(\mathrm{t}_{-}\) respectively for cations and anions where \(\mathrm{t}_{+}+\mathrm{t}_{-}=1\).A given aqueous salt solution, volume \(\mathrm{V}\), is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of a salt \(j\). A pair of electrodes is placed in the solution, \(\mathrm{d}\) metres apart. An electric potential, \(\mathrm{V}\) volts, is applied across the solution and an electric current \(\mathrm{I}\) is recorded. An electric charge \(\mathrm{Q}\) coulombs (\([\mathrm{A s}]\)) is passed through the solution. The electric current (unit \([\mathrm{A}]\)) is the rate of transport of charge, \(\mathrm{dQ} / \mathrm{dt}\).The speed of ion \(\mathrm{i}\) through the solution \(\mathrm{v}_{\mathrm{i}}\) is given by the ratio of the distance travelled to the time taken. Thus \(\mathrm{v}_{\mathrm{i}}\) is a measure of the distance travelled in one second; \(\mathrm{v}=(\mathrm{a} / \mathrm{t})\left[\mathrm{ms}^{-1}\right]\). A more interesting property is the velocity of ion \(\mathrm{i}\) in an electric field gradient measured using the ratio, volt/metre (Or, \(\mathrm{V} / \mathrm{m}\)). Thus electric mobility \(\mathrm{u}_{\mathrm{i}}\) has the unit, \(\left[\mathrm{m} \mathrm{s}^{-1} / \mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]=\left[\mathrm{v}_{\mathrm{i}} \mathrm{E}^{-1}\right]=\left[\mathrm{m}^{2} \mathrm{~s}^{-1} \mathrm{~V}^{-1}\right]\).The molar conductivity \(\Lambda\) of a salt solution is given by the ratio, \(\left(\kappa / c_{j}\right)\left\{=\left[S \mathrm{~m}^{2} \mathrm{~mol}^{-1}\right]\right\}\). In fact the majority of research publications describe the dependence of \(\Lambda\) on the concentration of salt in a specified solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\). For salt solutions both cations and anions contribute to \(\Lambda\); the transport number of an ion \(j\) describes the fraction of current carried by the \(j\) ion. Thus \(\mathrm{t}_{\mathrm{j}}=\left|\mathrm{z}_{\mathrm{j}}\right| \, \mathrm{c}_{\mathrm{j}} \, \mathrm{v}_{\mathrm{j}} / \sum\left|\mathrm{z}_{\mathrm{i}}\right| \, \mathrm{c}_{\mathrm{i}} \, \mathrm{v}_{\mathrm{i}}\).An electric current (i.e. a flow of electric charge) through a system is impeded by the electrical resistance. The ohm (symbol \(\Omega\)) is the unit of electrical resistance being the ratio of electric potential (unit = volt) to electric current (unit = ampere) Then, ohm = volt/ampere (d)In other words, Electrical resistance/ohm=[electric potential gradient /volt]/[electric current/ampere] . From equation (c) ohm (symbol \(\Omega\)) \[=\mathrm{V} \mathrm{} \mathrm{A}^{-1}=\mathrm{m}^{2} \mathrm{~kg} \mathrm{~s}^{-3} \mathrm{~A}^{-2}\]The property (electrical) resistance is a measure of the impedance to the flow of electric charge. This somewhat negative outlook is not consistent with the attitude of chemists who are interested in the ‘mechanism’ by which a system conducts electrical charge. Chemists prefer to discuss the property of electric conductance rather than resistance. The conductance is measured using the unit siemen, symbol S. A key component of electrical circuits is the electrical capacitance measured using the unit farad, symbol \(\mathrm{F}\).If the composition of an electrical conductor is uniform, the electrical resistance is directly proportional to its length \(\ell\) and inversely proportional to its cross sectional area, \(\mathrm{a}\). The material forming the conductor is characterized by its resistivity, \(\rho\). Thus resistance, \[\mathrm{R}=\rho \, \ell / \mathrm{a}\]Footnotes The electric charge on an electron \(=1.602 \times 10^{-19} \mathrm{C}=1.602 \times 10^{-19} \mathrm{~As}\). A given single wire carries a current of 1 ampere. Then in one second, the electric charge carried by that wire \(= 1.602 \times 10^{-19} \mathrm{~C}\) For one coulomb to pass a given point, the number of electrons passing \(=\frac{1}{1.602 \times 10^{-19}}\) In other words \(6.24 \times 10^{18}\) electrons pass by. P.W. Atkins and L. Jones, Chemistry; Molecules, Matter and Change, W. H. Freeman, New York, 1997,p.658. \([\mathrm{V}] \equiv\left[\mathrm{kg} \mathrm{m}{ }^{2} \mathrm{~s}^{-3} \mathrm{~A}^{-1}\right]=\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]\) \([\Omega]=\left[\mathrm{VA}^{-1}\right]=\left[\mathrm{S}^{-1}\right]\) \(\begin{aligned} &\kappa=\left[\mathrm{S} \mathrm{m}^{-1}\right]\\ &\left[\mathrm{A} \mathrm{} \mathrm{m}^{-2}\right]=\kappa \,\left[\mathrm{V} \mathrm{} \mathrm{m}^{-1}\right] \end{aligned}\) \(\mathbf{S}=\Omega^{-1}\) \(\mathrm{F}=\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-2}=\mathrm{As} \mathrm{} \mathrm{V}^{-1}=\mathrm{C} \mathrm{V}^{-1}\) \(\begin{gathered} {[\mathrm{S}] \equiv\left[\Omega^{-1}\right] \equiv\left[\mathrm{A} \mathrm{V}^{-1}\right] \equiv\left[\mathrm{m}^{-2} \mathrm{~kg}^{-1} \mathrm{~s}^{3} \mathrm{~A}^{2}\right]} \\ {[\Omega]=[\Omega \mathrm{m}] \,[\mathrm{m}] \,[\mathrm{m}]^{-2}} \end{gathered}\)This page titled 1.14.18: Electrochemical Units is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.19: Electrical Units

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.19%3A_Electrical_Units

In attempting to understand the properties of chemical substances, chemists divide chemistry into two parts. In one part, chemists are interested in understanding intramolecular forces which hold molecules together. For example, using quantum mechanics and associated theories of covalent bonding, chemists describe the cohesive forces holding carbon, hydrogen and nitrogen atoms together in cyanomethane, \(\mathrm{CH}_{3}\mathrm{CN}\). At ambient temperature and pressure, cyanomethane is a liquid. In the second part of the sub-division of chemistry, chemists describe the intermolecular forces which hold assemblies of molecules together in, for example, liquid and solid states; e.g. those forces which hold \(\mathrm{CH}_{3}\mathrm{CN}\) molecules together in the liquid state. Common experience tells us that intermolecular forces are weaker than intramolecular forces. When we heat \(\mathrm{CH}_{3}\mathrm{CN}(\ell)\) at ambient pressure, the liquid boils at a characteristic temperature to form a vapour. The intermolecular separation dramatically increases but the covalent bonds within \(\mathrm{CH}_{3}\mathrm{CN}\) do not break. [Of course, these bonds break at very high temperatures - thermolysis.] Here the emphasis centres on intermolecular cohesion. But this cannot be the whole story. If cohesion is the only force operating, molecules would collapse into each other in some nuclear catastrophe. This does not happen. Opposing the forces of cohesion are repulsive forces. In fact everyday experience leads to the idea of "size"; 'size is repulsive'.Molecules contain charged particles; protons (with positive electric charge) and electrons (with negative electric charge-- by convention). Intermolecular forces are understandable in terms of equations describing electrical interactions between electrically charged particles. An SI base unit is the ampere; symbol = \(\mathrm{A}\).The SI unit of electric charge is the coulomb (symbol = \(\mathrm{C}) defined as \(\mathrm{A s}.An electric current \(\mathrm{I}\) is driven through an electrical resistance \(\mathrm{R}\), by an electric potential gradient across the resistance. An ammeter measures the electric current \(\mathrm{I}\). The voltmeter records the electric potential gradient, \(\Delta \mathrm{E}\) across the resistance. The property called resistance \(\mathrm{R}\) is given by Ohm’s Law; \[\Delta \mathrm{E}=\mathrm{I} \, \mathrm{R}\]In the IUPAC system the unit of resistance is ohm [symbol \(\Omega \equiv \mathrm{VA}^{-1}\) ]. The electric potential difference is measured in volts, symbol \(\mathrm{V}\).In a simple electric circuit, a small battery is connected across a parallel plate capacitance. No current flows in this circuit. The battery produces a set of equal in magnitude but opposite in sign electric charges on the two plates. A capacitance stores electric charge. In practice the extent to which a capacitance stores charge depends on the chemical substance between the two plates. This substance is characterised by its electric permittivity; symbol = \(\varepsilon\). Where a vacuum exists between the two plates, the electric permittivity equals \(\varepsilon_{0}\).The permittivity of a liquid is measured by comparing capacitance \(\mathrm{C}\) when the gap between the plates is filled with this liquid and with capacitance \(\mathrm{C}_{0}\) when the gap is "in vacuo". Then \[\varepsilon_{\mathrm{r}}=\varepsilon / \varepsilon_{0}=\mathrm{C} / \mathrm{C}_{0}\]or \[\mathrm{C}=\varepsilon_{\mathrm{r}} \, \mathrm{C}_{\mathrm{o}}\]For all substances, \(\varepsilon_{\mathrm{r}}\) is greater than unity. In other words, with increase in \(\varepsilon_{\mathrm{r}}\) so the electrical insulating properties of the system increase. At this stage, we have not offered a molecular explanation of the properly called \(\varepsilon_{\mathrm{r}}\) but we have indicated that \(\varepsilon_{\mathrm{r}}\) be can measured.Molecule \(i\) and molecule \(j\) are separated by a distance \(\mathrm{r}\); we assert that \(\mathrm{r} >>\) molecular radii of molecules \(i\) and \(j\). Our discussion centres on the assertion that a force (symbol \(\mathrm{X}\)) exists between the two molecules. Moreover, this force depends on the distance of separation \(\mathrm{r}\). Thus \[X=f(r)\]The force \(\mathrm{X}\) between two electric charges \(\mathrm{q}_{1}\) and \(\mathrm{q}_{2}\) distance \(\mathrm{r}\) apart ‘in vacuo’ is given by equation (e); (Couloumb's Law). \[\mathrm{X}=\mathrm{q}_{1} \, \mathrm{q}_{2} / 4 \, \pi \, \varepsilon_{0} \, \mathrm{r}^{2}\]Two ions, \(i\) and \(j\), have charge numbers \(\mathrm{z}_{i}\) and \(\mathrm{z}_{j}\) respectively [for \(\mathrm{K}^{+}\), \(\mathrm{z}_{\mathrm{j}}=+1\); for \({\mathrm{SO}_{4}}^{2-}\), \(\mathrm{z}_{j} = -2\)]. For two ions ‘in vacuo’, the interionic force is given by equation (f). \[\mathrm{F}_{\mathrm{ij}}=\left(\mathrm{z}_{\mathrm{i}} \mathrm{e}\right) \,\left(\mathrm{z}_{\mathrm{j}} \mathrm{e}\right) / 4 \pi \, \varepsilon_{0} \, \mathrm{r}^{2}\]But pairwise potential energy, \[\mathrm{U}_{i j}=-\int_{\mathrm{ij}=\infty}^{\mathrm{r}} \mathrm{F}_{\mathrm{ij}} \, \mathrm{dr}\]Hence, \[U_{i j}=\left(z_{i} e\right) \,\left(z_{j} e\right) / 4 \pi \, \varepsilon_{0} \, r\]Equation (h) yields the interaction potential energy between a pair of ions. The result is an energy expressed in joules. However, there are often advantages in considering an Avogadro number (i.e. a mole) of such pairwise interactions. \[\left.\mathrm{U}_{\mathrm{ij}} / \mathrm{J} \mathrm{mol} \mathrm{mol}_{\mathrm{A}}^{-1}=\mathrm{N}_{\mathrm{i}} \mathrm{e}\right) \,\left(\mathrm{z}_{\mathrm{j}} \mathrm{e}\right) / 4 \pi \, \varepsilon_{0} \, \mathrm{r}\]We consider two classes of ion-ion interactions:\[\mathrm{U}_{\mathrm{ij}} / \mathrm{J} \mathrm{mol}^{-1}=-\mathrm{N}_{\mathrm{A}} \,\left|\mathrm{Z}_{\mathrm{i}} \, \mathrm{Z}_{\mathrm{j}}\right| \mathrm{e}^{2} / 4 \pi \, \varepsilon_{0} \, \mathrm{r}\]Hence \(\mathrm{U}_{ij}\) has a (\(1/\mathrm{r}\)) dependence on distance apart.Electric field strength, \(\mathrm{E}\) is the force exerted on unit charge at the point in question. At distance \(\mathrm{r}\) from charge \(\mathrm{q}\), \[\mathrm{E}=\mathrm{q} / 4 \pi \, \varepsilon_{0} \, \mathrm{r}^{2}\]An important topic in Chemistry concerns the effect of solvents on ion-ion interactions. Here we assume that solvents are characterised by their relative permittivities, \(\varepsilon_{\mathrm{r}}\). In a solvent the pairwise cation-anion interaction energy is given by equation (l). \[\mathrm{U}_{\mathrm{ij}} / \mathrm{J}=-\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}}\right| \mathrm{e}^{2} / 4 \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}\]\[\mathrm{U}_{\mathrm{ij}} / \mathrm{J} \mathrm{mol} \mathrm{m}^{-1}=-\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{Z}_{\mathrm{j}}\right| \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e}^{2} / 4 \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}\]As commented above, \(\varepsilon_{\mathrm{r}}\) is always greater than unity. Hence for a given system at fixed distance apart \(\mathrm{r}\), \(\mathrm{U}_{ij}\) increases (becomes less negative) with increase in \(\varepsilon_{\mathrm{r}}\). With increase in \(\varepsilon_{\mathrm{r}}\), the ions are increasingly insulated and so at given distance \(\mathrm{r}\) the stabilisation of the cation-anion pair is less marked.A given molecule comprises an assembly of positive and negative charges. Consider a point 0, distance \(\mathrm{r}\) from this assembly. We are concerned with the electric field strength at point 0, a short distance from the dipole moment. In the previous section we assumed that this assembly is simply characterised by the electric charge (i.e. \(z_{j} \, e\) for ion \(j\)). However, in those cases where the overall charge is zero, a measurable electric field is detected at point 0. In 1912 Peter Debye showed that this field could be accounted for as a first approximation by characterising a molecule by its dipole moment. In the next approximation the electric field at 0 can also be accounted by an additional contribution from a distribution of charges within a molecule called a quadrupole, and in the next approximation by an additional contribution from a distribution called an octupole.In a homonuclear diatomic molecule such as \(\mathrm{H}_{2}\) and \(\mathrm{Cl}_{2}\), the positive nuclei are embedded in charge clouds describing the distribution of negatively charged electrons. For such molecules the "centres" of positive charges and negative charges coincide. But for the molecule \(\mathrm{HCl}\) the electron distribution favours the more electronegative chlorine atom. Hence the centres of positive and negative charges, magnitude \(+\mathrm{q}\) and \(-\mathrm{q}\), are separated by a dipole length \(\ell\). The molecule has a dipole moment, a characteristic and permanent property of an \(\mathrm{HCl}\) molecule. The (molecular) dipole moment \(\mu\) is given by the product ‘\(\mathrm{q} \, \ell\)'. A dipole moment has both magnitude and direction; it is a vector.Footnotes The classic reference in this subject is: J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York 1954; corrected printing ,1964. The ampere is that constant current flowing in two parallel straight conductors, having negligible cross section, one metre apart in vacuo which produces a force between each metre of length equal to \(2 \time 10^{-7} \mathrm{~N}\). When a current of one A flows for one second, the total charge passed is one coulomb. In practice, a current of \(1 \mathrm{~A}\) is very high and the common unit is milliampere (symbol: \(\mathrm{mA}\)). The starter motor in a conventional car requires a peak current of around \(30 \mathrm{~A}\). Electric charge on a single proton, \(\mathrm{e}=1.602 \times 10^{-19} \mathrm{C}\). Faraday, \(\mathrm{F}=\mathrm{N}_{\mathrm{A}} \, \mathrm{e}=9.649 \times 10^{4} \mathrm{C} \mathrm{mol}^{-1}\) Just to keep up with the way the units are developing we note: - electric current coulomb \(\mathrm{C} = \mathrm{A}\) s electric potential gradient volt \(\mathrm{V}=\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}=\mathrm{J} \mathrm{C}^{-1}\) (\(\mathrm{J =}\) joule). Thus volt expressed as \(\mathrm{J C}^{-1}\) is energy per coulomb of electric charge passed. This link between electric potential and energy is crucial. electrical resistance ohm \(\Omega=\mathrm{V} \mathrm{A}^{-1}\) Ohm's Law is a phenomenological law. Continuing our concern for units. electrical capacitance: unit = farad \(\mathrm{F} \equiv \mathrm{As} \mathrm{} \mathrm{V}^{-1}\) electric permittivity \(\varepsilon\); unit = \(\mathrm{F m}^{-1}\) electric permittivity of a vacuum, \(\varepsilon_{0}=8.854 \times 10^{-12} \mathrm{~F} \mathrm{~m}^{-1}\) relative permittivity \(\varepsilon_{\mathrm{r}}=\varepsilon / \varepsilon_{0}\); unit \(= 1\) Older literature calls \(\epsilon_{\mathrm{r}\), the "dielectric constant". But this property is not a constant for a given substance such as water (\(\ell\)). Thus \(\varepsilon\) and \(\varepsilon_{\mathrm{r}}\) depend on both temperature and pressure; \(\varepsilon\) and \(\varepsilon_{\mathrm{r}}\) for a given liquid depend on electric field strength and frequency of AC current applied to the capacitance. The quantity \(\left(4 \, \pi \, \varepsilon_{0} \, 10^{-7}\right)^{-1 / 2}\) equals \(2.998 \times 10^{8} \mathrm{m s}^{-1}\) which is the speed of light. We check the units. If \(\mathrm{X}\) is a force, the unit for \(\mathrm{X}\) is newton (symbol \(\mathrm{N}\)). Then the right-hand side should simplify to the same unit. Electric charge is expressed in \(\mathrm{C}[= \mathrm{A s}]\); \(\varepsilon_{0}\) has units of \(\mathrm{F} \mathrm{m}{ }^{-1}\left[=\mathrm{As} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right]\). Then \(\mathrm{X}=[\mathrm{C}] \,[\mathrm{C}] / \, \,\left[\mathrm{A} \mathrm{s} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right] \,[\mathrm{m}]^{2}\) But \([\mathrm{V}]=\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]\) Then \(\mathrm{X}=\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] /\left[\mathrm{A} \mathrm{s} \mathrm{J}^{-1} \mathrm{~A} \mathrm{~s} \mathrm{~m}\right]=\left[\mathrm{J} \mathrm{m}^{-1} \mathrm{l}=[\mathrm{N}]\right.\) We check that our units are correct. If \(\mathrm{U}_{ij}\) is an energy expressed in joules, the terms on the right-hand side should reduce to joules. \(\mathrm{U}_{\mathrm{ij}}=[\mathrm{A} \mathrm{s}] \,[\mathrm{As}] / \, \,\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right] \,[\mathrm{m}] =\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] /\left[\mathrm{As} \mathrm{J}^{-1} \mathrm{As}\right]=[\mathrm{J}]\) \[\begin{aligned} \mathrm{E} &=[\mathrm{C}] / \, \,\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right] \,\left[\mathrm{m}^{2}\right] \\ &=[\mathrm{A} \mathrm{s}] /[\mathrm{A} \mathrm{s} \mathrm{V} \mathrm{m}]=\left[\mathrm{V} \mathrm{m}^{-1}\right] \\ &=\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1} \mathrm{~m}^{-1}\right]=\left[\mathrm{J} \mathrm{m}^{-1}\right] /[\mathrm{A} \mathrm{s}]=\left[\mathrm{N} \mathrm{C}^{-1}\right] \end{aligned}\] Thus electric field strength is expressed in \(\mathrm{Vm}^{-1}\) or \(\mathrm{N} \mathrm{C}^{-1}\); the latter is clearly a force per unit charge. The classic text is:- P. Debye, Polar Molecules, Chemical Catalog Co., New York 1929 (available as Dover paperback). We do not consider here interactions involving quadrupoles, octupoles, etc. These molecular properties are reviewed by A. D. Buckingham, Quart. Rev., 1959, 13, 183. Dipole moment, \(\mu=\mathrm{q} \, \ell=[\mathrm{C}] \,[\mathrm{m}]\) Thus \(\mu=[\mathrm{Cm}]\), coulomb metre . Dipole moments are normally quoted using the unit, debye. [The unit is named in honour of Peter Debye.] \(1 \mathrm{D}=3.336 \times 10^{-30} \mathrm{Cm}\).This page titled 1.14.19: Electrical Units is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.2: Excess Thermodynamic Properties - Liquid Mixtures

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A given liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong \mathrm{p}^{0}\)) contains \(\mathrm{i}\)-liquid chemical substances. The chemical potential of liquid component \(\mathrm{j}\) is given by equation (a) where \(\mu_{\mathrm{j}}^{*}(\ell)\) is the chemical potential of liquid component \(\mathrm{j}\) at the same \(\mathrm{T}\) and \(\mathrm{p}\). \[\mu_{\mathrm{j}}(\operatorname{mix})=\mu_{\mathrm{j}}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}\right)\]Here \(\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \mathrm{f}_{\mathrm{j}}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\).If the thermodynamic properties of the liquid mixture are ideal, \[\mu_{j}(\operatorname{mix} ; \mathrm{id})=\mu_{\mathrm{j}}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right)\]The excess chemical potential for liquid substance \(\mathrm{j}\), \[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{j}}\right)\] DSFThis page titled 1.14.2: Excess Thermodynamic Properties - Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.20: Electric Conductivities of Salt Solutions- Dependence on Composition

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At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the molar conductivity of given salt solution Λ depends on the concentration of salts. This subject has an extensive scientific literature. One of the challenges is to calculate \(\Lambda\) for given salt solution knowing the properties of the pure solvent and the salt at specified \(\mathrm{T}\) and \(\mathrm{p}\). A key quantity is the limiting molar conductivity \(\Lambda^{\infty}\) defined for a given salt solution by equation (a). \[\operatorname{Lt}\left(c_{j} \rightarrow 0\right) \Lambda=\Lambda^{\infty}\]Moreover as Kohlrausch showed in 1876 ( > 125 years ago) a given \(\Lambda^{\infty}\) can be expressed as the sum of limiting molar ionic conductances, \(\lambda_{\mathrm{j}}^{\infty}\). Thus \[\Lambda^{\infty}=\sum_{j=1}^{j=i} \lambda_{j}^{\infty}\]A difficult theoretical task is to estimate \(\lambda_{\mathrm{j}}^{\infty}\) for a given ion at defined \(\mathrm{T}\) and \(\mathrm{p}\) and specified solvent. Rather more progress has been made in predicting quantitatively the dependence of \(\left(\Lambda-\Lambda^{\infty}\right)\) on concentration of salt in a given solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\) assuming that the ions in solution are characterised by their electric charges and radii. Indeed quantitative treatments of the electrical conductivities of salt solutions have attracted enormous interest and provided a challenge to scientists with good mathematical abilities. Here we summarise briefly the essence of treatments published by Onsager and by Fuoss. The account given below is based on that set out by N. K. Adam.A relaxation effect and an electrophoretic effect contribute to the magnitude of \(\left(\Lambda-\Lambda^{\infty}\right)\) for a real salt solution for which \(\Lambda<\Lambda^{\infty}\). In a real solution under the influence of an applied electric field, anions and cations move in opposite direction. The word ‘move’ does not reflect the complexity of the real situation. In a real solution and in the absence of an applied electric field, the ions move in random directions, Brownian motion, as a consequence of the thermal energy of the system. In some sense, ions and solvent molecules are jostling continuously. When an electric potential gradient is applied across the solution, the previously random motion of ions is now biased in a particular direction depending on the ionic charge. If the solution is ‘infinitely dilute’, the velocity of a given ion is characteristic of the ion, solvent, temperature and pressure.In a real solution the mobility decreases with increase in concentration of salt. Two retarding effects are identified, relaxation and electrophoretic effects. The latter emerges from the fact that a given \(j\) ion moves against the flow of counter-ions together with associated solvent molecules. In a real solution and in the absence of an applied electric field, a given \(j\) ion is at the centre of an ion atmosphere which has an electric charge equal in magnitude but opposite in sign to that of the \(j\) ion. Under the impact of an applied electric field the \(j\)-ion moves away from the centre of the ion atmosphere. The latter pulls the \(j\)-ion back towards this centre. In other words the \(j\)-ion is retarded by this relaxation effect. The latter term reflects the fact that the retardation depends on the rate at which the electric charge density in the ion atmosphere grows as the \(j\) ion moves through the solution and decays in the wake of the \(j\) ion.A given salt solution at temperature \(\mathrm{T}\) and pressure \(p\) contains a sa }\)]. In solution, the motion of ions is quite random, a pattern usually described as Brownian motion. If however an electric field is applied across the solution the movement of ions is biased in a given direction depending on the sign of the charge on the \(j\) ion. The electrical mobility \(\mathrm{u}_{j}\) of ion \(j\) describes the velocity of ion-\(j\) in an electric field gradient measured in [\(\mathrm{V m}^{-1}\)]. In the absence of ion-ion charge-charge interaction , the electrical mobility \(\mathbf{u}_{j}^{\infty}\) is characteristic of the \(j\) ion, solvent, temperature and pressure. The superscript ‘\(\infty\)’ identifies that to all intents and purposes the \(j\) ion is in an infinitely dilute solution. However in a real solution, concentration \(\mathrm{c}_{i}\) in salt \(i\), the \(j\) ion is surrounded by an ‘ion atmosphere’ which has an electric charge equal in magnitude but opposite in sign to that on the \(j\) ion.The ion atmosphere is modelled as a series of shells, thickness \(\mathrm{dr}\) distance \(\mathrm{r}\) from the centre of the \(j\) ion. The electrical charge \(\mathrm{q}_{j}\) on a shell distance \(\mathrm{r}\) from the \(j\) ion is given by equation (c). \[q_{j}=4 \, \pi \, r^{2} \, \rho \, d r\]Here \(\rho\) is the electric charge density, measured in ‘coulombs per cubic metre’. As a result of the electric field gradient operating on the \(j\) ion the electric force \(\mathrm{F}\) expressed in newtons operating on this shell is given by equation (d). \[\mathrm{F}=4 \, \pi \, \mathrm{r}^{2} \, \rho \, \mathrm{E} \, \mathrm{dr}\]According to Stokes Law, a sphere having radius \(\mathrm{r}\) and moving with velocity \(v\) through a liquid having (shear) viscosity \(\eta\) is subject to a viscous resistance \(\mathrm{R}_{\eta}\), a force expressed in newtons and given by equation (e). \[\mathrm{R}_{\eta}=6 \, \pi \, \eta \, \mathrm{r} \, \mathrm{V}\]If the speed of the liquid stream increases by \(\mathrm{dv}\) when the radius of the shell defining the ion atmosphere increases by \(\mathrm{dr}\), the viscous resistance increases by \((6 \, \pi \, \eta \, r \, d v)\). If the motion of the \(j\) ion through the solution is steady, the increase in viscous resistance to movement of the \(j\) ion equals the electrical force (see equation d). Therefore \[6 \, \pi \, \eta \, r \, d v=4 \, \pi \, r^{2} \, \rho_{j} \, E \, d r\]The charge density \(\rho_{j}\) is obtained by combining equations (u) and (x) of Topic 680. Thus, \[\rho_{\mathrm{j}}=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right) \, \exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right)}{\left.\left.4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\right) 1+\kappa \, \mathrm{a}_{\mathrm{j}}\right)} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\kappa \, \mathrm{r})}{\mathrm{r}}\]Then with \(\ell^{-1}=\kappa\), \[p_{j}=-\frac{z_{j} \, e \, \exp \left(a_{j} / \ell\right) \, \exp (-r / \ell)}{4 \, \pi \, \ell \,\left(a_{j}+\ell\right) \, r}\]From equation (f), \[\mathrm{dv}=\frac{2}{3} \, \frac{\rho_{\mathrm{j}}}{\eta} \, \mathrm{E} \, \mathrm{r} \, \mathrm{dr}\]Hence \[\mathrm{dv}=-\frac{2 \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \exp \left(\mathrm{a}_{\mathrm{j}} / \ell\right)}{12 \, \pi \, \eta \, \ell \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)} \, \mathrm{E} \, \exp (-\mathrm{r} / \ell) \, \mathrm{dr}\]Equation (j) is integrated between limits (i) \(r=\sigma\) to \(r = \infty\), and (ii) \(v = 0\) and \(v_{1}\) where \(v_{1}\) is the stream velocity of the solution outside the ion atmosphere of the \(j\) ion. Then \[\mathrm{v}_{1}=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E} \, \exp \left(\mathrm{a}_{\mathrm{j}} / \ell\right)}{6 \, \pi \, \eta \, \ell \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)} \, \int_{\mathrm{a}_{\mathrm{j}}}^{\infty} \exp (-\mathrm{r} / \ell) \, \mathrm{dv}\]Hence, \[\mathrm{v}_{1}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E}}{6 \, \pi \, \eta \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)}\]For dilute solutions, \(\mathrm{a}_{\mathrm{j}}<\ll \ell\) such that the stream velocity of the solution outside the ion atmosphere is given by equation (m)Therefore \[\mathrm{v}_{1}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E}}{6 \, \pi \, \eta \, \ell}\]We shift the reference. The solvent does not physically move when we measure the electrical conductivity of a solution. Therefore the impact of the electrophoretic effect is to retard the \(j\)-ion in solution. The decrease in electrical mobility of the \(j\) ion is given by equation (n). \[-\left(\Delta \mathrm{u}_{\mathrm{j}}\right)_{\mathrm{clectrpbor}}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{6 \, \pi \, \eta \, \ell}\]In the limit of infinite dilution, a given \(j\)-ion proceeds through an aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\) under the influence of an applied electric field gradient. The impediment to its progress arises from the solvent molecules. However in a real salt solution, the \(j\) ion is surrounded by its ion atmosphere which has an electric charge equal in magnitude and opposite on sign. In the absence of an applied electric field the ion atmosphere is spherically symmetric about the \(j\) ion. In a real solution, the migrating ion is not at the centre of the ion atmosphere, the latter therefore retarding the migrating ion. This retardation is called the relaxation effect on the grounds that the build-up of the ion atmosphere preceeding the ion and the decay in the wake of the ion is characterised by a relaxation time.The relaxation effect can be understood in terms of irreversible thermodynamics. Thus the flow of cations and anions in opposite directions are coupled. The stronger the coupling the greater is the retardation of the migrating ions. The first treatment of this coupling of flows and forces was developed by Onsager who published a reasonably successful description of the impact of this coupling on ionic mobilities. The analogue of equation (n) describing the relaxation effect takes the following form where \(\mathrm{w}\) is a correction factor depending on the type of electrolyte. \[-\left(\Delta u_{j}\right)=\frac{e^{3} \, w \, u_{j}^{\infty}}{24 \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, k \, T \, \ell}\]Here \(\ell\) is the radius of the ion atmosphere surrounding the \(j\) ion ; equation (p) where the concentration of \(j\) ions \(\mathrm{c}_{j}\) is expressed in \(\mathrm{mol dm}^{-3}\). \[\ell=\frac{10^{3} \, 4 \, \pi \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}{8 \, \pi \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{I}}\]For dilute solutions \[I=(0.5) \, \sum_{i=1}^{i=j} c_{i} \, z_{i}^{2}\]In summary two retarding effects, electrophoretic and relaxation, mean that the molar conductivity of a given aqueous salt solution is less than the molar conductivity of the corresponding solution at infinite dilution, \(\Lambda^{\infty}\). The outcome is the famous Debye-Huckel-Onsager Equation for molar conductivities. For a 1:1 salt (e.g. \(\mathrm{KBr}\)) in aqueous solution at \(298.15 \mathrm{~K}\) and ambient pressure, the molar conductivity \(\Lambda\) is given by equation (r). \[\Lambda=\Lambda^{\infty}-\left(0.229 \, \Lambda^{\infty}+60.2\right) \,\left(c_{j} / c_{r}\right)^{1 / 2}\]Footnotes L. Onsager, Physik. Z.,1926,27,388. L. Onsager, Trans. Faraday Soc.,1927,23,341. L. Onsager and R. M. Fuoss, J. Phys. Chem.,1932,36,2689. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 1950, 2nd edn. Revised and enlarged N. K. Adam, Physical Chemistry, Oxford, 1956. \[\begin{aligned} &v_{j}=\left[\mathrm{m} \mathrm{s}^{-1}\right] \\ &u_{j}=\left[\mathrm{m} \mathrm{s}^{-1}\right] /[\mathrm{V} \mathrm{m} \end{aligned}\] \(\mathrm{q}_{\mathrm{j}}= \, \,\left[\mathrm{m}^{2}\right] \,\left[\mathrm{Cm}^{-3}\right] \,[\mathrm{m}]=[\mathrm{C}]\) \(4 \, \pi \, r^{2} \, \rho \, E \, d r= \, \,\left[\mathrm{m}^{2}\right] \,\left[\mathrm{C} \mathrm{m}^{-3}\right] \,\left[\mathrm{V} \mathrm{m}^{-1}\right] \,[\mathrm{m}] =\left[\mathrm{J} \mathrm{m}^{-1}\right]=[\mathrm{N}]\) \(\mathrm{R}_{\eta}= \, \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}] \,\left[\mathrm{m} \mathrm{s}^{-1}\right]=\left[\mathrm{kg} \mathrm{m} \mathrm{s}^{-22}\right]=[\mathrm{N}]\) \(\rho_{j}=\frac{ \,[\mathrm{C}] \,}{ \, \,\left[\mathrm{Fm}^{-1}\right] \, \,} \, \frac{\left[\mathrm{Fm}^{-1}\right] \, \,[\mathrm{m}]^{2} \,}{[\mathrm{m}]} =\left[\mathrm{C} \mathrm{m}^{-3}\right]\) \(\mathrm{v}_{1}=\frac{[\mathrm{l}] \,[\mathrm{C}] \,\left[\mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]}{[\mathrm{l}] \,[\mathrm{l}] \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}]} \mathrm{v}_{1}=\frac{[\mathrm{l}] \,[\mathrm{C}] \,\left[\mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]}{[\mathrm{l}] \,[\mathrm{l}] \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}]} \(\Delta \mathrm{u}_{\mathrm{j}}=\frac{\left[\mathrm{m} \mathrm{s}^{-1}\right]}{\left[\mathrm{V} \mathrm{} \mathrm{m}^{-1}\right]}=\left[\mathrm{m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}\right]\)\(-\left(\Delta \mathrm{u}_{\mathrm{j}}\right)_{\text {relax }}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{ \,\left[\mathrm{Fm}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{m}]} \, \mathrm{u}_{\mathrm{j}}^{\infty}\) \(=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{[\mathrm{F}] \,[\mathrm{J}]}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-2}\right] \,\left[\mathrm{kg} \mathrm{m} \mathrm{m}^{2} \mathrm{~s}^{-2}\right]} \, \mathrm{u}_{\mathrm{j}}^{\infty}\)\(= \, \mathrm{u}_{\mathrm{j}}^{\infty}\) For an advanced treatment, see J. O’M. Bockris and A.K.N Reddy, Modern Electrochemistry: Ionics, Plenum Press, New York, 2nd. edn.,1998, chapter 4. P. W. Atkins, Physical Chemistry, Oxford University Press, 1982, 2nd. edn., p.900. \(\Lambda=\left[\Omega^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\right] ; 0.229= ; 60.2=\left[\Omega^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\right]\)This page titled 1.14.20: Electric Conductivities of Salt Solutions- Dependence on Composition is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.21: Donnan Membrane Equilibria

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A given experimental system comprises two compartments, I and II, separated by a membrane. The two compartments contain aqueous solutions at common temperature and pressure. The experimental system is set up by placing in compartment I an aqueous salt solution; e.g. \(\mathrm{NaCl}(\mathrm{aq})\) having concentration \(\mathrm{c}_{1} \mathrm{~mol dm}^{-3}\). However compartment II contains a salt, \(\mathrm{R}^{+}\mathrm{Cl}^{-} (\mathrm{aq})\), concentration \(\mathrm{c}_{2} \mathrm{~mol dm}^{-3}\). The membrane is permeable to both \(\mathrm{Na}^{+} and \(\mathrm{Cl}^{-}\) ions but not to \(\mathrm{R}^{+}\) cations. The sodium and chloride ions spontaneously diffuse across the membrane until the two solutions are in thermodynamic equilibrium. We represent the equilibrium system as follows where | | represents the membrane.[I] \[ \mathrm{Na}^{+}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{cq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{eq}}||[\mathrm{II}] \mathrm{Na}^{+}(\alpha)^{\mathrm{eq}} \mathrm{R}^{+}\left(\mathrm{c}_{2}\right)^{\mathrm{eq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}}\]The solutions on both sides are electrically neutral. A thermodynamic analysis is somewhat complicated if account is taken of the role of ion-ion interactions. However the essential features of the argument are revealed if we identify the activities of the ions as equal to their concentrations.Hence at equilibrium at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\begin{aligned} &{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Na}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}} \\ &+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}= \\ &{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\alpha_{\mathrm{Na+}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}}} \\ &+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{2}+\alpha\right)_{\mathrm{Cl-}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}} \end{aligned}\]But \[\left(c_{1}-\alpha\right)_{\mathrm{Na+}}^{\mathrm{eq}}=\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}-}^{\mathrm{eq}}\]Or, \[\left[\left(c_{1}-\alpha\right)_{\mathrm{Na}+}^{\mathrm{eq}}\right]^{2}=\alpha_{\mathrm{Na}+}^{\mathrm{eq}} \,\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}} \mathrm{CI}^{-}\]Then, \[\frac{\alpha^{\mathrm{eq}}}{\mathrm{c}_{1}}=\frac{\mathrm{c}_{1}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}}\]The latter is Donnan’s Equation. The ratio \(\left(\alpha^{\mathrm{eq}} / \mathrm{c}_{1}\right)\) tends to be smaller the larger is \(\mathrm{c}_{2}\). This conclusion is confirmed by experiment.We have simplified the algebra by writing \(\mathrm{R}^{+} \mathrm{Cl}^{-}\) as the salt in compartment II. In practice the Donnan equilibrium finds major application where salt \(\mathrm{RCl}\) is a macromolecule.Footnotes \(\mathrm{c}_{1}^{2}-2 \, \alpha \, \mathrm{c}_{1}+\alpha^{2}=\alpha \, \mathrm{c}_{2}+\alpha^{2}\)Then, \(c_{1}^{2}-2 \, \alpha \, c_{1}=\alpha \, c_{2}\)Or, \(\alpha=\frac{\mathrm{c}_{1}^{2}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}}\) F. G. Donnan et al, J. Chem. Soc.,1911, 1554; 1914,1941. F. G. Donnan, Chem. Rev.,1924, 1,73. F. G. Donnan and E. A. Guggenheim, Z. Physik Chem. A, 1932,162,346.This page titled 1.14.21: Donnan Membrane Equilibria is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.22: Descriptions of Systems

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Important themes in thermodynamics involve (i) properties (variables) which can be measured (e.g. volumes and/or densities) and (ii) thermodynamic variables which are rigorously defined (e.g. enthalpies). In these terms a measured property (e.g. density) is the reporter of the chemical properties or processes taking place in a system. So it is always important to ask if the “reporter” can be interrogated for the required information. In fact there is often a limit to the amount of information which a given reporter offers to the investigator. These important limitations should be borne in mind. An example makes the point.A system is prepared by placing \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of substance in a closed vessel at fixed \(\mathrm{T}\) and \(\mathrm{p}\). [The superscript ‘0’ implies at zero time.] We explore two possible descriptions of this system. Perhaps two samples were analysed by two independent laboratories.The first laboratory reports that the system is simple and contains the single substance \(\mathrm{X}\).Gibbs energy \[\mathrm{G}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mu_{\mathrm{X}}^{*}(\ell)\]and volume \[\mathrm{V}(\mathrm{A})=\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{X}}^{*}(\ell)\]Here \(\mu_{X}^{*}(\ell)\) and \(\mathrm{V}_{\mathrm{x}}^{*}(\ell)\) are the chemical potential and molar volume of the pure chemical substance \(\mathrm{X}\).The second laboratory identifies two substances \(\mathrm{X}\) and \(\mathrm{Y}\) in chemical equilibrium such that the equilibrium amounts of substances \(\mathrm{X}\) and \(\mathrm{Y}\) are respectively \(\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}}\).Gibbs energy \[\mathrm{G}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mu_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mu_{\mathrm{Y}}^{\mathrm{eq}}\]and volume \[\mathrm{V}(\mathrm{B})=\mathrm{n}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{X}}^{\mathrm{eq}}+\mathrm{n}_{\mathrm{Y}}^{\mathrm{eq}} \, \mathrm{V}_{\mathrm{Y}}^{\mathrm{eq}}\]Here \(\mu_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mu_{\mathrm{Y}}^{\mathrm{eq}}\) are the equilibrium chemical potentials; \(\mathrm{V}_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mathrm{V}_{\mathrm{Y}}^{\mathrm{eq}}\) are the equilibrium partial molar volumes.Description A is “primitive” and Description B is “sophisticated”. Both Gibbs energies and volumes are functions of state so that \(\mathrm{V}(\mathrm{A})=\mathrm{V}(\mathrm{B})\) and \(\mathrm{G}(\mathrm{A})=\mathrm{G}(\mathrm{B})\). The chemical potential of substance \(\mathrm{X}\) describes the change in \(\mathrm{G}\) when \(\delta n_{X}\) moles of \(\mathrm{X}\) are added. This chemical potential is insensitive to the changes taking place in the equilibrium system;\(\mu_{X}(\mathrm{~A})=\mu_{X}(\mathrm{~B})\). Consequently, measurement of volume \(\mathrm{V}\) [and if it were possible of \(\mathrm{G}\)] would not distinguish between the two descriptions. Similarly, measurement of \(\mathrm{H}\) (if it were possible) would not distinguish between the two descriptions.Footnotes L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd edn., 1970,p.16.This page titled 1.14.22: Descriptions of Systems is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.23: Enzyme-Substrate Interaction

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We consider the formation in aqueous solution of an enzyme –substrate complex \(\mathrm{ES}\) by an enzyme \(\mathrm{E}\) and substrate \(\mathrm{S}\). The system is prepared using \(\mathrm{n}^{0} (\mathrm{E})\) moles of enzyme and \(\mathrm{n}^{0} (\mathrm{S}\)) moles of substrate; equation (a)The upper limit of the extent of interaction \(\xi\) is controlled by whichever is the smallest amount, either \(\mathrm{n}^{0}(\mathrm{E})\) or \(\mathrm{n}^{0}(\mathrm{S})\). The latter two variables determine the total amount of \(\mathrm{ES}(\mathrm{aq})\) which can be formed in the limit of tight binding.The approach described above can be extended to more complicated schemes involving multip-step reactions. In the following we consider the case where enzyme \(\mathrm{E}\) converts substrate \(\mathrm{A}\) into product \(\mathrm{D}\). The system is prepared using \(\mathrm{n}^{0}(\mathrm{E})\) moles of enzyme and \(\mathrm{n}^{0}(\mathrm{A})\) moles of substrate \(\mathrm{A}\) such that there are two intermediates \(\mathrm{EB}(\mathrm{aq})\) and \(\mathrm{EC}(\mathrm{aq})\), product \(\mathrm{D}\) being liberated from the bound state in the final step. The equilibrium state can be represented by the following scheme. \[\begin{aligned} &\mathrm{E}(\mathrm{aq}) \quad+\mathrm{A}(\mathrm{aq}) \Leftrightarrow \mathrm{EB}(\mathrm{aq}) \Leftrightarrow \mathrm{EC}(\mathrm{aq}) \Leftrightarrow \mathrm{E}(\mathrm{aq})+\mathrm{D}(\mathrm{aq})\\ &\mathrm{n}^{0}(\mathrm{E})-\xi_{1}+\xi_{3} \mathrm{n}^{0}(\mathrm{~A})-\xi_{1} \quad \xi_{1}-\xi_{2} \quad \xi_{2}-\xi_{3} \quad \mathrm{n}^{0}(\mathrm{E})-\xi_{1}+\xi_{3} \xi_{3} \end{aligned}\]The key point is that at equilibrium the amounts of enzyme \(\mathrm{E}(\mathrm{aq})\) identified at both ends of the reaction must be the same. Further a mass balance shows that the total amount of enzyme present equals \(\mathrm{n}^{0}(\mathrm{E}) \mathrm{~mol}\). Other features are interesting; \(\xi_{3}\) must be zero if \(\xi_{2}\) is zero. The method can be applied to more complicated reaction schemes including those where the path from reactant to product involves parallel reactions.This page titled 1.14.23: Enzyme-Substrate Interaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.24: Equation of State- General Thermodynamics

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We take up the challenge of seeking an equation of state for all chemical substances. We confine attention to closed systems containing one chemical substance. We also confine our attention to systems at equilibrium where the affinity for spontaneous change is zero. A calculus operation allows us to relate \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\). \[\left(\frac{\partial p}{\partial V}\right)_{T} \,\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial T}{\partial p}\right)_{V}=-1\]By definition, the equilibrium isobaric expansivity. \[\alpha_{p}(A=0)=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, A=0}\]The equilibrium isothermal compressibility, \[\kappa_{\mathrm{T}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\]Hence, \[\left(\frac{\partial p}{\partial T}\right)_{V, A=0}=\frac{\alpha_{p}(A=0)}{\kappa_{T}(A=0)}\]\(\left(\frac{\partial p}{\partial T}\right)_{V, A=0}\) is the equilibrium isochoric thermal pressure coefficient. Equation (d) shows that this property can be obtained from the experimentally accessible, \(\alpha_{p}(A=0)\) and \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\). In fact the coefficient can be directly measured, at least for liquids. From the Master Equation where the affinity for spontaneous change is zero, \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}\]At constant \(\mathrm{T}\), \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}-\mathrm{p}\]Using a Maxwell Equation, \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p}\]All terms on the right hand side of equation (g) are experimentally accessible. Moreover this equation applies to all systems, solids, liquids and gases. By definition, \[\beta_{\mathrm{V}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]Then, \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \, \beta_{\mathrm{V}}-\mathrm{p}\]Equation (i) is a Thermodynamic Equation of State. Equation (j) is another Thermodynamic Equation of State. \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{V} \,\left(1-\mathrm{T} \, \alpha_{\mathrm{p}}\right)\]For many condensed phases the product \(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{v}}\) is much larger than the pressure \(\mathrm{p}\). Hence \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \cong \mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}\]The partial differential \(\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\) is the internal pressure, \(\pi_{j}\) for liquid \(j\). Originally the term ‘internal pressure’ referred to the product , \(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{v}}\). The closely related ratio of molar thermodynamic energy of vaporisation to molar volume \(\left[\frac{\Delta_{\mathrm{vap}} \mathrm{U}^{0}}{\mathrm{~V}^{*}(\ell)}\right]\) is the cohesive energy density, \[\text { c.e.d. }=\left[\frac{\Delta_{\mathrm{vap}} \mathrm{U}^{0}}{\mathrm{~V}^{*}(\ell)}\right]\]The rational behind this definition notes that \(\Delta_{\text {vap }} \mathrm{U}^{0}\) defines the change in thermodynamic energy when one mole of a given substance passes from the liquid to the vapour state, breaking strong cohesive forces in the liquid. By dividing by the molar volume of the liquid we normalise this change to a fixed volume.Internal pressures are interesting parameters. Nevertheless despite their thermodynamic basis, treatments of chemical properties in terms of internal pressures receive only sporadic attention. One feels they should be more informative but it is not always clear how one draws conclusions from analysis of experimental data using these properties.Footnotes \(\alpha_{p}=\frac{1}{\left[m^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right]}{[\mathrm{K}]}=\left[\mathrm{K}^{-1}\right]\) \(\kappa_{\mathrm{T}}=\frac{1}{\left[\mathrm{~m}^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right]}{\left[\mathrm{N} \mathrm{m}^{-3}\right]}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=[\mathrm{Pa}]^{-1}\) \(\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)=\frac{\left[\mathrm{N} \mathrm{m}^{-2}\right]}{[\mathrm{K}]}=\left[\mathrm{Pa} \mathrm{K}{ }^{-1}\right]\) A small amount of liquid sample is held in a sample cell sealed by a piston; the latter is linked to a device which allows a known pressure to be applied to the sample. The sample cell is held in a thermostat; the temperature of the latter is tightly controlled. The temperature is changed by a small amount; \(\Delta \mathrm{T}\). The volume of the liquid in the sample cell (normally) increases. The applied pressure is changed by a small amount \(\Delta \mathrm{p}\) in order to recover the original volume. Then for a given liquid at defined \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\), we have the ratio \((\Delta \mathrm{p} / \Delta \mathrm{T})\). J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd. edn., 1982, p.12. \(\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\frac{[\mathrm{J}]}{\left[\mathrm{m}^{3}\right]}=\left[\mathrm{N} \mathrm{m}{ }^{-2}\right]=[\mathrm{Pa}]\) Links between calorimetry and equations of state are discussed by S. L. Randzio, Chem. Soc. Rev., 1995, 24, 359. \(\mathrm{T}/\mathrm{K} = 298.15\)Liquid \(\pi_{\mathrm{i} /10^{8} \mathrm{Pa} \quad \mathrm{c.e.d}./10^{8} \mathrm{Pa}\) Methanol \(2.930 \quad 8.600\) DMSO \(5.166 \quad 7.047\) Water \(1.013 \quad 22.98\)This page titled 1.14.24: Equation of State- General Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.25: Equation of State- Perfect Gas

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.25%3A_Equation_of_State-_Perfect_Gas

A closed system contains \(\mathrm{n}_{j}\) moles of a gaseous substance \(j\). No chemical reaction takes place in the system. The system is at equilibrium where the affinity for spontaneous change is zero. The system is characterised by the thermodynamic energy \(\mathrm{U}\). The system is displaced to a neighbouring equilibrium state by a change in entropy \(\mathrm{dS}\) and a change in volume \(\mathrm{dV}\). The change in thermodynamic energy is given by the Master Equation. \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}\]At equilibrium the isothermal dependence of thermodynamic energy on volume is given by equation (b). \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}-\mathrm{p}\]In attempting to understand from a chemical standpoint the properties of gases, liquid mixtures and solutions, a common approach formulates a set of properties which classify a given system as ideal. The definition of an ideal (or, perfect) system is made with practical chemistry in mind. When examining the properties of gases there is merit in identifying the properties of a perfect gas. [No real gas is perfect!] If the gaseous substance \(j\) is a perfect gas, the following conditions are met at all temperatures and pressures.Here \(\mathrm{R}\) is the gas constant, \(8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\). Conditions (A) and (B) are equivalent. In most cases condition (B) is quoted because the equation links three practical properties, \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\).Footnotes \[\begin{aligned} &\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\frac{[\mathrm{J}]}{\left[\mathrm{m}^{3}\right]}=\frac{[\mathrm{N} \mathrm{m}]}{\left[\mathrm{m}^{3}\right]}=\left[\mathrm{N} \mathrm{m}{ }^{2}\right]=[\mathrm{Pa}] \\ &\mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}{ }^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\ &\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=[\mathrm{J}] \end{aligned}\] From definition (A) and equation (b), \[p=T \,\left(\frac{\partial S}{\partial V}\right)_{T}\]We use a Maxwell equation; \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]Hence from equations (i) and (ii), \[p=T \,\left(\frac{\partial p}{\partial T}\right)_{v}\]From definition (B), \[\left(\frac{\partial \mathrm{p}}{\partial T}\right)_{\mathrm{V}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} / \mathrm{V}=\mathrm{p} / \mathrm{T}\]Equations (iii) and (iv) are the same. Hence definition (B) is the integrated form of definition \(\mathrm{A}\). The gas constant \(\mathrm{R}\) is experimentally determined.This page titled 1.14.25: Equation of State- Perfect Gas is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.26: Equation of State - Real Gases, van der Waals, and Other Equations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.26%3A_Equation_of_State_-_Real_Gases%2C_van_der_Waals%2C_and_Other_Equations

The properties of gases pose a formidable challenge for chemists who seek to understand their \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties. Chemists adopt an approach which starts by defining the properties of a (hypothetical) ideal gas (Topics 1220 and 2588). The fact that the properties of a given real gas are not ideal is understood in terms of intermolecular interactions.In understanding the properties of real liquid mixtures and real solutions, the classic approach identifies the properties of the corresponding systems where the thermodynamic properties are defined as ‘ideal’. In the next stage the reasons why the properties of real liquid mixtures and real solutions are not ideal are discussed in terms of the nature and strength of intermolecular interactions. This general approach mimics the approach used to explain why the properties of real gases are not those of a defined ideal gas (see Topic 2588). In this context the van der Waals equation describing the differences between the properties of a real gas and an ideal gas sets the stage for theories describing the differences between the properties of real and ideal liquid mixtures and the properties of real and ideal solutions. Nevertheless we develop here the argument by considering the properties of a single gas, chemical substance \(j\).The starting points are two statements concerning an ideal gas.In a real gas the molecular volume is not negligible. Also cohesive intermolecular forces mean that the pressure exerted on the containing vessel is less than in the case of an ideal gas. Therefore the equation of state requires that the pressure \(\mathrm{p}\) is incremented by a quantity proportional to the density or, by a quantity inversely proportional to the volume. The famous van der Waals equation takes the following form where \(\mathrm{V}_{j}\) is the molar volume of gas \(j\) at pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\).\[\left(\mathrm{p}+\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{2}}\right) \,\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)=\mathrm{R} \, \mathrm{T}\]The van der Waals equation has played an important role in the development of theories describing fluids; i.e. both liquids and gases. The equation has merit in that it involves just two constants, both characteristic of given chemical substance. Further as McGlashan notes, the equation never leads to physical nonsense and does not predict physically absurd results. Similarly Rowlinson comments that the equation is easy to manipulate and never predicts physically absurd results. Chue comments that despite its simplicity the van der Waals equation ‘comprehends’ both liquid and gaseous states.However other authors are not so enthusiastic. For example, Prigogine and Defay comment that the equation is ‘mainly of qualitative interest’. Similarly Denbigh states that the a-parameter ‘does not have a sound theoretical basis and interpretation of the a-parameter in terms of intermolecular attraction is ‘intuitive’.Perhaps the expectation that the \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties of all gases and liquids can be accounted for using two parameters characteristic of each chemical substance is too optimistic. Nevertheless there is merit in reviewing the van der Waals equation.Equation (a) can be written as an equation for pressure \(\mathrm{p}\).\[\mathrm{p}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}-\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{2}}\]This form of equation (a) highlights the role of the parameter \(\mathrm{b}\) in describing the effect of molecular size and the role of parameter a in describing inter-molecular cohesion.A plot of \(\mathrm{p}\) as function of \(\mathrm{V}_{j}\) at fixed temperature has an extremum at the point defined by equation (c).\[\frac{\mathrm{R} \, \mathrm{T}}{\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)^{2}}=\frac{2 \, \mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{3}}\]Or,\[\mathrm{T}=\frac{2 \, \mathrm{a} \,\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)^{2}}{\mathrm{R} \, \mathrm{V}_{\mathrm{j}}^{3}}\]Hence using equations (a) and (d) we obtain an equation relating pressure \(\mathrm{p}\) to \(\mathrm{V}_{j}\) in terms of the two parameters\[\mathrm{p}=\mathrm{a} \,\left[\frac{\mathrm{V}_{\mathrm{j}}-2 \, \mathrm{b}}{\mathrm{V}_{\mathrm{j}}^{3}}\right]\]In a family of curves showing \(\mathrm{p}\) as a function of \(\mathrm{V}_{j}\), one curve has a point of inflexion with a horizontal tangent where both \(\frac{\partial p}{\partial V_{m}} \text { and } \frac{\partial^{2} p}{\partial V_{m}^{2}}\) are zero. This point is the critical point. Interesting features based on the van der Waals equation characterise this point.At the critical point,\[V_{j}^{c}=3 \, b\]\[p^{c}=a / 27 \, b^{2}\]and\[\mathrm{T}^{\mathrm{c}}=8 \, \mathrm{a} / 27 \, \mathrm{R} \, \mathrm{b}\]A classic plot describes the properties of a fixed amount of carbon dioxide in terms of isotherms showing the dependence of pressure on volume. This plot reported by \(\mathrm{T}\). Andrews in 1870 showed that \(\mathrm{CO}_{2}(\mathrm{g})\) cannot be liquefied solely by the application of pressure at temperatures above \(304.2 \mathrm{~K}\). The latter is the critical temperature \(\mathrm{T}^{\mathrm{c}}\) for \(\mathrm{CO}_{2}\), the critical pressure \(\mathrm{p}_{c}\) being the pressure required to liquefy \(\mathrm{CO}_{2}\) at this temperature. The molar volume at pressure \(\mathrm{p}^{\mathrm{c}}\) and temperature \(\mathrm{T}^{\mathrm{c}}\) is the critical molar volume \(\mathrm{V}_{\mathrm{j}}^{\mathrm{c}}\).The critical volume is obtained using the Law of Rectilinear Diameters originally described by L. Caillete and E. Mathias. The law requires that the mean density \(\rho_{\mathrm{j}}(\mathrm{T})\) of gas and liquid states of a given chemical substance \(j\) at common temperature \(\mathrm{T}\) is a linear function of \(\mathrm{T}\) Thus,\[\rho_{\mathrm{j}}(\mathrm{T})=\rho_{\mathrm{j}}\left(\mathrm{T}^{\mathrm{c}} / \mathrm{K}\right)+\alpha \,(\mathrm{T} / \mathrm{K})\]The parameters \(\rho_{\mathrm{j}}\left(\mathrm{T}^{\mathrm{c}} / \mathrm{K}\right)\) and \(\alpha\) are characteristic of chemical substance \(j\); \(\rho_{\mathrm{j}}\left(\mathrm{T}^{\mathrm{c}} / \mathrm{K}\right)\) is the critical density of chemical substance \(j\), leading to the critical molar volume \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{T}^{\mathrm{c}}\right)\) at critical temperature \(\mathrm{T}^{\mathrm{c}}\) and critical pressure \(\mathrm{p}^{\mathrm{c}}\). Above the critical temperature the plot of pressure \(\mathrm{p}\) and against molar volume at a given temperature \(\mathrm{T}\) is a smooth curve. Below \(\mathrm{T}^{\mathrm{c}}\) and at low pressures, chemical substance \(j\) is a gas. With increase in pressure a stage is reached where a given system comprises two phases, gas and liquid. With further increase in pressure the system comprises a liquid. There is no sharp transition between liquid and gaseous states in contrast to that observed on melting a solid. In other words, gas and liquid states for chemical substance \(j\) form a continuity of states.The Law of Corresponding States is an interesting concept, following an observation by J. D. van der Waals in 1881.The pressure, volume and temperature for a given gas \(j\) are expressed in terms of the critical pressure \(\mathrm{p}_{\mathrm{j}}^{\mathrm{c}}\), volume \(\mathrm{V}_{\mathrm{j}}^{\mathrm{c}}\), and temperature \(\mathrm{T}_{\mathrm{j}}^{\mathrm{c}}\) using three proportionality constants, \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) respectively. Thus\[\mathrm{p}=\beta_{1} \, \mathrm{p}_{\mathrm{j}}^{\mathrm{c}}\]\[\mathrm{V}_{\mathrm{j}}=\beta_{2} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{c}}\]\[\mathrm{T}_{\mathrm{j}}=\beta_{3} \, \mathrm{T}_{\mathrm{j}}^{\mathrm{c}}\]Hence from equation (a),\[\left(\beta_{1} \, p_{j}^{c}+\frac{a}{\left(\beta_{2} \, V_{j}^{c}\right)^{2}}\right) \,\left(\beta_{2} \, V_{j}^{c}-b\right)=R \, \beta_{3} \, T_{j}^{c}\]Using equations (f), (g) and (h) for \(\mathrm{p}_{j}^{c}\), \(\mathrm{V}_{j}^{c}\) and \(\mathrm{T}_{j}^{c}\) the following equation \(j\)is obtained from equation (m).\[\left(\beta_{1}+\frac{3}{\beta_{2}^{2}}\right) \,\left(3 \, \beta_{2}-1\right)=8 \, \beta_{3}\]Equation (n) is the van der Waals reduced Equation of State for gas \(j\); \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) being the reduced pressure, volume and temperature respectively. Significantly there are no parameters in equation (n) which can be said to be characteristic of a given chemical substance. In other words the equation has a universal character.The van der Waals equations prompted the development of many equations of state.The van der Waals equation can be modified in two simple ways. In one modification it is assumed that \(V_{j} \gg b\). The assumption is that attractive intermolecular processes are dominant Hence,\[\left[\mathrm{p}+\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{2}}\right] \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}\]Or,\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}-\left(\mathrm{a} / \mathrm{V}_{\mathrm{j}}\right)\]In another approach it is assumed that repulsive intermolecular forces are dominant. Thus,\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{b}\]One criticism of the van der Waal equation is that no account is taken of the possibility that parameters a and b can depend on temperature. Clausius suggested the following equation in which intermolecular attraction is described as inversely proportional to temperature; \(a\), \(b\) and \(c\) are three constants characteristic of a gas \(j\). \[\left(\mathrm{p}+\frac{\mathrm{a}}{\mathrm{T} \,\left(\mathrm{V}_{\mathrm{j}}+\mathrm{c}\right)^{2}}\right) \,\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)=\mathrm{R} \, \mathrm{T}\]Nevertheless the advantages gained by recognising that attraction might be dependent on temperature are outweighed by problems associated with using this equation.In this approach, the term \(\left(\mathrm{V}_{j}+c\right)\) in the Clausius equation is replaced by \(\mathrm{V}_{j}\). Then\[\left(\mathrm{p}+\frac{\mathrm{a}}{\mathrm{T} \, \mathrm{V}_{\mathrm{j}}^{2}}\right) \,\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)=\mathrm{R} \, \mathrm{T}\]Or,\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}+\mathrm{p} \, \mathrm{b}-\frac{\mathrm{a}}{\mathrm{T} \, \mathrm{V}_{\mathrm{j}}}+\frac{\mathrm{a} \, \mathrm{b}}{\mathrm{T} \, \mathrm{V}_{\mathrm{j}}^{2}}\]The van der Waals, Clausius and Berthelot equations are the forerunners of a large family of cubic equations of state; i.e. equations of state that are cubic polynomials in molar volume.Analysis of experimental data prompted the development of the following equation using critical pressure \(\mathrm{p}_{\mathrm{j}}^{\mathrm{c}}\) and temperature \(\mathrm{T}_{\mathrm{c}}\).\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T} \,\left[1+\frac{\mathrm{a}}{128} \, \frac{\mathrm{p} \, \mathrm{T}_{\mathrm{j}}^{\mathrm{c}}}{\mathrm{p}_{\mathrm{j}}^{\mathrm{c}} \, \mathrm{T}} \,\left(1-6 \, \frac{\left(\mathrm{T}_{\mathrm{j}}^{\mathrm{c}}\right)^{2}}{\mathrm{~T}^{2}}\right)\right]\]This modification suggested in 1899 by C. Dieterici attempts to account for the fact that molecules at the wall of a containing vessel have higher potential energy than molecules in the bulk gas. The following equation was proposed.\[\mathrm{p} \,\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)=\mathrm{R} \, \mathrm{T} \, \exp \left(-\frac{\mathrm{a}}{\mathrm{R} \, \mathrm{T} \, \mathrm{V}_{\mathrm{j}}}\right)\]The following virial equation was proposed in 1885 by Thiesen.\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T} \,\left[1+\frac{\mathrm{B}(\mathrm{T})}{\mathrm{V}_{\mathrm{j}}}+\frac{\mathrm{C}(\mathrm{T})}{\mathrm{V}_{\mathrm{j}}^{2}}+\ldots \ldots\right]\]Following a suggestion in 1901 by H. K. Onnes, B(T), C(T),… are called virial coefficients. A modern account of equations of state is given in reference.Boyle’s Law requires that the product \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\) at fixed temperature is independent of pressure. In the case of hydrogen, the product \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\) at \(273 \mathrm{~K}\) increases with increase in pressure. However for many gases (e.g. nitrogen and carbon dioxide) the product \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\) at fixed \(\mathrm{T}\) decreases with increase in pressure, passes through a minimum and then increases. For a given gas the minimum moves to lower pressures with increase in temperature until at high temperatures no minimum is observed. This is the Boyle temperature, which is characteristic of a given gas [11a]. The van der Waals equation offers an explanation of the pattern. Thus from equation (b),\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T} \,\left(\frac{\mathrm{V}_{\mathrm{j}}}{\mathrm{V}_{\mathrm{j}}-\mathrm{b}}\right)-\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}} \label{x}\]Equation \ref{x} is differentiated with respect to pressure at constant temperature. If the plot of \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\) against \(\mathrm{p}\) passes through zero at temperature \(\mathrm{T}_{\mathrm{B}}\), then \(\mathrm{T}_{\mathrm{B}}\) is given by equation (y).\[\mathrm{T}_{\mathrm{B}}=\mathrm{a} / \mathrm{R} \, \mathrm{b}\]Therefore in terms of Equation (a), a low Boyle temperature is favoured by small \(\mathrm{a}\) and large \(\mathrm{b}\) parameters.Footnotes For a reproduction of a portrait of Johannes van der Waals see D. Kondepudi and I. Prigogine, Modern Thermodynamics, Wiley , Colchester, 1998, page 15. \(\begin{aligned} &V_{j}=\left[m^{3} \mathrm{~mol}^{-1}\right] \quad b=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]\\ &\left(\mathrm{p}+\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{2}}\right)=\left(\left[\mathrm{Nm}^{-2}\right]+\frac{\mathrm{a}}{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{2}}\right) \end{aligned}\)Then, \(\begin{aligned} &\mathrm{a}=\left[\mathrm{N} \mathrm{m} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{2}=\left[\mathrm{N} \mathrm{mol}^{-2} \mathrm{~m}^{4}\right]\\ &\mathrm{R} \, T=\left[\mathrm{J} \mathrm{mol} \mathrm{K}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}\)Then \(\begin{aligned} &\left(\left[\mathrm{N} \mathrm{m}^{-2}\right]+\frac{\left[\mathrm{N} \mathrm{mol}^{-2} \mathrm{~m}^{4}\right]}{\left[\mathrm{mol}^{-1} \mathrm{~m}^{3}\right]^{2}}\right) \,\left[\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]-\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]\right] \\ &=\left(\left[\mathrm{Nm}^{-2}\right]+\left[\mathrm{Nm}^{-2}\right]\right) \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}\) M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 176. J. S. Rowlinson, Liquids and Liquid Mixtures, Butterworths, London, 2nd edn., 1969, page 66. S. H. Chue, Thermodynamics, Wiley, Chichester, 1977, page 136. I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D.H.Everett, Longmans Green, London, 1954, p. 145. K. Denbigh, The Principles of Chemical Equilibrium, Cambridge University Press. 3rd. edn. 1971, page 119. \(\frac{\mathrm{dp}}{\mathrm{dV}}=-\frac{\mathrm{R} \, \mathrm{T}}{\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)^{2}}+\frac{2 \, \mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{3}}\) At an extremum, \(\frac{\mathrm{dp}}{\mathrm{dV}_{\mathrm{j}}}=0\); then, \(\frac{\mathrm{R} \, \mathrm{T}}{\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)^{2}}=\frac{2 \, \mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{3}}\) \(\begin{aligned} &\left(p+\frac{a}{V_{j}^{2}}\right) \,\left(V_{j}-b\right)=R \, 2 \, a \, \frac{\left(V_{j}-b\right)^{2}}{V_{j}^{3}} \\ &p+\frac{a}{V_{j}^{2}}=2 \, a \, \frac{\left(V_{j}-b\right)}{V_{j}^{3}} \end{aligned}\)Or, \(\mathrm{p}=\mathrm{a} \,\left[\frac{2 \, \mathrm{V}_{\mathrm{j}}-2 \, \mathrm{b}-\mathrm{V}_{\mathrm{j}}}{\mathrm{V}_{\mathrm{j}}^{3}}\right]\) T.Andrews, Phil. Mag.,1870,,39,150. See for example, We rewrite equation (e) in the following form. \(p=a \,\left[V_{j}-2 \, b\right] \, V_{j}^{-3}\)Or, \(\mathrm{p}=\mathrm{a} \,\left\{\left[\mathrm{V}_{\mathrm{j}}-2 \, \mathrm{b}\right] \,\left[\mathrm{V}_{\mathrm{j}}^{-3}\right]\right\}\)Then \(\begin{aligned} \frac{\mathrm{dp}}{\mathrm{dV}} &=\mathrm{a} \,\left[\mathrm{V}_{\mathrm{j}}^{-3}+\left(\mathrm{V}_{\mathrm{j}}-2 \, \mathrm{b}\right) \,(-3) \, \mathrm{V}_{\mathrm{j}}^{-4}\right] \\ &=\mathrm{a} \,\left[\mathrm{V}_{\mathrm{j}}^{-3}-3 \, \mathrm{V}_{\mathrm{j}}^{-3}+6 \, \mathrm{b} \, \mathrm{V}_{\mathrm{j}}^{-4}\right] \\ &=\mathrm{a} \,\left[-2 \, \mathrm{V}_{\mathrm{j}}^{-3}+6 \, \mathrm{b} \, \mathrm{V}_{\mathrm{j}}^{-4}\right] \end{aligned}\)\(\frac{d p}{d V_{j}}=0\) where \(\left[-2 \, \mathrm{V}_{\mathrm{j}}^{-3}+6 \, \mathrm{b} \, \mathrm{V}_{\mathrm{j}}^{-4}\right]=0\) Or, \(\frac{1}{V_{j}^{3}}=\frac{3 \, b}{V_{j}^{4}}\) Or, \(V_{j}^{c}=3 \, b\) From equation (e), \(p^{c}=\frac{a \, b}{27 \, b^{3}}\) or, \(\mathrm{p}^{\mathrm{c}}=\frac{\mathrm{a}}{27 \, \mathrm{b}^{2}}\)Then from equation (d), \(T^{c}=\frac{2 \, a \,(3 \, b-b)^{2}}{R \, 27 \, b^{3}}=\frac{8 \, a \, b^{2}}{27 \, R \, b^{3}}=\frac{8 \, a}{27 \, R \, b}\)\[\begin{aligned} &{\mathrm{V}_{\mathrm{j}}^{\mathrm{c}}=3 \, \mathrm{b}= \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]} \\ &\mathrm{p}^{c}=\frac{\left[\mathrm{N} \mathrm{m}^{4} \mathrm{~mol}^{-2}\right]}{ \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{2}}=\left[\mathrm{N} \mathrm{m}^{4} \mathrm{~m}^{-6} \mathrm{~mol}^{-2} \mathrm{~mol}^{2}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right] \\ &\mathrm{T}^{\mathrm{c}}=\frac{ \,\left[\mathrm{N} \mathrm{m} \mathrm{mol}^{-2}\right]}{ \,\left[\mathrm{J} \mathrm{mol} \mathrm{m}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}=\frac{[\mathrm{N} \mathrm{m}]}{[\mathrm{N} \mathrm{m}]} \,[\mathrm{K}]=[\mathrm{K}] \end{aligned}\] \(\left[\beta_{1} \, \frac{\mathrm{a}}{27 \, \mathrm{b}^{2}}+\frac{\mathrm{a}}{\left(\beta_{2} \, 3 \, \mathrm{b}\right)^{2}}\right] \,\left[\beta_{2} \, 3 \mathrm{~b}-\mathrm{b}\right]=\mathrm{R} \, \frac{8 \, \mathrm{a} \, \beta_{3}}{27 \, \mathrm{R} \, \mathrm{b}}\) F. T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965, page 169. S. I. Sandler, H. Orbey and B.-I. Lee , Models for Thermodynamics and Phase Equilibria Calculations , ed. S. I. Sandler, Marcel Dekker, New York, 1994, pp.87-186.\[\begin{aligned} &\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T} \, \frac{\mathrm{V}_{\mathrm{j}}}{\mathrm{V}_{\mathrm{j}}-\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}} \\ &{\left[\frac{\partial\left(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=\left[\mathrm{R} \, \mathrm{T} \,\left(\frac{1}{\left.\mathrm{~V}_{\mathrm{j}}-\mathrm{b}\right)}\right)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\mathrm{V}_{\mathrm{j}}}{\left(\mathrm{V}_{\mathrm{j}}-\mathrm{b}\right)^{2}}\right)+\frac{\mathrm{a}}{\mathrm{V}_{\mathrm{j}}^{2}}\right] \,\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}} \end{aligned}\]Then at \(\left[\frac{\partial\left(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=0\),\[\left[R \, T \,\left(\frac{V_{j}-b-V_{j}}{\left(V_{j}-b\right)^{2}}\right)+\frac{a}{V_{j}^{2}}\right]=0\]Or, \(R \, T=\frac{a}{b} \, \frac{\left(V_{j}-b\right)^{2}}{V_{j}^{2}}\)If the minimum occurs where \(\mathrm{p}\) is zero (i.e. where \(\mathrm{V}_{j}\) is infinitely large), \(\mathrm{R} \, \mathrm{T}_{\mathrm{B}}=\mathrm{a} / \mathrm{b}\)This page titled 1.14.26: Equation of State - Real Gases, van der Waals, and Other Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.27: Euler's Theorem

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This theorem emerges from theories concerned with differential equations. The theorem finds many applications in thermodynamics. In particular the theorem concerned with homogeneous functions of the first degree is important. This theorem can be stated as follows.\[\mathrm{f}(\mathrm{k} \, \mathrm{x}, \mathrm{k} \, \mathrm{y}, \mathrm{k} \, \mathrm{z})=\mathrm{k} \, \mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}) \label{a}\]By way of illustration we consider a liquid mixture volume \(\mathrm{V}\) prepared using \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles of liquid 1 and 2. If we had used \(2 . \mathrm{n}_{1}\) and \(2 . \mathrm{n}_{2}\) moles of liquids 1 and 2, then the final volume would have been \(2 . \mathrm{V}\). The important theorem allows us to set down the following descriptions.For a system comprising \(\mathrm{i}\)-chemical substances, it follows that\[\mathrm{V}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\label{b}\]where partial molar volume\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \label{c}\]It should be noted that some thermodynamic monographs, when citing Equation \ref{b}, include the phrase ‘at constant temperature and pressure’. Other monographs do not include this phrase on the grounds that the isobaric - isothermal condition is included in the definition of the partial derivative in Equation \ref{c}. In practice nothing is lost by including this phrase simply to indicate that the analysis is concerned with the properties of systems in the \(\mathrm{T} – \mathrm{p}\) - composition domain.A similar analysis in the context of Gibbs energies leads to the following two equations and the definition of chemical potentials.\[G=\sum_{j=1}^{j=i} n_{j} \, \mu_{j} \label{d}\]where chemical potential\[\mu_{j}=\left(\frac{\partial G}{\partial n_{j}}\right)_{T, p, n(i \neq j)} \label{e}\]Footnotes R. J. Tykodi, J. Chem. Educ.,1982,59,557.This page titled 1.14.27: Euler's Theorem is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.28: First Law of Thermodynamics

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The first law of thermodynamics centres on the concept of energy. In its broadest sense, the law requires that the energy of the universe is constant. This is a rather overwhelming statement. A more attractive statement requires that the (internal) thermodynamic energy \(\mathrm{U}\) of a chemistry laboratory is constant: \[\mathrm{U} = \text { constant }\]The latter statement is the principle of conservation of energy; energy can be neither created nor destroyed. All that a chemist can do, during an experiment using a closed reaction vessel, is to watch energy ‘move ‘ between system and surroundings. As a consequence of equation (a), we state that \[\Delta \mathrm{U}(\text { system })=-\Delta \mathrm{U}(\text { surroundings })\]We can not know the actual energy of a closed system although we know that it is an extensive property of the system. In describing energy changes we need a convention. So we use the acquisitive convention describing all changes in terms of how a system is affected. Thus the statement \(\Delta \mathrm{U} < 0\) means that the energy of a given system decreases during a given process; e.g. chemical reaction.Footnote In principle it is possible to know the total energy of a given system using a scale in conjunction with Einstein’s famous equation, \(\mathrm{E} = \mathrm{m} \, \mathrm{c}^{2}\). However, the mass corresponding to \(1 \mathrm{~kJ}\) is only about \(10^{-14} \mathrm{~kg}\).This page titled 1.14.28: First Law of Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.29: Functions of State

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For a system containing one chemical substance we define the volume using equation (a). \[\mathrm{V}=\mathrm{V}[\mathrm{T}, \mathrm{p}, \mathrm{n},]\]The variables in the square brackets are the Independent Variables. The term `independent' means that, within limits, we can change \(\mathrm{T}\) independently of the pressure and \(\mathrm{n}_{j}\); change \(\mathrm{p}\) independently of \(\mathrm{T}\) and \(\mathrm{n}_{\mathrm{j}}\); and \(\mathrm{n}_{\mathrm{i}}\) independently of \(\mathrm{T}\) and \(\mathrm{p}\). There are some restrictions in our choice of independent variables. At least one variable must define the amount of all substances in the system and one variable must define the `hotness' of the system.Actually there is merit in writing equation (a) in terms of three intensive variables which in turn define, for example the, the molar volume of liquid chemical substance 1 at specified temperature and pressure, \(\mathrm{V}_{1}^{*}(\ell)\). \[\mathrm{V}_{1}^{*}(\ell)=\mathrm{V}(\ell)\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}=1\right]\]If the chemical composition of a given closed system is specified in terms of two chemical substances 1 and 2, four independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) define the dependent variable \(\mathrm{V}\). Thus \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\]For a system containing i-chemical substances where the amounts can be independently varied, the dependent variable \(\mathrm{V}\) is defined by the following equation. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \mathrm{n}_{\mathrm{i}}\right]\]In a general analysis, we start out with a closed system having Gibbs energy at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), molecular composition (organisation) \(\xi\) and affinity for spontaneous change \(\mathrm{A}\). We define the Gibbs energy as follows. \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \boldsymbol{\xi}]\]In the state defined by equation (e), there is an affinity for spontaneous change (chemical reaction) \(\mathrm{A}\). We imagine that starting with the system in the state defined by equation (e), it is possible to change the pressure and perturb the system to a series of neighbouring states for which the affinity for spontaneous change remains constant. The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant affinity is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\).Returning to the original state characterised by \(\mathrm{T}\), \(\mathrm{p}\) and \(\xi\), we imagine it is possible to perturb the system by a change in pressure in such a way that the system remains at fixed extent of reaction \(\xi\). The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\xi\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\). We explore these dependences of \(\mathrm{G}\) on pressure at fixed temperature and at (i) fixed composition \(\xi\) and (ii) fixed affinity for spontaneous change, \(\mathrm{A}\). \((\partial G / \partial p)_{\mathrm{T}, \mathrm{A}}\) and \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\) are related using a standard calculus operation. Thus at fixed temperature, \[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}}\]This interesting equation shows that the differential dependence of Gibbs energy (at constant temperature) on pressure at constant affinity for change does NOT equal the corresponding dependence at constant extent of chemical reaction. This is inequality is not surprising. But our interest is drawn to the case where the system under discussion is, at fixed temperature and pressure, at thermodynamic equilibrium where \(\mathrm{A}\) is zero, \(\mathrm{d} \xi / \mathrm{dt}\) is zero, the Gibbs energy is a minimum AND significantly \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. We conclude that \[\mathrm{V}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \mathrm{A}=0}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi(\mathrm{eq})}\]This rather long winded argument confirms that volume \(\mathrm{V}\) is a state variable, the dependence of \(\mathrm{G}\) on pressure for differential displacement at constant '\(\mathrm{A} =0\)' and \(\sim^{e} q\) being identical. These comments may seem trivial but the point is made if we go on to consider the volume of a system as a function of temperature at constant pressure. We again use a calculus operation to derive the relationship in equation (h). \[\left[\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{p}}\]We are not surprised to discover that in general terms the differential dependence of \(\mathrm{V}\) on temperature at constant affinity does not equal the differential dependence of \(\mathrm{V}\) on temperature at constant composition/organisation. Indeed unlike the simplification we used in connection with equation (e), we cannot assume that the volume of reaction \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p} p}\) is zero at equilibrium. In other words for a closed system at thermodynamic equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) {where \(\mathrm{A}=0, \xi=\xi^{\mathrm{eq}} \text { and } \mathrm{d} \xi / \mathrm{dt}=0\)}, there are two thermal expansions.We consider a closed system in equilibrium state I defined by the set of variables, \(\left\{\mathrm{T}[\mathrm{I}], \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{I}]\right\}\). The equilibrium composition is represented by \(\xi^{\mathrm{eq}}[\mathrm{I}]\) at zero affinity for spontaneous change. This system is perturbed to nearby state at constant pressure .\[\Delta \mathrm{V}(\mathrm{A}=0)=\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}]\] We record the equilibrium thermal expansion, \[\mathrm{E}(\mathrm{A}=0)=\left[\frac{\mathrm{V}(\mathrm{II})-\mathrm{V}(\mathrm{I})}{\Delta \mathrm{T}}\right]\] The equilibrium expansivity, \[\alpha(\mathrm{A}=0)=\mathrm{E}) \mathrm{A}=0) / \mathrm{V}\] In order for the system to move from one equilibrium state, I with composition \(\xi^{\mathrm{eq}}[\mathrm{I}]\) to another equilibrium state, II with composition \(\xi^{\mathrm{eq}}[\mathrm{II}]\), the chemical composition and /or molecular organisation changes. The term `expansion' indicates the isobaric dependence of volume on temperature, \[\mathrm{E}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1}\right]\] \(\mathrm{E}\) is an extensive variable. The corresponding volume intensive variable is the expansivity, \(\alpha\). \[\alpha=\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{p}\] where \(\alpha=\frac{1}{\left[\mathrm{~m}^{3}\right]} \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~K}}\right]=\left[\mathrm{K}^{-1}\right]\)Hence we define the `frozen' expansion, \(\mathrm{E}(\sim=\text { fixed })\). An alternative name is the instantaneous expansion because, practically, we would have to change the temperature at such a high rate that there is no change in molecular composition or molecular organisation in the system. Thus, \[\mathrm{E}(\xi=\text { fixed })=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\]Further \[\alpha(\xi=\text { fixed })=\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\]Similar comments apply to isothermal compressibilities, \(\kappa_{\mathrm{T}}\); there are two limiting properties, \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) and \(\kappa_{\mathrm{T}}(\xi)\). In order to measure \(\kappa_{\mathrm{T}}(\xi)\) we have to change the pressure in a infinitely short time.The entropy \(\mathrm{S}\) at fixed composition is given by the partial differential \(-\left(\frac{\partial G}{\partial T}\right)_{p, 5}\) and, at constant affinity of spontaneous change by \(-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}}\). At equilibrium where \(\mathrm{A} = 0\), the equilibrium entropy, \(S(A=0)=-\left(\frac{\partial G}{\partial T}\right)_{p, A=0}\). We carry over the argument described above but now concerned with a system characterised by \(\mathrm{T}, \mathrm{p}, \xi\) which is perturbed by a change in temperature. We consider two pathways, at constant \(\mathrm{A}\) and at constant \(\xi\). \[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]But at equilibrium, \(\mathrm{A}\) which equals \(-\left[\frac{\partial G}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\) is zero and so \(\mathrm{S}(\mathrm{A}=0)=\mathrm{S}\left(\xi^{\mathrm{eq}}\right)\). Then just as for volumes, the entropy of a system is not a property concerned with pathways between states; entropy is a function of state.Another important link involving Gibbs energy and temperature is provided by the Gibbs-Helmholtz equation. We explore the relationship between changes in (\(\mathrm{G}/\mathrm{T})\) at constant affinity and fixed \(\xi\) following a change in temperature. Thus, \[\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]But at equilibrium \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Then \(\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\).In other words the variable, enthalpy is a function of state. This is not the case for isobaric heat capacities. Thus, \[\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]We cannot assume that the triple product term in equation (q) is zero. Hence there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) and the `frozen' isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)\). In other words, an isobaric heat capacity is not a function of state because it is concerned with a pathway between states.Footnotes The phrase `independent variable' is important. With reference to the properties of an aqueous solution containing ethanoic acid, the number of components for such a solution is 2, water and ethanoic acid. The actual amounts of ethanoic acid, water, ethanoate and hydrogen ions are determined by an equilibrium constant which is an intrinsic property of this sytem at a given \(\mathrm{T}\) and \(\mathrm{p}\). From the point of view of the Phase Rule, the number of components equals two. For the same reason when we consider the volume of a system containing \(\mathrm{n}\) moles of water we do not take account of evidence that water partly self-dissociates into \(\mathrm{H}^{+} (\mathrm{aq})\) and \(\mathrm{OH}^{-} (\mathrm{aq})\) ions. In terms of the Phase Rule, we note that for two components (C = 2) and one phase (P = 1) , the number of degrees of freedom F equals equals two. These degrees of freedom refer to intensive variables. Hence for a solution where chemical substance 1 is the solvent and chemical substance 2 is the solute, the system is defined by specifying by the three (intensive) degrees of freedom, \(\mathrm{T}, \mathrm{p} and, for example, solute molality.This page titled 1.14.29: Functions of State is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.3: Excess Thermodynamic Properties- Aqueous Solutions

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A given aqueous solution, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong \mathrm{p}^{0}\)), contains \(\mathrm{i}\)-solutes, with \(\mathrm{n}_{\mathrm{j}}\) moles of each solute \(\mathrm{j}\), and \(\mathrm{n}_{1}\) moles of water(\(\ell\)).The Gibbs energy of the solution is given by equation (a). \[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]For a solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)), in vast molar excess, \[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\sum_{\mathrm{j}=1}^{\mathrm{s}} \mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]We assert that the system is at thermodynamic equilibrium. For each solute \(\mathrm{j}\), \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{\mathrm{j}}\) and the reference chemical potential for solute \(\mathrm{j}\) in a solution where \(\mathrm{m}_{\mathrm{j}} = 1 \mathrm{~mol kg}^{-1}\) and the thermodynamic properties of the solute are ideal. Then, \[\left\{\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right\} \quad \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]where \(\operatorname{limit}\left(m_{j} \rightarrow 0\right) \gamma_{j}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\).For the solvent we express the properties in terms of a practical osmotic coefficient, \(\phi\). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \(\operatorname{limit}\left(\sum_{j=1}^{j=i} m_{j} \rightarrow 0\right) \phi=1.0\)For the solution, \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}\right] \\ &+\sum_{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]If the thermodynamic properties of the solution are ideal, \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}\right] \\ &+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]By definition the solution excess Gibbs energy of the solution, \[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]\(\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) \text { is expressed in }\left[\mathrm{J} \mathrm{kg}^{-1}\right]\).Then \[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,(1-\phi) \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]\[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) / \mathrm{R} \, \mathrm{T}=(1-\phi) \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)\]For a solution containing a single solute \(\mathrm{j}\), \[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) / \mathrm{R} \, \mathrm{T}=\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \, \mathrm{m}_{\mathrm{j}}\]If the thermodynamic properties of the solution are ideal, the chemical potential of the solute is given by equation (k). \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Equation (c) describes the properties of solute \(\mathrm{j}\) in a real solution. By definition the excess chemical potential \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is given by equation (l). \[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mu_{\mathrm{j}}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\]Then, \[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]Often an excess chemical potential \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is written in the form \(\mathrm{G}_{\mathrm{j}}^{\mathrm{E}}\). In the case of the solvent, water(\(\ell\)) the corresponding equations for the chemical potentials in solutions having either real or ideal thermodynamic properties are given by equations (n) and (o). \[\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\[\mu_{1}^{\mathrm{E}}(\mathrm{aq})=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Footnotes For further comments see—This page titled 1.14.3: Excess Thermodynamic Properties- Aqueous Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.30: Heat, Work, and Energy

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Thermodynamics asserts that the energy of a closed system increases ifSeparation of the heat term from the work term is extremely important in the context of the Second Law of Thermodynamics. Heat flows spontaneously from high to low temperatures, the word ‘spontaneous’ being absolutely crucial in the context of the Second Law.There are many ways in which the surroundings can do work on a system. At this stage we note the distinction which is drawn between the three variables \(\mathrm{U}\), \(\mathrm{q}\) and \(\mathrm{w}\). [The point is emphasized by use of upper and lower case letters.] The variables \(\mathrm{q}\) and \(\mathrm{w}\) describe pathways which can result in a change in thermodynamic energy. We make the point by rewriting equation (a) to show the change in thermodynamic energy on going from state I to state II. Thus, \[\Delta \mathrm{U}=\mathrm{U}(\mathrm{II})-\mathrm{U}(\mathrm{I})=\mathrm{q}+\mathrm{w}\]If for example \(\Delta \mathrm{U} = 100 \mathrm{~J}\), this can be a consequence of many pathways between state I and state II: e.g.Hence \(\mathrm{U}\) is a function of state (or, state variable) although \(\mathrm{q}\) and \(\mathrm{w}\) are not state variables. This is a triumph of the First Law of Thermodynamics. The task faced by chemists is to identify and describe quantitatively the actual pathway accompanying, for example, a given chemical reaction. Equation (a) signals the energy difference \(\Delta \mathrm{U}\) between two states which might involve a comparison of the energies at the start and finish of a chemical reaction in a closed system. In developing our argument there is merit in considering the change in energy of the original system following a small change along the overall reaction pathway.We consider a closed reaction vessel containing ethyl ethanoate (aq; \(0.1 \mathrm{~mol}\)) and NaOH(aq; excess) . Spontaneous chemical reaction leads to hydrolysis of the ester to form EtOH(aq). The change in thermodynamic energy \(\Delta \mathrm{U}\) equals \(\mathrm{U}(\mathrm{II}) − \mathrm{~U}(\mathrm{I})\). We subdivide the total chemical reaction into small steps where the change in composition, (i.e. \(\mathrm{d}\xi\)) is accompanied by a change in thermodynamic energy \(\mathrm{dU}\). \[\Delta \mathrm{U}=\int_{\text {state } \mathrm{I}}^{\text {state II }} \mathrm{dU}\]If the volume of the system changes by the differential amount \(\mathrm{dV}\) such that the pressure within the closed system equals the confining pressure \(\mathrm{p}\), \[\mathrm{w}=-\mathrm{p} \, \mathrm{dV}\]Then, \[\mathrm{dU}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}\]We write equation (e) in the following form; \[\mathrm{q}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}\]The right hand side of equation (f) contains the differential changes in two extensive state variables, \(\mathrm{U}\) and \(\mathrm{V}\). Consequently heat \(\mathrm{q}\) is precisely defined by the changes in thermodynamic energy and volume at pressure \(\mathrm{p}\). The ‘equivalence ‘ of heat and work was first demonstrated in many experiments carried out in the 19th Century by James Joule, the son of a brewer (Salford, England). Joule showed that by doing work on a thermally isolated system the temperature of the latter increases. In other words, doing work on a system is equivalent to passing heat into the system.The SI unit of energy is the joule, symbol \(\mathrm{J}\); \(\mathrm{J} \equiv \mathrm{kg} \mathrm{m}^{2} \mathrm{~s}^{-2}\) Sometimes one reads that thermodynamics is not concerned with ‘time’. However the concept of energy and the unit of energy involves ’time’. Of course the origins of these concepts are classical mechanics and accompanying discussion of potential and kinetic energies. \(\mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}{ }^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}]\) The fundamental link between heat and work was established by Joule. Interestingly the link between heat and work was apparent previously to A. Haller who suggested that human bodies are heated by the friction between solid particles in the blood passing through the capillaries in the lungs; see comments byThis page titled 1.14.30: Heat, Work, and Energy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.31: Helmholtz Energy

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.31%3A_Helmholtz_Energy

Gibbs energy is defined with practical chemistry in mind because the definition centres on closed systems held at constant temperature and constant pressure. A similar interest prompt definition of the Helmholtz energy, symbol \(\mathrm{F}\). By definition, \[\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S}\]Physicists use the term ‘Helmholtz Function’. The old term ‘Helmholtz free energy’ is not encouraged. If a closed system is displaced to a neighboring state, the differential change in Helmholtz energy is given by equation (b). \[\mathrm{dF}=\mathrm{dU}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}\]But the differential change in thermodynamic energy is given by the Master Equation. By incorporating the latter into equation (b), we obtain equation (c), the memorable “all-minus” equation. \[\begin{aligned} &\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \\ &\mathrm{A} \, \mathrm{d} \xi \geq 0 \end{aligned}\]At fixed temperature and fixed volume (isothermal and isochoric conditions), \[\mathrm{dF}=-\mathrm{A} \, \mathrm{d} \xi\]All spontaneous processes at fixed temperature and fixed volume lower the Helmholtz energy of a closed system. In practical terms, for a closed system held at constant volume and temperature, chemical reaction (molecular reorganization) lowers the Helmholtz energy of the system. We presume that the pressure inside the reaction vessel will change, decreasing for some systems and increasing for other systems. As it stands thermodynamics offers no generalization concerning how the pressure changes. In fact if we want to use the Helmholtz energy as an indicator of the direction of spontaneous change we would build the reaction vessel with thick steel walls. This is a practical possibility and so the Helmholtz energy is a practical thermodynamic potential. For equilibrium transformations (i.e. at constant A = zero), \[S=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{T}}\right)_{\mathrm{V}, \mathrm{A}=0}\]Similarly \[\mathrm{p}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}, \mathrm{A}=0}\] Since the product \(\mathrm{T} \, \mathrm{S}\) is the linked energy, equation (a) shows that \(\mathrm{F}\) is the ‘free energy’ of the system. An interesting literature discusses the analysis of experimental data in terms of an isochoric condition. A similar set of equations describes the differential change in \(\mathrm{F}\) at constant molecular composition (molecular organization). Thus \[\mathrm{S}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{T}}\right)_{\mathrm{V}, \xi} \text { and, } \mathrm{p}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}, \xi}\]This page titled 1.14.31: Helmholtz Energy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.32: Hildebrand Solubility Parameter

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.32%3A_Hildebrand_Solubility_Parameter

The cohesive energy density (c.e.d.) of a liquid is defined by equation (a). \[\text { c.e.d. }=\Delta_{\text {vap }} \mathrm{U}^{0} / \mathrm{V}^{*}(\ell)\]\(\Delta_{\text {vap }} \mathrm{U}^{0}\) is the change in thermodynamic energy when one mole of a given chemical substance passes from the liquid to the vapor state. The square root of the c.e.d. for liquid \(j\) is the Hildebrand solubility parameter for that liquid. \[\delta=(\text { c.e.d. })^{1 / 2}\]\(\delta\) can be expressed in many units but following the original definition the customary unit is \(\left(\mathrm{cal}^{1 / 2} \mathrm{~cm}^{-3 / 2}\right)\). Property \(\delta\) provides an estimate of cohesion within a given liquid. The idea goes a little further in terms of understanding solubilities. A clever idea is based on the following argument.Consider two liquids \(\mathrm{A}\) and \(\mathrm{B}\). We want to take a small sample of liquid \(\mathrm{A}\) (as a solute) and dissolve in liquid \(\mathrm{B}\) as the solvent. Within liquid \(\mathrm{A}\) the intermolecular interactions \(\mathrm{A} \ldots \(\mathrm{A}\) are responsible for the cohesion within this chemical substance. Similarly within liquid \(\mathrm{B}\), \(\mathrm{B} - \mathrm{~B}\) intermolecular forces are responsible for the cohesion within liquid \(\mathrm{B}\). If \(\mathrm{B} - \mathrm{~B}\) interactions are much stronger than \(\mathrm{A} - \mathrm{~A}\) and \(\mathrm{A} - \mathrm{~B}\) intermolecular interactions it is likely that \(\mathrm{A}\) will not be soluble in liquid \(\mathrm{B}\). Similarly if \(\mathrm{A} - \mathrm{~A}\) interactions are stronger than \(\mathrm{B} - \mathrm{~B}\) and \(\mathrm{A} - \mathrm{~B}\) interactions it is likely that \(\mathrm{A}\) will not be soluble in liquid \(\mathrm{B}\). If substance \(\mathrm{A}\) is to be soluble in liquid \(\mathrm{B}\), their cohesive energy densities should be roughly equal.This page titled 1.14.32: Hildebrand Solubility Parameter is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.33: Infinite Dilution

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.33%3A_Infinite_Dilution

The term ’infinite dilution‘ is often encountered in reviewing the properties of solutions. However some caution has to be exercised when this term is used. There is merit in distinguishing between the properties of aqueous solutions containing simple neutral solutes and those containing salts because the impact of solute - solute interactions plays an important role in the analysis. Further we need to distinguish the properties of solutes and solvents.The chemical potential of solute \(j\), \(\mu_{j}(\mathrm{aq})\) is related to the composition of the solution, molality \(\mathrm{m}_{j}\), at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) which is assumed to be close to the standard pressure. \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Or \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]For simple solutes in aqueous solutions, \(\ln \left(\gamma_{j}\right)\) is a linear function of the molality \(\mathrm{m}_{j}\). \[\ln \left(\gamma_{\mathrm{j}}\right)=\chi \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Here \(\chi\) is a function of temperature and pressure. \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \chi \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]We note therefore that \[\operatorname{limit}\left(\mathrm{m}_{j} \rightarrow 0\right) \mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=-\infty\]Hence with increasing dilution of the solution, solute \(j\) is increasingly stabilised, \(\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) decreasing to ‘\(- \infty\)’ in an infinite amount of solvent Using the Gibbs-Helmholtz Equation, equation (a) yields equation (f). \[-\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) / \mathrm{T}^{2}=-\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) / \mathrm{T}^{2}+\mathrm{R} \,\left(\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right)_{\mathrm{p}}\]\(\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the partial molar enthalpy of solute \(j\). \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right)_{\mathrm{p}}\]Using equation (c), \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \,(\partial \chi / \partial \mathrm{T})_{\mathrm{p}}\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]As the solution becomes more dilute and approaches infinite dilution so \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) in the limit of infinite dilution approaches the partial molar enthalpy of solute \(j\) in the reference solution \(\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\).where the partial molar enthalpy is identified as \(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}). Granted the latter conclusion based on equation (h), this equation offers information concerning the form of the plot of \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) against \(\mathrm{m}_{j}\). \[\left[\partial \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) / \partial \mathrm{m}_{\mathrm{j}}\right]=-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{m}^{0}\right)^{-1} \,(\partial \chi / \partial \mathrm{T})_{\mathrm{p}}\]In other words the gradient of the plot of \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) against \(\mathrm{m}_{j}\) is finite, the gradient being determined by the sign of \((\partial \chi / \partial T)_{p}\).The partial molar isobaric heat capacity of the solute \(j\) is given by the differential of equation (h) with respect to temperature. \[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pj}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\mathrm{T}^{2} \,(\partial \chi / \partial \mathrm{T})_{\mathrm{p}}\right)\right]\]\(\operatorname{Limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{C}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is a finite quantity, \(\mathrm{C}_{\mathrm{p} j}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\). In other words the limiting partial molar isobaric heat capacity of the solute \(\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) equals the standard partial molar isobaric heat capacity, \(\mathrm{C}_{\mathrm{pj}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\). \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{C}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pj}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]A similar conclusion is reached when we turn our attention to partial molar volumes recognizing that for solute \(j\), \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \,(\partial \chi / \partial \mathrm{p})_{\mathrm{T}}\]Therefore, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]The limiting value of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is a finite quantity such that the limiting (i.e. infinite dilution) value of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\), namely \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) equals the standard partial molar volume, \(\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\).The interesting question arises as to why the limiting values of partial molar enthalpies, volumes and isobaric heat capacities are real (and important) properties but limiting chemical potentials are not. We start again with equation (b) recalling that partial molar entropy \(\mathrm{S}_{\mathrm{j}}=-\left(\partial \mu_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\). \[\begin{aligned} &\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\ &\quad \mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-\mathrm{R} \, \ln \left(\gamma_{\mathrm{j}}\right)-\mathrm{R} \, \mathrm{T} \,\left(\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right)_{\mathrm{p}} \end{aligned}\]But \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)=\text { minus infinity }\]Then \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}_{\mathrm{j}}=\text { plus infinity }\]With increase in dilution \(\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) tends to the asymptotic limit, plus infinity. For a solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\) prepared using \(1 \mathrm{~kg}\) of water, the Gibbs energy is given by equation (r). \[\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]Or, \[\begin{aligned} \mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}\right.&=1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\right] \end{aligned}\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mu_{1}^{*}(\ell) / \mathrm{M}_{1}\]We consider a dilute 1:1 salt solution, confining the analysis to a consideration of the impact of the Debye - Huckel Limiting Law (DHLL). For a salt solution, molality \(\mathrm{m}_{j}\), \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)\]Or, using the DHLL \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\ &\quad \mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{S}_{\mathrm{\gamma}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \end{aligned}\]From equation (u), \[\left[\frac{\partial \mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})}{\partial \mathrm{m}_{\mathrm{j}}}\right]=\frac{2 \, \mathrm{R} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \ln \left(\gamma_{\pm}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]\]As for a non-ionic solute, \[\operatorname{limit}\left(m_{j} \rightarrow 0\right)\left[\frac{\partial \mu_{j}(a q ; T ; p)}{\partial m_{j}}\right]=\infty\]From the Gibbs - Helmholtz equation and equation (v), \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{S}_{\mathrm{H}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]where \[\mathrm{S}_{\mathrm{H}}=\left(\partial \mathrm{S}_{\gamma} / \partial \mathrm{T}\right)_{\mathrm{p}}\]Further, \[\begin{aligned} {\left[\partial \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) / \partial \mathrm{m}_{\mathrm{j}}\right] } &=\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{S}_{\mathrm{H}} \, /\left(\mathrm{m}_{\mathrm{j}} \, \mathrm{m}^{0}\right)^{1 / 2} \\ &+2 \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{m}_{\mathrm{j}}\right) \end{aligned}\]Thus the gradient of a plot of \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq})\) against \(\mathrm{m}_{j}\) has infinite slope in the \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\). A similar pattern emerges in the case of partial molar volumes of the salt.A similar analysis can be undertaken with respect to the partial molar properties of the solvent and apparent molar thermodynamic properties of salts and neutral solutes.Footnote M. Spiro, Educ. Chem.,1966,3,139. J. E. Garrod and T. H. Herrington, J. Chem. Educ.,1969,46,165.This page titled 1.14.33: Infinite Dilution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.34: Internal Pressure

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According to the Thermodynamic Equation of State, \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p}\]The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) (with units, \(\mathrm{N m}^{-2}\)) is the internal pressure \(\pi_{\mathrm{int}}\). \[\pi_{\mathrm{int}}=\mathrm{T} \, \beta_{\mathrm{V}}-\mathrm{p}\]\(\pi_{\mathrm{int}\) describes the sensitivity of energy \(\mathrm{U}\) to a change in volume. A high \(\phi_{\mathrm{int}\) implies strong inter-molecular cohesion. For many liquids, \(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}>>\mathrm{p}\) such that \[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}} \cong \mathrm{T} \, \beta_{\mathrm{V}}\]\(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}\) is sometimes called the thermal pressure. By definition, for \(\mathrm{n}\) moles of a perfect gas, \[\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T}\]Then \[\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R}\]Or, \[\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} / \mathrm{V}=\mathrm{p}\]From equation (a), for a perfect gas, \(\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\) is zero.The internal pressure for water(\(\ell\)) presents an interesting puzzle. From equations (a) and (c), it follows that \[\pi_{\mathrm{int}}=\mathrm{T} \,\left(\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\right)-\mathrm{p}\]But at the temperature of maximum density (TMD), \(\alpha_{p}\) is zero. So near the TMD, \(\pi_{\mathrm{int}\) is zero. We understand this pattern if we think about hydrogen bonding. In order to form a strong hydrogen bond between two neighboring water molecules the O-H---O link has to be close to if not actually linear. In other words the molar volume for water(\(\ell\)) is larger than the molar volume of a system comprising close-packed water molecules. Consequently hydrogen bonding has a strong ‘repulsive’ component to intermolecular interaction. However once formed hydrogen bonding has a strong cohesive contribution to intermolecular forces. Hence for water between \(273\) and \(298 \mathrm{~K}\) cohesive and repulsive components of hydrogen bonding play almost competitive roles. Using a calculus operation, \(\left(\frac{\partial p}{\partial T}\right)_{V}=-\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial p}{\partial V}\right)_{T}\) For equilibrium properties, \[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\] Some authors use the term ‘isochoric thermal pressure coefficient’ for the property, \(\left(\frac{\partial p}{\partial T}\right)_{V}\) For details of original proposals concerning internal pressures see the following references. Internal pressures are quoted in the literature using many units. Here we use \(\mathrm{N m}^{-2}\). We list some internal pressures and relative permitivities at \(298.15 \mathrm{~K}\).The above details are taken from M. R. J. Dack, J.Chem.Educ.,1974,51,231;see also For a discussion of effects of solvents on rates of chemical reactions with reference to internal pressures, see For comments on solvent polarity and internal pressures see J. E. Gordon, J. Phys. Chem.,1966,70,2413. For comments on internal pressures of binary aqueous mixtures see D. D. Macdonald, Can. J Chem.,1976,54,3559; and references therein. For comments on effect of internal pressure on conformational equilibria see R. J. Ouellette and S. H. Williams, J. Am. Chem.Soc.,1971,93,466. For details concerning the dependence of internal pressure of water and \(\mathrm{D}_{2}\mathrm{O}\), see M. J. Blandamer, J. Burgess and A.W.Hakin, J. Chem. Soc. Faraday Trans. 1, 1987, 83, 1783. For comments on the calculation of excess internal pressures for binary liquid mixtures using equation (h) see W. Marczak, Phys.Chem.Chem.Phys.,2002,4,1889.This page titled 1.14.34: Internal Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.35: Internal Pressure: Liquid Mixtures: Excess Property

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The thermodynamic equation of state takes the form shown in equation (a). \[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p}\]The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) is the internal pressure, \(\pi_{\mathrm{int}}\) (with units, \(\mathrm{N m}^{-2}\)). A calculus operation relates three interesting partial derivatives in the context of \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties; equation (b). \[\left(\frac{\partial p}{\partial T}\right)_{V}=-\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial p}{\partial V}\right)_{T}\]For a given liquid at defined \(\mathrm{T}\) and \(\mathrm{p}\), the isobaric (equilibrium) thermal expansion, \(\mathrm{E}_{\mathrm{p}}\) equals \((\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\). The isothermal (equilibrium) compression \(\mathrm{K}_{\mathrm{T}}\) is defined by \(-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}\). According to equations (a) and (b), \(\pi_{\mathrm{int}}\) is given by equation (c). \[\pi_{\mathrm{int}}=\left(\mathrm{T} \, \mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}\right)-\mathrm{p}\]For the purpose of the analysis described here, equation (c) describes the equilibrium molar properties of a given binary liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The internal pressure is a non-Gibbsian property of a liquid. Nevertheless it is interesting to compare internal pressures of real and the corresponding ideal binary liquid mixture. In other words we require an equation for the internal pressure of binary liquid mixture \(\pi_{\mathrm{int}}^{\mathrm{id}}\) having thermodynamic properties which are ideal. Marczak uses equation (c) in which the corresponding molar properties of the mixture, mole fraction composition \(\mathrm{x}_{2}\), \(\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) and \(\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) are given by the mole fraction weighted properties of the pure liquids. \[\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell)\]\[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)\]Equations (d) and (e) can be generalised to multi-component liquid mixtures. From equation (c) for a binary liquid mixture having thermodynamic properties which are ideal, the internal pressure \(\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\) is given by equation (f). \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)}{\mathrm{K}_{\mathrm{Tm}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)}-\mathrm{p}\]Or using equations (d) and (e), \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{~T} \, \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}-\mathrm{p}\]Equation (g) is re-written to establish \(\pi_{\mathrm{int,i}}^{*}(\ell)\) as a term on the r.h.s. of the latter equation for \(\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)\). \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=-\mathrm{p}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{~T} \, \mathrm{x}_{\mathrm{i}} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell) \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell) / \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}\]But according to equation (c) , for the pure liquid–\(\mathrm{i}\), \[\pi_{\mathrm{im}, \mathrm{i}}^{*}(\ell)+\mathrm{p}=\mathrm{T} \, \mathrm{E}_{\mathrm{pi}}^{*}(\ell) / \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)\]Hence from equation (h), \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=-\mathrm{p}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \,\left[\pi_{\mathrm{int,i}}^{*}(\ell)+\mathrm{p}\right] \, \mathrm{K}_{\mathrm{T}_{1}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}^{*}}(\ell)}\]In other words, \[\begin{aligned} \pi_{\mathrm{idt}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=&-\mathrm{p}+\frac{\mathrm{x}_{1} \, \pi_{\mathrm{int1} 1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}+\frac{\mathrm{x}_{1} \, \mathrm{p} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\sum_{\mathrm{i}=2}^{*} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \\ &+\frac{\mathrm{x}_{2} \, \pi_{\mathrm{int}, 2}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{p} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)} \end{aligned}\]Hence, \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\frac{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \pi_{\mathrm{int}, \mathrm{i}}^{*}(\ell) \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}{\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}\]By definition, for liquid component \(\mathrm{k}\), \[\Psi_{\mathrm{k}}=\frac{\mathrm{x}_{\mathrm{k}} \, \mathrm{K}_{\mathrm{Tk}}^{*}(\ell)}{\sum \mathrm{x}_{\mathrm{i}} \, \mathrm{K}_{\mathrm{Ti}}^{*}(\ell)}\]In other words, \[\pi_{\mathrm{int}}^{\mathrm{id}}\left(\mathrm{x}_{2}\right)=\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \psi_{\mathrm{i}} \, \pi_{\mathrm{int,i}}^{*}(\ell)\]The corresponding excess internal pressure at mole fraction \(\mathrm{x}_{2}\), \[\pi_{\mathrm{int}}^{\mathrm{E}}\left(\mathrm{x}_{2}\right)\] is defined by equation (o). \[\pi_{\mathrm{int}}^{\mathrm{E}}\left(\mathrm{x}_{2}\right)=\pi_{\mathrm{int}}\left(\mathrm{x}_{2}\right)-\sum_{\mathrm{i}=1}^{\mathrm{i}=2} \psi_{\mathrm{i}} \, \pi_{\mathrm{int}, \mathrm{i}}^{*}(\ell)\]Marczak reports \(\pi_{\text {int }}^{E}\left(X_{2}\right)\) as a function of mole fraction \(\mathrm{x}_{2}\) for two binary liquid mixtures at \(298.15 \mathrm{~K}\); W. Marczak, Phys. Chem. Chem. Phys.2002,4,1889.This page titled 1.14.35: Internal Pressure: Liquid Mixtures: Excess Property is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.36: Irreversible Thermodynamics

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According to the Second Law of Thermodynamics, the change in entropy \(\mathrm{dS}\) is related to the affinity for spontaneous change \(\mathrm{A}\) using equation (a). \[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]In these terms chemists usually have in mind a chemical reaction driven by the affinity \(\mathrm{A}\) for spontaneous chemical reaction producing extent of reaction \(\mathrm{d}\xi\). We generalize the law in the following terms. \[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+\mathrm{d}_{\mathrm{i}} \mathrm{S} ; \quad \mathrm{d}_{\mathrm{i}} \mathrm{S} \geq 0\]\(\mathrm{d}_{\mathrm{i}\mathrm{S}}\) is the change in entropy of the system by virtue of spontaneous processes in the system. Comparison of equations (a) and (b) yields the following equation. \[\mathrm{T} \, \mathrm{d}_{\mathrm{i}} \mathrm{S}=\mathrm{A} \, \mathrm{d} \xi \geq 0\]We introduce two new terms. A quantity \(\mathrm{P}[\mathrm{S}]\) describes the rate of entropy production within the system; a quantity \(\sigma[\mathrm{S}]\) describes the corresponding rate of entropy production in unit volume of the system. \[\mathrm{P}[\mathrm{S}]=\mathrm{d}_{\mathrm{i}} \mathrm{S} / \mathrm{dt}=\int_{\mathrm{V}} \sigma[\mathrm{S}] \, \mathrm{dV} \geq 0\]We combine equations (c) and (d). \[\mathrm{P}[\mathrm{S}]=\frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}} \geq 0\]But if \(\mathrm{dn}_{j}\) is the change in amount of chemical substance \(j\) in the system, \(\mathrm{dn} \mathrm{j}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} \, \mathrm{d} \xi\). Then, \[P[S]=\frac{d_{i} S}{d t}=\frac{A}{T} \, \frac{1}{v_{j}} \, \frac{d n_{j}}{d t} \geq 0\]We develop this equation into a form which has wider significance. We assume that the system is homogeneous such that for a system volume \(\mathrm{V}\), \[\sigma[S]=P[S] / V\]Then \[\sigma[\mathrm{S}]=\frac{1}{\mathrm{~V}} \, \frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{1}{\mathrm{~V}} \, \frac{\mathrm{dn}_{\mathrm{j}}}{\mathrm{dt}} \geq 0\]But the concentration of chemical substance \(j\), \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\). Then, \(\mathrm{dc}_{\mathrm{j}}=\mathrm{dn}_{\mathrm{j}} / \mathrm{V}\). Hence from equation (h), \[\sigma[S]=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{\mathrm{dc}_{\mathrm{j}}}{\mathrm{dt}} \geq 0\]The quantity \(\left(1 / v_{j}\right) \, d c_{j} / d t\) describes the change in composition of the system, the flow of the system from reactants to products. In these terms we identify a chemical flow, \(\mathrm{J}_{\mathrm{ch}}\). \[\mathrm{J}_{\mathrm{ch}}=\left(\mathrm{l} / \mathrm{v}_{\mathrm{j}}\right) \, \mathrm{dc}_{\mathrm{j}} / \mathrm{dt}\]Then, \[\sigma[S]=(A / T) \, J_{c h} \geq 0\]Or, \[\mathrm{T} \, \sigma[\mathrm{S}]=\mathrm{A} \, \mathrm{J}_{\mathrm{ch}} \geq 0\]The latter equation has an interesting feature; \(\mathrm{T} \, \sigma[\mathrm{S}]\) is given by the product of the affinity for spontaneous chemical reaction (the driving force) and the accompanying flow. Indeed \(\mathrm{T} \, \sigma[\mathrm{S}]\) is related to the rate of entropy production in the system. Thermodynamics takes us no further. We make an extrathermodynamic leap and suggest that the flow is proportional to the force; i.e. the stronger the driving force the more rapid the chemical flow from reactants to products.In general terms phenomenological equations start out from the basis of a linear model described by a phenomenological law of the general form, \(\mathrm{J} = \mathrm{L} \, \(\mathrm{~X}\) where \(\mathrm{J}\) is the flow and \(\mathrm{X}\) is the conjugate force such that the product \(\mathrm{J} \, \mathrm{~X}\) yields the rate of entropy production. These laws are based on experiment. Many such phenomenological laws have been proposed. Some examples are listed below.In 1890 Nernst suggested this approach to chemical kinetics. Unfortunately chemists have no method for measuring the affinity; there is no affinity meter. Instead chemists use the Law of Mass Action.This page titled 1.14.36: Irreversible Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.37: Irreversible Thermodynamics: Onsager Phenomenological Equations

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The major thrust of the account presented in these Topics concerns reversible processes in which system and surroundings are in thermodynamic equilibrium. When attention turns to non-equilibrium processes, the thermodynamic treatment is necessarily more complicated. Here we examine several aspects of irreversible thermodynamics for near equilibrium in open systems. In other words there is a strong ‘communication’ between system and surroundings. With increasing displacement of a given system from equilibrium the thermodynamic analysis becomes more complicated and controversial. Here we confine attention to processes in near-equilibrium states. The key assumption is that equations describing relationships between thermodynamic properties are valid for small elemental volumes, the concept of local equilibrium. Hence we can in a description of a given system identify the energy per unit volume and the entropy per unit volume in the context respectively of the first and second laws of thermodynamics.With respect to a small volume \(\mathrm{dv}\) of a given system the change in entropy \(\mathrm{ds}\) is given by the change in entropy \(\mathrm{d}_{\mathrm{i}} \mathrm{s}\) by virtue of processes within a small volume \(\mathrm{dv}\) and by virtue of exchange with the rest of the system, des. The rate of change of \(\mathrm{d}_{\mathrm{i}} \mathrm{s}\), namely \(\mathrm{d}_{\mathrm{i}} \mathrm{s} / \mathrm{dt}\) is the local entropy production and is determined by the following condition. \[\sigma \equiv \mathrm{d}_{\mathrm{i}} \mathrm{s} / \mathrm{dt} \geq 0\]For systems close to thermodynamic equilibrium, the entropy production per unit volume \(\sigma\) can be expressed as the sum of products of forces \(\mathrm{X}_{\mathrm{k}}\) and conjugate flows, \(\mathrm{J}_{\mathrm{k}}\). Thus for \(\mathrm{k}\) flows and forces, \[\sigma=\sum_{\mathrm{k}} \mathrm{X}_{\mathrm{k}} \, \mathrm{J}_{\mathrm{k}}\]The condition ‘conjugate’ is important in the sense that for each flow \(\mathrm{J}_{k}\) there is a conjugate force \(\mathrm{X}_{k}\). For near equilibrium systems a given flow is a linear function of the conjugate force, \(\mathrm{X}_{k}\). Then, \[\mathrm{J}_{\mathrm{k}}=\sum_{\mathrm{j}} \mathrm{L}_{\mathrm{kj}} \, \mathrm{X}_{\mathrm{k}}\]The property \(\mathrm{L}_{\mathrm{kj}}\) is a phenomenological coefficient describing the dynamic flow and conjugate force.In simple systems there is only one flow and one force such that the flow is directly proportional to the force. A classic example is Ohm’s law which can be written in the following form. \[\mathrm{I}=(1 / \mathrm{R}) \, \mathrm{V}\]Thus \(\mathrm{I}\) is the electric current, the rate of flow of electric charge for a system where the driving force is the electric potential gradient \(\mathrm{V}\). The relevant property of the system under consideration is the resistance \(\mathrm{R}\) or, preferably, its conductance \(\mathrm{L} (= 1/\(\mathrm{R}\)).A similar phenomenological law is Fick’s Law of diffusion relating the rate of diffusion of chemical substance \(j\), \(\mathrm{J}_{j}\) to the concentration gradient \(\mathrm{dc}_{j}/\mathrm{dx}\) where \(\mathrm{D}_{j}\) describes the property of diffusion. Thus \[\mathrm{J}_{\mathrm{j}}=\mathrm{D}_{\mathrm{j}} \,\left(\mathrm{dc}_{\mathrm{j}} / \mathrm{dx}\right)\]The Law of Mass Action is a similar phenomenological law. In other words throughout chemistry (and indeed all sciences) there are phenomenological laws which do not, for example, follow from the first and second laws of thermodynamics.Following on a proposal by Lord Rayleigh relating to mechanical properties, in 1931 Onsager extended the ideas discussed above to include all forces and flows. For a system involving two flows and forces we may write the following two equations to describe near –equilibrium systems. \[\mathrm{J}_{1}=\mathrm{L}_{11} \, \mathrm{X}_{1}+\mathrm{L}_{12} \, \mathrm{X}_{2}\]\[\mathrm{J}_{2}=\mathrm{L}_{21} \, \mathrm{X}_{1}+\mathrm{L}_{22} \, \mathrm{X}_{2}\]This formulation recognises that force \(\mathrm{X}_{2}\) may also produce a coupled flow \(\mathrm{J}_{1}\). In each case the products \(\mathrm{L}_{11} \, \mathrm{X}_{1}, \mathrm{~L}_{12} \, \mathrm{X}_{2}, \mathrm{~L}_{21} \, \mathrm{X}_{1}\) and \(\mathrm{L}_{22} \, \mathrm{X}_{2}\) involve conjugate flows and forces such that the product, \(\mathrm{J}_{\mathrm{i}} \, \mathrm{X}_{\mathrm{i}}\) has the dimension of the rate of entropy production. The cross terms \(\mathrm{L}_{12}\) and \(\mathrm{L}_{21}\) are the coupling coefficients such that for example, force X2 produces flow J1.The key theoretical advance made by Onsager was to show that for near-equilibrium states the matrix of coefficients is symmetric. Then, for example, \[\mathrm{L}_{12}=\mathrm{L}_{21}\]The point can be developed by considering a system involving two flows and two forces. According to equation (b) \[\sigma=\mathrm{J}_{1} \, \mathrm{X}_{1}+\mathrm{J}_{2} \, \mathrm{X}_{2}\]Hence from equations (f) and (g) \[\sigma=\mathrm{L}_{11} \, \mathrm{X}_{1}^{2}+\left(\mathrm{L}_{12}+\mathrm{L}_{21}\right) \, \mathrm{X}_{1} \, \mathrm{X}_{2}+\mathrm{L}_{22} \, \mathrm{X}_{2}^{2}>0\]It also follows that \[\mathrm{L}_{11} \, \mathrm{X}_{1}^{2} \geq 0 \quad ; \quad \mathrm{L}_{22} \, \mathrm{X}_{2}^{2} \geq 0\]And, \[\mathrm{L}_{11} \, \mathrm{L}_{22} \geq \mathrm{L}_{12}^{2}\]These phenomena illustrate the application of the equations discussed above. A membrane separates two salt solutions; an electric potential E and a pressure gradient are applied across the membrane. There are two flows;The dynamics of the system are described by the dissipation function \(\phi\) given by equation (m), the sum of products of flows and forces. \[\phi=\mathrm{J}_{\mathrm{V}} \, \Delta \mathrm{p}+\mathrm{I} \, \mathrm{E}\]The dynamics of the system are described by two dynamic equations, \[\mathrm{J}_{\mathrm{V}}=\mathrm{L}_{11} \, \Delta \mathrm{p}+\mathrm{L}_{12} \, \mathrm{E}\]\[\mathrm{I}=\mathrm{L}_{21} \, \Delta \mathrm{p}+\mathrm{L}_{22} \, \mathrm{E}\]Onsager’s law requires that, \[\mathrm{L}_{12}=\mathrm{L}_{21}\]In an experiment we set \(\mathrm{E}\) at zero. Then \[\mathrm{L}_{11}=\left(\frac{\mathrm{J}_{\mathrm{v}}}{\Delta \mathrm{p}}\right)_{\mathrm{E}=0}\]However the electric current is not zero. According to equation (o), \[\mathrm{I}=\mathrm{L}_{21} \, \Delta \mathrm{p}\]In other words, there is a coupled flow of ions. Katchalsky and Curran discuss numerous experiments which illustrate this type of coupling of flows and forces. A. Katchalsky and P. F. Curran, Non-Equilibrium Thermodynamics in Biophysics, Harvard University Press, 1965. P. Glandsorff and I. Prigogine, Thermodynamics of Structure Stability and Fluctuations, Wiley-Interscience, London,1971. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, New York, 1977. P Gray and S. K. Scott, Chemical Oscillations and Instabilities, Oxford,1990. B. Lavenda, Thermodynamics of Irreversible Processes, MacMillan Press, London, 1978. D. Kondepudi and I. Prigogine, Modern Thermodynamics, Wiley, New York, 1998. L. Onsager, Phys. Rev.,1931,38,2265. D. G. Miller, Chem. Rev.,1960,60,15.This page titled 1.14.37: Irreversible Thermodynamics: Onsager Phenomenological Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.38: Joule-Thomson Coefficient

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An important property of a given gas is its Joule-Thomson coefficient. These coefficients are important from two standpoints;A given closed system contains one mole of gaseous chemical substance \(\mathrm{j}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The molar enthalpy of the gas \(\mathrm{H}_{\mathrm{j}}\) describes its molar enthalpy defined by equation (a). \[\mathrm{H}_{\mathrm{j}}=\mathrm{H}_{\mathrm{j}}[\mathrm{T}, \mathrm{p}]\]Then, \[\mathrm{dH}_{\mathrm{j}}=\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp}\]Hence at constant enthalpy, \(\mathrm{H}\), \[\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}=-\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp}\]Or, \[\left(\frac{\partial T}{\partial p}\right)_{H}=-\left(\frac{\partial H_{\mathrm{j}}}{\partial p}\right)_{T} \,\left(\frac{\partial T}{\partial H_{j}}\right)_{p}\]The Joule-Thomson coefficient for gas \(\mathrm{j}\), \(\mu_[\mathrm{j}}\) is defined by equation (e). \[\mu_{\mathrm{j}}=\left(\frac{\partial T}{\partial p}\right)_{\mathrm{H}(\mathrm{j})}\]For all gases (except helium and hydrogen) at \(298 \mathrm{~K}\) and moderate pressures \(\mu_{\mathrm{j} > 0\). At room temperature and ambient pressure, \(\mu_{\mathrm{j}}\) is \(0.002 \mathrm{~K Pa}^{-1}\) for nitrogen and \(0.025 \mathrm{~K Pa}^{-1}\) for 2,2-dimethylpropane.Further the isobaric heat capacity for chemical substance \(\mathrm{j}\) is defined by equation (f). \[\mathrm{C}_{\mathrm{pj}}=\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Hence from equations (d), (e) and (f), \[\mu_{\mathrm{j}}=-\frac{(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{T}}}{\mathrm{C}_{\mathrm{pj}}}\]Then, \[\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mu_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}\]Equation (h) marks an important stage in the analysis. For example, \(\mathrm{C}_{\mathrm{pj}} > 0\). From the definition of enthalpy \(\mathrm{H}_{\mathrm{j}}\), \[\mathrm{U}_{\mathrm{j}}=\mathrm{H}_{\mathrm{j}}-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}\]Equation (i) is differentiated with respect to \(\mathrm{V}_{\mathrm{j}} at fixed \(\mathrm{T}\). Thus, \[\[\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}-\mathrm{V}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}-\mathrm{p}\]Or, \[\left(\frac{\partial U_{j}}{\partial V_{j}}\right)_{T}=\left(\frac{\partial p}{\partial V_{j}}\right)_{T} \,\left[\left(\frac{\partial H_{j}}{\partial V_{j}}\right)_{T} \,\left(\frac{\partial V_{j}}{\partial p}\right)_{T}-V_{j}\right]-p\]Then, \[\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{T} \,\left[\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}-\mathrm{V}_{\mathrm{j}}\right]-\mathrm{p}\]Using equation (h), \[\left(\frac{\partial U_{j}}{\partial V_{j}}\right)_{T}=-\left(\frac{\partial p}{\partial V_{j}}\right)_{T} \,\left[\mu_{j} \, C_{p j}+V_{j}\right]-p\]An important application of equation (m) concerns the case where chemical substance \(\mathrm{j}\) is a perfect gas. In this case, \[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}\]Or, \[\mathrm{p}=\mathrm{R} \, \mathrm{T} \, \frac{1}{\mathrm{~V}_{\mathrm{j}}}\]Hence, \[\left(\frac{\partial p}{\partial V_{j}}\right)_{T}=-R \, T \, \frac{1}{V_{j}^{2}}=-\frac{p}{V_{j}}\]Then from equation (m), \[\left(\frac{\partial U_{j}}{\partial V_{j}}\right)_{T}=\frac{p}{V_{j}} \,\left[\mu_{j} \, C_{p j}+V_{j}\right]-p\]Or, \[\left(\frac{\partial U_{j}}{\partial V_{j}}\right)_{T}=\frac{p \, \mu_{j} \, C_{p j}}{V_{j}}\]But \[\operatorname{limit}(\mathrm{p} \rightarrow 0) \frac{\mathrm{p} \, \mu_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}}{\mathrm{V}_{\mathrm{j}}}=0\]Then \[\operatorname{limit}(\mathrm{p} \rightarrow 0)\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}=0\]A definition of a perfect gas is that \(\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}}\) is zero. Then all real gases are perfect in the \(\operatorname{limit}(\mathrm{p} \rightarrow 0)\). James Prescott Joule William Thomson; Later Lord Kelvin Some authors refer to the Joule-Thomson coefficient; e.g. E. B. Smith, Basic Chemical Thermodynamics, Clarendon Press, Oxford, 1982, 3rd. edn., page 119. Other authors refer to the Joule –Kelvin Effect; e.g. E. F. Caldin, Chemical Thermodynamics, Clarendon Press, Oxford, 1958,page 81. Other authors refer to either the Joule-Thomson or Joule-Kelvin Effect; e.g. M. H. Everdell, Introduction to Chemical Thermodynamics, English Universities Press, London 1965, page 57. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London 1979, page 94. Benjamin Thompson; later Count von Rumford, married Lavoisier’s widow.This page titled 1.14.38: Joule-Thomson Coefficient is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.39: Kinetic Salt Effects

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The chemical potential of a given solute \(\mathrm{j}\) in an aqueous solution is related to the concentration \(\mathrm{c}_{\mathrm{j}}\) using equation (a) where \(\mathrm{c}_{\mathrm{r}}\) is a reference concentration, \(1 \mathrm{~mol dm}^{-3}\), and yj is the solute activity coefficient. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{aq} ; \mathrm{id}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)\]By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right) \mathrm{y}_{\mathrm{j}}=1.0\]In the application of equation (a) to the rates of chemical reactions in solution, transition state theory is used. In the case of a second order bimolecular reaction involving solutes \(\mathrm{X}(\mathrm{aq})\) and \(\mathrm{Y}(\mathrm{aq})\), the reaction proceeds as described by equation (c). \[\mathrm{X}(\mathrm{aq})+\mathrm{Y}(\mathrm{aq}) \Leftarrow \Rightarrow \mathrm{TS}^{\neq} \rightarrow \text { products }\]An equilibrium between reactants and transition state, \(\mathrm{TS}^{\neq}\) is described by an equilibrium constant \(\mathrm{K}^{\neq}\). Hence, \[\Delta^{\neq} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{\neq}\right)=\mu_{\neq}^{0}(\mathrm{aq})-\mu_{\mathrm{X}}^{0}(\mathrm{aq})-\mu_{\mathrm{Y}}^{0}(\mathrm{aq})\]At equilibrium, \[\mu^{\mathrm{eq}}(\mathrm{X} ; \mathrm{aq})+\mu^{\mathrm{eq}}(\mathrm{Y} ; \mathrm{aq})=\mu^{\mathrm{eq}}(\mathrm{TS} ; \mathrm{aq})\]Using equation (a), \[\mathrm{K}^{\neq}=\frac{\mathrm{c}^{\neq}(\mathrm{aq}) \, \mathrm{y}^{\neq}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{c}_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq})}\]According to \(\mathrm{TS}\) theory rate constant \(\mathrm{k}\) is related to \(\mathrm{K}^{\neq}\) using equation (g) where \(\kappa\) is a transmission coefficient, customarily set to unity. Then, \[\mathrm{k}=\mathrm{K} \,(\mathrm{k} \, \mathrm{T} / \mathrm{h}) \, \mathrm{K}^{\neq} \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq}) / \mathrm{y}_{\neq}(\mathrm{aq})\]In the event that the thermodynamic properties of the aqueous solution are ideal, equation (g) simplifies to equation (h). \[\mathrm{k}(\mathrm{id})=\kappa \,(\mathrm{k} \, \mathrm{T} / \mathrm{h}) \, \mathrm{K}^{\neq}\]For a real system, \[\mathrm{k}=\mathrm{k}(\mathrm{id}) \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq}) / \mathrm{y}_{\neq}(\mathrm{aq})\]The Bronsted-Bjerrum analysis concerns rates of chemical reaction between ions having electric charges, \(\mathrm{z}_{\mathrm{x}} \, \mathrm{e}\) and \(\mathrm{z}_{\mathrm{y}} \, \mathrm{e}\) where the transition state has charge z ⋅ e ≠ ( z e z e) X Y = ⋅ + ⋅ .In most applications, the activity coefficients are related to the ionic strength of the solution using the Debye - Huckel Limiting Law. For reactant \(\mathrm{j}\), \[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=-\mathrm{S}_{\mathrm{Y}} \, \mathrm{z}_{\mathrm{j}}^{2} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\]Then, \[\ln (\mathrm{k})=\ln (\mathrm{k}(\mathrm{id}))+\ln \left(\mathrm{y}_{\mathrm{X}}\right)+\ln \left(\mathrm{y}_{\mathrm{Y}}\right)-\ln \left(\mathrm{y}_{z}\right)\]\[\ln (\mathrm{k})=\ln (\mathrm{k}(\mathrm{id}))-\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[\mathrm{z}_{\mathrm{X}}^{2}+\mathrm{z}_{\mathrm{Y}}^{2}-\left(\mathrm{z}_{\mathrm{X}}+\mathrm{z}_{\mathrm{Y}}\right)^{2}\right]\]Or, \[\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))=\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[2 \, \mathrm{z}_{\mathrm{X}} \, \mathrm{Z}_{\mathrm{Y}}\right]\]Equation (m) forms the basis of the classic and oft-quoted plot of \([\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))]\) against \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) in which the slope is determined by the product of charge numbers, \(\mathrm{z}_{\mathrm{x}} \, \mathrm{z}_{\mathrm{y}}\); [1;see Footnote, page 429].An interesting feature was noted by Rosseinsky. Equation (m) can be written in a quite general form for a reaction involving \(\mathrm{n}\) ions. Then, \[\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))=\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \, \sum_{\mathrm{i}}^{\mathrm{n}} \sum_{\mathrm{j}}^{\mathrm{n}} \mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \quad(\mathrm{i} \neq \mathrm{j})\]For chemical reaction involving cations and anions , cases can arise where the double sum in equation(n) is zero. Hence the rate constant will be independent of ionic strength. Rosseinsky cites the following reaction as a case in point. \[2 \mathrm{Mn}^{2+}(\mathrm{aq})+\mathrm{MnO}_{4}^{-} \text {(aq) } \rightarrow \mathrm{Mn}_{3} \mathrm{O}_{4}^{3+}\] S. A. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York, 1941, pp. 427-429. D. R. Rosseinsky, J. Chem. Phys.,1968,48, 4806. D. R. Rosseinsky and M. J. Nicol, Trans. Faraday Soc.,1965,61, 2718.This page titled 1.14.39: Kinetic Salt Effects is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.4: Extensive and Intensive Variables

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The terms, variables and properties are synonymous. Nevertheless a given thermodynamic property of a system can be classified as either intensive or extensive.Intensive Properties. The magnitude of an intensive variable does NOT depend on the amount of chemical substance in a given closed system; e.g. density.Extensive Properties. The magnitude of an extensive variable depends on the amount of chemical substances in a closed system; e.g. volume. Let us ask – is temperature an intensive or extensive variable? Consider two conical flasks. Flask A contains \(10 \mathrm{~cm}^{3}\) of water(\(\ell\)) at \(298 \mathrm{~K}\). Flask B contains \(5 \mathrm{~cm}^{3}\) of water(\(\ell\)) at \(298 \mathrm{~K}\). The contents of Flask are poured into Flask B.Footnote O. Redlich ( J. Chem.Educ.,1970,42,154) presents a provocative discussion of the distinction between intensive and extensive variables.This page titled 1.14.4: Extensive and Intensive Variables is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.40: Laws of Thermodynamics

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A remarkable feature of the subject called thermodynamics is the extent to which it is founded on four laws: Zeroeth, First, Second and Third. These laws summarize elegantly the results of experiments. Actually these are not laws in the sense of being laid down by government or by religious doctrine. Rather the laws are axioms. As McGlashan notes each axiom is a ‘rule of the game’. These axioms refer to state variables such as temperature, pressure, energy and entropy. At this level the laws are not of immediate interest to chemists. However chemists have discovered how to ‘tell’ these axioms about chemical substances and chemical reactions.The First Law invokes the concepts of energy and energy change. The law states that the energy of the universe is constant. In a realistic sense, at least for chemists, the law states that the energy of a chemical laboratory is constant. Then if the energy of system held in a reaction vessel increases, an equivalent amount of energy is lost from the rest of the laboratory. Then \[\Delta \mathrm{U}(\text { system })+\Delta \mathrm{U}(\text { surroundings })=0\]The Second Law of thermodynamics invokes the concepts of entropy and entropy change. In summary the law states that heat cannot flow spontaneously from low to high temperatures. The elegant studies carried out by James Prescott Joule (1818 -1889) were crucial to the development of thermodynamics. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London 1979. L. Woodcock and L. Lue, Chem. Britain, 2001, August, p. 38.This page titled 1.14.40: Laws of Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.41: Lewisian Variables

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A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. If the thermodynamic properties of the liquid mixture are ideal the volume of the mixture is given by the sum of products of amounts and molar volumes (at the same \(\mathrm{T}\) and \(\mathrm{p}\)); equation (a). \[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]If the thermodynamic properties of the mixture are not ideal, the volume of the (real) mixture is given by equation (b). \[V(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]\(\mathrm{V}_{1}(\operatorname{mix})\) and \(\mathrm{V}_{2}(\operatorname{mix})\) are the partial molar volumes of chemical substances 1 and 2 defined by equations (c) and (d). \[\mathrm{V}_{1}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{l}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]\[\mathrm{V}_{2}(\operatorname{mix})=\left(\frac{\partial V}{\partial n_{2}}\right)_{T_{, p, n}}\]The similarities between equations (a) and (b) are obvious and indicate an important method for describing the extensive properties of a given system. This was the aim of G. N. Lewis who sought equations of the form show in equation (b). In general terms we identify an extensive property \(\mathrm{X}\) of a given system such that the variable can be written in the general form shown in equation (e). \[\mathrm{X}=\mathrm{n}_{1} \, \mathrm{X}_{1}+\mathrm{n}_{2} \, \mathrm{X}_{2}\]where \[\mathrm{X}_{1}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]\[\mathrm{X}_{2}(\operatorname{mix})=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]Other than the composition variables, the conditions on the partial differentials in equations (f) and (g) are intensive properties;Lewisian partial molar variables can be used to describe the thermodynamic energy \(\mathrm{U}\), entropy \(\mathrm{S}\) and volume \(\mathrm{V}\) together with their Legendre transforms, Helmholtz energy, enthalpy and Gibbs energy. With respect to other thermodynamic properties of a closed system, the case for identifying similar Lewisian partial molar properties has to be established. It turns out that partial molar expansions [e.g. \(\mathrm{E}_{\mathrm{p} j}(\mathrm{T}, \mathrm{p})\)] and partial molar compressions [e.g. \(\mathrm{K}_{\mathrm{T} j}(\mathrm{T}, \mathrm{p})\)] for chemical substance \(j\) in a closed single phase system are Lewisian but partial molar isentropic compressions are not .This page titled 1.14.41: Lewisian Variables is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.42: L'Hospital's Rule

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In several important cases, analysis of thermodynamic properties of solutions (and liquid mixtures) requires consideration of a term having the general form \(x \, \ln (x)\) where \(x\) is an intensive composition variable; e.g. molality, concentration or mole fraction. The accompanying analysis requires an answer to the question --- what value does the product \(x \, \ln (x)\) take in the limit that \(x\) tends to zero. But \(\operatorname{limit}(x \rightarrow 0) \ln (x)=-\infty\). The thermodynamic analysis has to take account of the answer to this question. In fact most accounts assume that the answer to the above question is ‘zero’. Confirmation that the latter statement is correct emerges from application of L’Hospital’s Rule (G. F. A. de l’Hospital, 1661-1704, marquis de Saint-Mesme). This rule allows the evaluation of terms having indeterminate forms. Most applications of this method usually involve the ratio of two terms each being a function of \(x\).If \(\mathrm{f}(\mathrm{x}) / \mathrm{F}(\mathrm{x})\) approaches either [0/0] or \([\infty / \infty]\) when \(x\) approaches a, and \(\mathrm{f}^{\prime}(\mathrm{x}) / \mathrm{F}^{\prime}(\mathrm{x})\) [where \(\mathrm{f}^{\prime}(\mathrm{x})\) and \(\mathrm{F}^{\prime}(\mathrm{x})\) are first derivatives of \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{F}(\mathrm{x})\)] approaches a limit as \(x\) approaches a, then \(\mathrm{f}(\mathrm{x}) / \mathrm{F}(\mathrm{x})\) approaches the same limit.If \(f(x)=x^{2}-1\) and \(F(x)=x-1\)then \(\frac{f(x)}{F(x)}=\frac{x^{2}-1}{x-1}\) and \(\frac{f^{\prime}(x)}{F^{\prime}(x)}=\frac{2 \, x^{2}}{1}\)then \[\operatorname{limit}(x \rightarrow 1) \frac{f^{\prime}(x)}{F^{\prime}(x)}=2\]Hence, \[\operatorname{limit}(x \rightarrow 1) \frac{f(x)}{F(x)}=2\]This rule can be proved using three assumptions.In the present context the terms under consideration have a different form. With reference to the term, \(x \, \ln (x)\), \[f(x)=\ln (x) \text { and } F(x)=1 / x\]Then \(\mathrm{f}^{\prime}(\mathrm{x})=1 / \mathrm{x}\) and \(\mathrm{F}^{\prime}(\mathrm{x})=-1 / \mathrm{x}^{2}\).Hence, \(\mathrm{f}^{\prime}(\mathrm{x}) / \mathrm{F}^{\prime}(\mathrm{x}) = -\mathrm{x}\).Thus \[\operatorname{limit}(x \rightarrow 0) \mathrm{f}^{\prime}(x) / \mathrm{F}^{\prime}(\mathrm{x})=0\]Hence, \[\operatorname{limit}(x \rightarrow 0) x \, \ln (x)=0\]This page titled 1.14.42: L'Hospital's Rule is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.43: Master Equation

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Two important laws of thermodynamics describe spontaneous change in a closed system.First Law \[\mathrm{dU}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}\]Second Law \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0\]Heat \(\mathrm{q}\) is common to these equations which we combine. The result is a very important equation. \[\begin{aligned} &\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \\ &\mathrm{A} \, \mathrm{d} \xi \geq 0 \end{aligned}\]We use the description ‘Master Equation’. A case can be made for the statement that chemical thermodynamics is based on this Master Equation.The Master Equation describes the differential change in the thermodynamic energy of a closed system. \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi\]where \[\mathrm{T}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{S}}\right)_{\mathrm{V}, \xi}\]and \[\mathrm{p}=-\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{s}_{, \xi}}\]Symbol \(\xi\) represents the chemical composition of the system (and quite generally molecular organization).The thermodynamic energy of a closed system containing \(\mathrm{k}\) chemical substances is defined by the independent variables \(\mathrm{S}\), \(\mathrm{V}\) and amounts of each chemical substance. \[\mathrm{U}=\mathrm{U}\left[\mathrm{S}, \mathrm{V}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \mathrm{n}_{\mathrm{k}}\right]\]We assert that we can independently add \(\delta \mathrm{n}_{j}\) moles of any one of the k chemical substances in the system and that the entropy \(\mathrm{S}\) and \(\mathrm{V}\) can change independently. Based on equation (f), the following (often called the Gibbs equation) is a key relationship. \[\mathrm{dU}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{S}}\right)_{\mathrm{V}, \mathrm{n}(\mathrm{i})} \, \mathrm{dS}+\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{S}, \mathrm{n}(\mathrm{i})} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn}_{\mathrm{j}}\]Here \(\mathrm{n}(\mathrm{i})\) represents the amounts of each of the \(\mathrm{k}\) chemical substances in the system. Hence from equation (e), \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]The importance of these equations is indicated by imagining a closed system held at constant volume (\(\mathrm{dV}=0\)) and entropy (\(\mathrm{dS}=0\)). Under these constraints \(\mathrm{dU}(\mathrm{S} \text { and } \mathrm{V} =\) constant) equals - \(\mathrm{A} \, \mathrm{d} \xi\). But according to equation(c), the product \(\mathrm{A} \, \mathrm{d} \xi\) is always positive for spontaneous reactions. Hence \(\mathrm{dU}(\mathrm{S} \text { and } \mathrm{V} =\) constant) is negative. In other words, all spontaneous chemical reactions in a closed system at constant \(\mathrm{S}\) and constant \(\mathrm{V}\) proceed in a direction which lowers the thermodynamic energy \(\mathrm{U}\) of the system. This conclusion is universal, independent of the type of chemical reaction and of the mechanism of chemical reaction. For this reason the thermodynamic energy is the thermodynamic potential function for processes in closed systems at constant \(\mathrm{S}\) and constant \(\mathrm{V}\).There is however a problem in terms of practical chemistry. We can envisage designing a reaction vessel which has constant volume. In fact we would probably use heavy steel walls because the conclusions reached above tell us nothing about a possible change in pressure as we face the challenge of holding the volume constant. But it is not obvious what we have to do to hold the entropy constant. Clearly the line of argument is important. Indeed a similar analysis based on the definitions of enthalpy \(\mathrm{H}\), Helmholtz energy \(\mathrm{F}\) and Gibbs energy \(\mathrm{G}\) leads to the following three key equations for changes in enthalpy, Helmholtz energy and Gibbs energy respectively. \[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{S}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, d n_{j}\]\[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial F}{\partial n_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn}_{\mathrm{j}}\]\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{G}}{\partial n_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]The four partial derivatives with respect to \(\mathrm{n}_{j}\) in the four equations define the chemical potential, \(\mu_{j}\). \[\begin{aligned} \mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{v}, \mathrm{n}(i \neq \mathrm{j})} &=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{S}, \mathrm{p}, \mathrm{n}(i \neq \mathrm{j})} \\ &=\left(\frac{\partial \mathrm{F}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \end{aligned}\]For example, \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn}_{\mathrm{j}}\]In context of chemistry, the latter equation is very important.Footnote An analogy is drawn with electric potential. In an electrical circuit, electric charge flows spontaneously from high to low electric potential.This page titled 1.14.43: Master Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.44: Maxwell Equations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.44%3A_Maxwell_Equations

Important relationships in thermodynamics are based on Maxwell Equations. Consider the state variable G for a given closed system characterized by the two independent variables, \(\mathrm{T}\) and \(\mathrm{p}\). Hence, \[\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T}\]or, \[\left(\frac{\partial[\partial \mathrm{G} / \partial \mathrm{T}]_{\mathrm{p}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]But at both fixed composition \(\xi\) and at equilibrium, \(\mathrm{A} = 0\), \(\mathrm{V}=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\) and \(\mathrm{S}=-[\partial \mathrm{G} / \partial \mathrm{T}]_{\mathrm{p}}\)Then \[\mathrm{E}_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\]For the most part we use this relationship in the context of an equilibrium displacement; i.e. at \(\mathrm{A} = 0\). Equation (c) shows that at equilibrium the isothermal dependence of entropy on pressure equals, with opposite signs, the isobaric dependence of volume on temperature. \(\mathrm{E}_{\mathrm{p}}\) is the isobaric expansion.This equation has practical importance. Suppose we require for either practical or theoretical reasons the dependence of the molar entropy of water(\(\ell\)), \(\mathrm{S}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\) on pressure at a given temperature. This has all the signs of being a difficult project. However the Maxwell Equation (c) shows that the information is obtained by measuring the dependence of molar volume \(\mathrm{V}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\) on temperature at constant pressure, a simpler approach to the problem. Equation (c) finds several important applications. One application concerns the isothermal dependence of enthalpy on pressure. We start with the equation, \(\mathrm{H}=\mathrm{G}-\mathrm{T} \, \mathrm{S}\). We are interested in the dependence of the properties of a given system on pressure at, for example, equilibrium, \(\mathrm{A} = 0\) and constant temperature. Then, \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]But at \(\mathrm{A} = 0, \mathrm{V}=(\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}}\). Using equation (c), \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{V}-\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]We see that the isothermal dependence of enthalpy on pressure is readily obtained knowing the volume of a system and its isobaric dependence on temperature. This is another interesting way in which Maxwell equations often simplify tasks facing chemists when probing the properties of systems. In fact equation (e) is fascinating bearing in mind that we can never know the enthalpy \(\mathrm{H}\) of a system but we can calculate in a straightforward manner using volumetric properties the isothermal dependence of enthalpy on pressure. In fact the integrated form of equation (e) is also useful. For a system at constant temperature [and at either constant composition \(\xi\) or at equilibrium, \(\mathrm{A} = 0\)], \[\mathrm{H}\left(\mathrm{T}, \mathrm{p}_{2}\right)-\mathrm{H}\left(\mathrm{T}, \mathrm{p}_{1}\right)=\int_{\mathrm{p}_{1}}^{\mathrm{p}_{2}}\left[\mathrm{~V}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\right] \, \mathrm{dp}\]Another important Maxwell Equation is based on the Helmholtz energy, \(\mathrm{F}\), of a closed system. \[\mathrm{F}=\mathrm{F}[\mathrm{V}, \mathrm{T}, \xi]\]For a closed system at fixed composition \(\xi\) (or at equilibrium when \(\mathrm{A} = 0\)) \[\left(\frac{\partial[\partial \mathrm{F} / \partial \mathrm{T}]_{\mathrm{V}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial[\partial \mathrm{F} / \partial \mathrm{V}]_{\mathrm{T}}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]But, \(\mathrm{S}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) and \(\mathrm{p}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\). Hence, \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]The right-hand-side of equation (i) involves the three practical properties, \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\). In summary, the isochoric dependence of pressure on temperature equals the isothermal dependence of entropy on volume.Two interesting Maxwell Equations develop from the Gibbs energy \(\mathrm{G}\). For a system at fixed pressure, \[\frac{\partial}{\partial T}\left(\frac{\partial G}{\partial \xi}\right)_{T}=\frac{\partial}{\partial \xi}\left(\frac{\partial G}{\partial T}\right)_{\xi}\]But \(\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \dot{\xi}}\right)_{\mathrm{T}, \mathrm{p}}\), and \(S=-\left(\frac{\partial G}{\partial T}\right)_{p, \xi}\), Then, \[\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=\left(\frac{\partial \mathrm{S}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]This interesting equation concerns the temperature dependence of the affinity for spontaneous reaction at fixed pressure and composition. In fact this dependence equals the isothermal-isobaric entropy of reaction, \((\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\).Also with respect to the Gibbs energy we explore the properties of a closed system at fixed temperature. Thus, \[\frac{\partial}{\partial p}\left(\frac{\partial G}{\partial \xi}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial G}{\partial p}\right)\]But, \(\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=-\mathrm{A}\), and \(\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\mathrm{V}\). Then, \[-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]In other words at constant composition the isothermal dependence of the affinity for spontaneous change on pressure equals (minus) the volume of reaction, \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\).Maxwell Equations are used in the analysis of parameters describing chemical equilibria. In general terms the limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) depends on pressure and the limiting volume of reaction. \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) depends on temperature. Further the entropy of reaction at temperature \(\mathrm{T}\), \(\Delta_{\mathrm{r}} \mathrm{S}^{\#}\) depends on pressure. These complexities signal more complexities in data analysis. Fortunately two Maxwell Equations assist the analysis. [Here ∆rS# refers to the difference between partial molar entropies of reactants and products in solution reference states at a pressure significantly different from the standard pressure.]The isothermal dependence of entropy of reaction on pressure is related to the isobaric dependence of limiting volume of reactions on temperature. \[\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{S}^{\#}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=-\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]Further the isothermal pressure dependence of the limiting enthalpy of reaction is related to the limiting volume of reaction and its isobaric temperature dependence. Thus, \[\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}-\mathrm{T} \,\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]The relationships offer a check of derived quantities and the numerical analysis when equilibrium constants are reported as functions of temperature and pressure. The beauty of thermodynamics is appreciated when one realizes that these relationships are precise. Discovery that a set of data and associated analyses do not conform to these equations does not disprove these Maxwell Equations. Rather one must conclude that analysis of the original experimental results is flawed. In fact Maxwell Equations offer an interesting exercise in units of derived and measured parameters. The isentropic expansion \(\mathrm{E}_{\mathrm{S}}\) is related to the isochoric dependence of entropy on pressure. From \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}]\), \[\partial^{2} U / \partial S \, \partial V=\partial^{2} U / \partial V \, \partial S\]Then, \[(\partial \mathrm{T} / \partial \mathrm{V})_{\mathrm{S}}=-(\partial \mathrm{p} / \partial \mathrm{S})_{\mathrm{V}}\]We invert the latter equation. Hence, \[\mathrm{E}_{\mathrm{S}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{s}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}}\]Two other Maxwell Equations are worthy of note. From, \[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\]At equilibrium and fixed composition, \[\left[\partial(\partial H / \partial S)_{p} / \partial p\right]_{\mathrm{S}}=(\partial T / d p)_{\mathrm{S}}\]and \[\left[\partial(\partial H / \partial \mathrm{p})_{\mathrm{s}} / \partial \mathrm{T}\right]_{\mathrm{p}}=(\partial \mathrm{V} / \mathrm{dS})_{\mathrm{p}}\]Then, \[(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}}\]From \((\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\). Then, \((\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}} \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\) Hence, \((\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}}=-(\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\) Then, \[(\partial \mathrm{p} / \partial \mathrm{T})_{\mathrm{V}}=(\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}}\] The extent of information available from thermodynamic partial derivatives is explored by: E. F. Caldin comments on 1010 possible relationships; Chemical Thermodynamics, Oxford, 1958 (page 158). H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, van Nostrand 1943. Extending the observation made by A. B. Pippard [The Elements of Classical Thermodynamics, Cambridge, 1957, p. 46], Maxwell Equations are dimensionally homogeneous in that cross-multiplication yields the following pairs of variables;The product of each pair is energy, with unit ‘Joule’. \[\begin{array}{r} \mathrm{T} \, \mathrm{S}=[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}^{-1}\right]=[\mathrm{J}] \\ \mathrm{p} \, \mathrm{V}=\left[\mathrm{Nm}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\ \mathrm{A} \, \xi=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \,[\mathrm{mol}]=[\mathrm{J}] \end{array}\] With reference to equation (o), \[\begin{aligned} {\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+[\mathrm{K}] \,\left[\frac{\mathrm{m}^{3} \mathrm{~mol}^{-1}}{[\mathrm{~K}]}\right] } \\ &=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \end{aligned}\] S. D. Hamann, Aust. J. Chem.,1984,37,867.This page titled 1.14.44: Maxwell Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.45: Moderation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.45%3A_Moderation

For a closed system, the dependence of chemical composition \(\xi\) on temperature \(\mathrm{T}\) at affinity \(\mathrm{A}\) and constant pressure is given by equation (a). \[\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left[\frac{\mathrm{A}+(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{\mathrm{T} \,(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}\right]\]Similarly for a closed system, the dependence of chemical composition \(\xi\) on pressure at fixed temperature is given by equation (b). \[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=\left[\frac{(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}\right]\]These two equations form the basis of ‘Laws of Moderation’ for closed systems at chemical equilibrium. These equations yield the sign for the two quantities \(\left(\frac{\partial \xi}{\partial T}\right)_{p, A=0}\) and \(\left(\frac{\partial \zeta}{\partial p}\right)_{T, A=0}\) which describe the change in composition when a system at equilibrium is perturbed to a neighboring equilibrium state.We recall that by definition \(\xi\) is positive for displacement in composition from reactants to products; \(\left(\frac{\partial V}{\partial \xi}\right)_{T, A=0}\) is the volume of reaction. If \(\left(\frac{\partial V}{\partial \xi}\right)_{T, A=0}\) is positive, \(\left(\frac{\partial \xi}{\partial p}\right)_{T, A=0}\) is negative because \(\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}<0\). According to equation (b), an increase in pressure favors a swing in the equilibrium position towards more reactants.Similarly it follows from equation (a) that an increase in temperature favors a swing in the equilibrium position towards more reactants for an exothermic reaction.Moderation is a striking example of the Second Law of Thermodynamics in action with reference to the direction of spontaneous changes in a closed system following changes in either \(\mathrm{T}\) or \(\mathrm{p}\). Here the stress on the word ‘closed’ reminds us that these laws of moderation do not apply to open system although the point is not always stressed. Therefore controversy often surrounds what is often called Le Chatelier’s Principle.Consider a closed system in which the following chemical equilibrium is established at defined \(\mathrm{T}\) and \(\mathrm{p}\). \[x X+y Y \Leftrightarrow z Z\]As often argued, if \(\delta \mathrm{n}_{\mathrm{Y}}\) moles of chemical substance \(\mathrm{Y}\) are added to the system, then the equilibrium amount of chemical substance \(\mathrm{Z}\) increases. In fact such moderation of composition only occurs if \(\sum_{j=1}^{j=i} v_{j}\) is zero for a chemical equilibrium involving i chemical substances. An interesting case concerns the Haber Synthesis. \[\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \Leftrightarrow 2 \mathrm{NH}_{3}(\mathrm{g})\]If in the equilibrium system mole fraction \(\mathrm{x}\left(\mathrm{N}_{2}\right) < 0.5\), addition of a small amount of \(\mathrm{N}_{2}(\mathrm{g})\) leads to an increase in the amount of ammonia. However if \(\mathrm{x}\left(\mathrm{N}_{2}\right) > 0.5\) addition of a small amount of \(\mathrm{N}_{2}(\mathrm{g})\) leads to dissociation of ammonia to form more \(\mathrm{N}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2}(\mathrm{g})\). This conclusion is called a Theorem of Moderation. Co-author MJB was taught that the outcome was “Nature’s Law of Cussedness” (\(\equiv\) Obstinacy). An exothermic reaction operates to generate heat so the system responds when the temperature is raised in the direction for which the process is endothermic. This line of argument is not good thermodynamics but does make the point. Another example of Nature’s Obstinacy; see. Note the switch in sign on the r.h.s of equations (a) and (b). I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans - Green, London, 1953, page 268.This page titled 1.14.45: Moderation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.46: Molality and Mole Fraction

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For a solution prepared using \(\mathrm{n}_{j}\) moles of solute and \(\mathrm{w}_{\mathrm{s}}\) kg of solvent , molality \[m_{j}=n_{j} / w_{s}\]Molality \(\mathrm{m}_{j}\) expressed in ‘\(\mathrm{mol kg}^{-1}\)’ is independent of temperature and pressure being defined by the masses of solvent and solute. The solvent may comprise a mixture of liquids, the composition of the solvent being described using mole fractions, weight-per-cent or volume-per-cent.For a closed system comprising \(\mathrm{n}_{1}, \mathrm{~n}_{2}, \mathrm{~n}_{3} \ldots \mathrm{~n}_{i}\) moles of each \(\mathrm{k}\) chemical substance, the mole fraction of chemical substance \(\mathrm{j}\), \[\mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \sum_{\mathrm{k}=1}^{\mathrm{k}=\mathrm{i}} \mathrm{n}_{\mathrm{k}}\]where \[\sum_{k=1}^{k=1} x_{k}=1\]Mole fraction xj is independent of temperature and pressure (in the absence of chemical reaction between the chemical substances in the system).This page titled 1.14.46: Molality and Mole Fraction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.47: Newton-Laplace Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.47%3A_Newton-Laplace_Equation

The Newton-Laplace Equation is the starting point for the determination of isentropic compressibilities of solutions using the speed of sound \(\mathrm{u}\) and density \(\rho\); equation (a). \[\mathrm{u}^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1}\]Densities of liquids and speeds of sound at low frequencies can be precisely measured .The isentropic condition means that as the sound wave passes through a liquid the pressure and temperature fluctuate within each microscopic volume but the entropy remains constant. The condition ‘at low frequencies‘ is important because at high frequencies ( e.g. \(> 100 \mathrm{~MHz}\)) there is a velocity dispersion and absorption of sound as the sound wave couples with molecular processes within the liquid.Several points emerge from a consideration of equation (a). For example one might ask --- is it just assumed that the correct term is \(\kappa_{\mathrm{S}}\) and not \(\kappa_{\mathrm{T}}\)? The point is that in their examination of the properties of aqueous solutions and aqueous mixtures authors often write something along the following lines -- ‘we used the Newton-Laplace equation to calculate \(\kappa_{\mathrm{S}}\) from measured speeds of sound’. One might then ask-- can one prove equation (a) and is the proof thermodynamic? Rowlinson states that the speed of sound defined by equation (a) is, and we quote, ‘of course a purely thermodynamic quantity’. This comment raises the issue as to whether or not the defined quantity equals the measured speed of sound.Intuitively the task of measuring the isothermal property \(\mathrm{K}_{\mathrm{T}}\left[=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}\right]\) might seem less problematic than measuring the isentropic property, \(\mathrm{K}_{\mathrm{s}}\left[=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}\right]\). \(\mathrm{K}_{\mathrm{T}}\) would be obtained by measuring the change in volume following an increase in pressure. However as Tyrer warned in 1914, the isothermal condition is difficult to satisfy and estimated compressions and compressibilities reported up to that time and in the majority of cases were certainly between isentropic and isothermal values. Tyrer did in fact measure \(\kappa_{\mathrm{T}}\) and calculated \(\kappa_{\mathrm{S}}\) using equation (b). \[\kappa_{\mathrm{S}}=\kappa_{\mathrm{T}}-\mathrm{T} \,\left[\alpha_{\mathrm{p}}\right]^{2} / \sigma\]Other authors have measured \(\kappa_{\mathrm{T}}\) directly by, for example, the volume increase on sudden decompression of a liquid from high to ambient pressure. Nevertheless the conventional approach uses the Newton-Laplace equation. Historically this subject has its origins in the attempts initiated in the 17th Century to measure the speed of sound in air.A sound wave traveling through a fluid produces a series of compressions and rarefactions. Consequently planes of molecules perpendicular to the direction of the sound waves are displaced. The displacement \(\varepsilon\) depends on both position \(\mathrm{x}\) and time \(t\). Thus \[\varepsilon=\varepsilon[\mathrm{x}, \mathrm{t}]\]The speed of the sound wave \(\mathrm{u}\) is related to the displacement ε using equation (d), the wave equation. \[\left(\partial^{2} \varepsilon / \partial x^{2}\right)=\left(1 / u^{2}\right) \,\left(\partial^{2} \varepsilon / \partial t^{2}\right)\]A classic analysis in terms of equation (d) and stress-strain relationships for an isotropic phase using Hooke’s Law yields equation (a). At this stage we could consider both the isothermal compressibility \(\kappa_{\mathrm{T}}\) and the isentropic compressibility \(\kappa_{\mathrm{S}}\). If equation (a) is correct then, either (a) the speed of sound can be calculated knowing \(\kappa_{\mathrm{S}}\) and \(\rho\), or (b) \(\kappa_{\mathrm{S}}\) can be calculated by measuring speed of sound \(\mathrm{u}\) and density \(\rho\).Another line of argument states that equation (a) defines the speed of sound in terms of \(\kappa_{\mathrm{S}}\) and density \(\rho\). The question arises -- is the speed of sound calculated using equation (a) equal to the measured speed of sound?The analysis up to and including equation (d) was familiar to Newton (I. Newton 1642-1727). Newton using Boyle's Law assumed that the fluid is an ideal gas and that the compressions and rarefactions are isothermal (and in a thermodynamic sense, reversible); Hence \[\mathrm{u}^{2}= \mathrm{p} / \rho\]Equation (e) was particularly important to Newton because the three quantities in equation (e) can independently determined for (dry) air. Using the density ρ for air at pressure p one can calculate the speed of sound in air. The agreement between observed and calculated speeds was, somewhat disappointingly, only fair but encouraging. The disagreement was an underestimate by 20% as was noted by Newton.The argument is interesting in the sense of testing if the analysis yields the measured speed of sound. Clearly the equations do not. An important contribution was made by Laplace who assumed that the compressions and rarefactions are perfect and isentropic; i.e. \(\mathrm{p} \, \mathrm{V}^{\gamma}=\) constant where \(\gamma\) is the ratio of isobaric and isochoric heat capacities. This is the assertion made by Laplace. The overall condition is isentropic for a gas at temperature T. The condition refers to macroscopic properties. Within each microscopic volume both temperature and pressure fluctuate but the entropy remains constant. [The equilibrium and isentropic conditions mean that there is no loss of heat on compression and no gain of heat on rarefaction when the sound wave passes through the system; everything is in phase.] Assuming that \(\gamma\) is independent of \(\mathrm{p}\), \[\mathrm{u}^{2}=\gamma \, \mathrm{p} / \rho\]The point is that Laplace knew \(\gamma\) for (dry) air at \(273 \mathrm{~K}\) and standard pressure equals 1.4. With this information Laplace obtained good agreement between theory and experiment for the speed of sound in air. In other words Laplace confirmed his assertion that for air (a fluid with low density, \(1.29 \times 10^{-3} \mathrm{~g cm}^{-3}\)) compressions and rarefactions are isentropic and not isothermal. Hence the fame of the Newton-Laplace equation which is based on an assertion. Laplace did not prove that the processes are isentropic but having shown agreement between theory and experiment one must conclude that the assertion is correct for air. Equation (a) is the Newton-Laplace Equation. The key point is that the equation emerges from an Equation of State for isentropic compressions of a particular gas, air. Indeed the success achieved by the Newton-Laplace equation in term of predicting the speed of sound in a gas is noteworthy. However we need to comment on the link between \(\kappa_{\mathrm{S}}\) measured directly and obtained from measurements of \(\kappa_{\mathrm{T}}\), \alpha_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{p}}\) using equation (b). We direct attention to a given closed system containing liquid water. From a practical standpoint, the difference between isothermal and isentropic compressibilities (cf. equation (b)) written here for the pure liquid water, \(\frac{\left[\alpha_{\mathrm{pl}}^{*}(\ell)\right]^{2} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{T}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}\). is reasonably accessible. The molar volume \(\mathrm{V}_{1}^{*}(\ell)\) is obtained from the density \(\rho_{1}^{*}(\ell); \alpha_{p 1}^{*}(\ell)\) is obtained from the dependence of density on temperature at fixed pressure.The molar isobaric heat capacity \(\mathrm{C}_{p 1}^{*}(\ell)\) is also experimentally accessible. The most frequently cited data set for \(\mathrm{V}_{1}^{*}(\ell)\) and \(\alpha_{p 1}^{*}(\ell)\) was published by Kell and Whalley in 1965; see also reference. The isothermal compressibility is less accessible . In 1967 Kell summarized the results obtained by Kell and Whalley and quoted that at 25 Celsius, \(\alpha_{\mathrm{p} 1}^{*}(\ell)=257.05 \times 10^{-6} \mathrm{~K}^{-1}\) and \(\kappa_{\mathrm{T} 1}^{*}(\ell)=45.24 \mathrm{Mbar}^{-1}\). In 1969 Millero and co-workers directly measured isothermal compressions of water(\(\ell\)) drawing comparisons with the estimates made by Kell and Whalley. They reported that for water(\(\ell\)) at 25 Celsius, \(\kappa_{\mathrm{Tl}}^{*}(\ell)=(45.94 \pm 0.06) \mathrm{Matm}^{-1}\). Millero et al. comment on the excellent agreement.In 1970, Kell addressed the issue which is of interest here. Equation (b) is the key to the debate because we obtain an estimate of \(\kappa_{\mathrm{S} 1}^{*}(\ell)\) from measured \(\kappa_{\mathrm{2} 1}^{*}(\ell), \alpha_{\mathrm{pl}}^{*}(\ell), \mathrm{~V}_{1}^{*}(\ell) \text { and } \mathrm{C}_{\mathrm{pl}}^{*}(\ell)\); i.e. \(\kappa_{\mathrm{S}}^{*}(\ell ; \text { density })\). Alternatively we obtain \(\kappa_{\mathrm{S} 1}^{*}(\ell)\) using equation (a); i.e. speed of sound and density yielding \(\kappa_{\mathrm{S}}\) (acoustic) . The key question is --- are \(\kappa_{\mathrm{S}}^{*}(\ell ; \text { density })\) and \(\kappa_{\mathrm{S}}\) (acoustic) equal? How confident are we that they are equal? There are no assumptions underlying the calculation of \(\kappa_{\mathrm{S}}^{*}(\ell ; \text { density })\). In the case of \(\kappa_{\mathrm{S}}\) (acoustic), the sound wave perturbs the system isentropically; cf. Laplace analysis. Kell comments that speeds of sound can be precisely measured and also a precise estimate of the defined \(\kappa_{\mathrm{S}}\) (acoustic) is obtained. Granted the validity of equation (a) one can re-express equation (b) as an equation for \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) in terms of measured \(\kappa_{\mathrm{S} 1}^{*}(\ell), \alpha_{\mathrm{p} 1}^{*}(\ell), \mathrm{~V}_{1}^{*}(\ell) \text { and } \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\). Examination of various sets of data showed that \(\kappa_{\mathrm{S}}\) (acoustic) has less systematic errors than \(\kappa_{\mathrm{S}}^{*}(\ell ; \text { density })\) but that they are effectively the same, a point confirmed by Fine and Millero. J. S. Rowlinson and F. L. Swinton, Liquid and Liquid Mixtures, Butterworths, London , 3rd. edn., 1982, pp. 16-17. J. O. Hirschfelder, C. F. Curtis and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, corrected printing 1964, chapters 5 and 11. \[\mathrm{u}^{2}=\frac{1}{\kappa_{\mathrm{s}}} \, \frac{1}{\rho}=\left[\mathrm{N} \mathrm{m}^{-2}\right] \, \frac{1}{\left[\mathrm{~kg} \mathrm{~m}^{-3}\right]}=\frac{\left[\mathrm{kg} \mathrm{m} \mathrm{s}^{-2} \mathrm{~m}^{-2}\right]}{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{2} \mathrm{~s}^{-2}\right]\]\[\mathrm{u}=\left[\mathrm{m} \mathrm{s}^{-1}\right]\] A.T. J. Hayward, Brit. J. Appl. Phys.,1967,18,965, A. T. J. Hayward, J. Phys. D: Appl. Physics, 1971.4,938. A.T. J .Hayward, Nature, 1969,221.1047 M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press,1973. D. Tyrer, J. Chem. Soc.,1914,105,2534. D. Harrison and E. A. Moelwyn-Hughes, Proc. R. Soc. London, Ser.A, 1957, 239. 230. L. A. K. Staveley, W. I. Tupman and K. Hart, Trans. Faraday, Soc.,1955,51,323. P. Costabel and L.Auger, in Science in the Nineteenth Century; ed. R. Taton, transl. A. J. Pomerans, Basic Books, New York,1961, p. 170. S. G. Starling and A. J. Woodhall, Physics, Longmans, London, 2nd edn.,1957. I. Newton, Philosophicae Naturalis Principia Mathematica, Vol.II, Sect VII, Prop.46, London,1687. S. Laplace, Ann. Chim. Phvs., 1816,3,328. D.-P. Wang and F. J. Millero, J. Geophys. Res.,1973,78,7122. G. S. Kell and E. Whalley, Philos. Trans. R. Soc. London, Ser.A.,1965,258,565. G. S. Kell, G. E. McLaurin and E. Whalley, Proc. R. Soc. London, Ser,.A, 1978, 360, 389. G. S. Kell, J. Chem. Eng. Data, 1967,12,66;1970,l5, 119. F. J. Millero, R.W. Curry and W. Drost-Hansen, J. Chem. Eng. Data,1969,14,422. G. S. Kell, in Water A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York,1973,volume I, chapter 10. R. A. Fine and F. J. Millero, J Chem. Phys.,1973,59,5529;1975,63,89.This page titled 1.14.47: Newton-Laplace Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.48: Open System

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The chemical thermodynamics of open systems is more complicated than that of closed systems because chemical substances exchange between system and surroundings, crossing the boundary of the system.Footnote D. Kondepudi and I. Prigogine, Modern Thermodynamics; From Heat Engines to Dissipative Structures, Wiley, New York, 1998. Clearly a treatment of the chemical thermodynamics of the human body has to take account of the fact that such systems are open. Farmers are very practical chemical thermodynamic experts because in feeding their livestock they judge if the animals they are feeding willFarmers do not leave these options to chance as they cope in very practical way with such open systems.This page titled 1.14.48: Open System is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.49: Osmotic Coefficient

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There is possible disadvantage in an approach using the mole fraction scale to express the composition of a solution. Grantedthe mole fraction scale for the solvent is not the most convenient method for expressing the composition of a given solution. Hence another equation relating \(\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of a solution finds favor.By definition, for a solution containing a single solute, chemical substance \(\mathrm{j}\), \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Or, in terms of the standard chemical potential for water at temperature \(\mathrm{T}\) and standard pressure \(\mathrm{p}^{0}\), \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp}\]\(\mathrm{M}_{1}\) is the molar mass of water; \(\phi\) is the practical osmotic coefficient which is characteristic of the solute, molality mj, temperature and pressure. By definition \(\phi\) is unity for ideal solutions at all temperatures and pressures. \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \text { at all T and } \mathrm{p}\]Further for ideal solutions, the partial differentials \((\partial \phi / \partial T)_{p}, \left(\partial^{2} \phi / \partial T^{2}\right)_{p} \text { and } (\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\) are zero. For an ideal solution, \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]We rewrite equation (d) in the following form: \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})-\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Hence with an increase in molality of solute in an ideal aqueous solution, the solvent is stabilized, being at a lower chemical potential than that for pure water. We contrast the chemical potentials of the solvent in real and ideal solutions using an excess chemical potential, \(\mu_{1}^{E}(a q ; T ; p)\); \[\begin{aligned} \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) &=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}) \\ &=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \end{aligned}\]The term \((1 - \phi)\) is often encountered because it expresses succinctly the impact of the solute on the properties of the solvent. At a given molality (and fixed temperature and pressure), \(\phi\) is characteristic of the solute.In the case of a salt \(\mathrm{j}\) which on complete dissociation forms ν ions the analogue of equation (a) takes the following form. \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\] N. Bjerrum, Z. Electrochem., 1907, 24,259. G. N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 2nd edn., 1961, chapter 22. Mole fractions of solvent \(\mathrm{x}_{1}\) for aqueous solutions having gradually increasing molality of solute \(\mathrm{m}_{j}. \[\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]- \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\]The definitions of ideal solutions expressed in equations (i) and (ii) are not in conflict. \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]Thus for an ideal solution these equations require that, \(v-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\ln \left(\mathrm{x}_{1}\right)\) But \[\ln \left(\mathrm{x}_{1}\right)=\ln \left[\mathrm{M}_{1}^{-1} /\left(\mathrm{M}_{1}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\right]=-\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)\]Bearing in mind that \(\mathrm{M}_{1} = 0.018 \mathrm{~kg mol}^{-1}\), then for dilute solutions \(\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)=\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\).This page titled 1.14.49: Osmotic Coefficient is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.5: Extent of Reaction

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For chemists, chemical reaction is the key thermodynamic process. By definition chemical reaction produces a change in composition of a closed system. The extent of chemical reaction is measured by a quantity \(\mathrm{d}\xi\), where the chemical composition is described by the symbol \(\xi\). An example makes the point.An aqueous solution is prepared at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains solute \(\mathrm{X}\). The latter undergoes spontaneous chemical reaction to form chemical substance \(\mathrm{Y}\).Thus[Time is a legitimate thermodynamic property.]A key concept states that spontaneous chemical reaction is driven by the affinity for spontaneous change, \(\mathrm{A}\). Then by definition equilibrium corresponds to the state where \(\mathrm{A} = 0\), and \(\mathrm{d}\xi / \mathrm{dt} = 0\).For a system containing \(\mathrm{i}\)-chemical substances, the chemical potential of chemical substance \(\mathrm{j}\) is given by equation (a). \[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial n_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]Then, \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]But, \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\]By comparison, \[A \, d \xi=-\sum_{j=1}^{j=i} \mu_{j} \, d n_{j}\]But \(\mathrm{dn}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} \, \mathrm{d} \xi\) where \(\mathrm{ν}_{\mathrm{j}}\) is positive for products and negative for reactants. Hence, \[A=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j}\]This remarkable equation relates the affinity for chemical reaction \(A\) with the chemical potentials of the chemical substances involved in the chemical reaction. Moreover at equilibrium, \(A\) is zero. Hence, \[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}=0\]We have a condition describing chemical equilibrium in terms of the chemical potentials of reactants and products at equilibrium.This page titled 1.14.5: Extent of Reaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.50: Osmotic Pressure

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A semi-permeable membrane separates an aqueous solution (where the mole fraction of water equals \(\mathrm{x}_{1}\)) and pure solvent at temperature \(\mathrm{T}\) and ambient pressure. Solvent water flows spontaneously across the membrane thereby diluting the solution. This flow is a consequence of the chemical potential of the solvent in the solution being lower than the chemical potential of pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). If a pressure (\(\mathrm{p} + \pi\)) is applied to the solution, the spontaneous process stops because the solution at pressure (\(\mathrm{p} + \pi\)) and the solvent at pressure \(\mathrm{p}\) are in thermodynamic equilibrium; \(\pi\) is the osmotic pressure. Thus at equilibrium, \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}+\pi)=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]Under this equilibrium condition solvent flows in both directions across the semi-permeable membrane but the net flow is zero. In the analysis presented here we take account of the fact, writing \(\mathrm{p}^{\prime}\) for (\(\mathrm{p} + \pi\)), the chemical potential of water in the aqueous solution is given by equation (b). \[\begin{aligned} &\mu_{1}^{\mathrm{eq}}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{\prime}\right)= \\ &\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0)+\mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \mathrm{K}_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{p}^{\prime}\right] \\ & \\ &+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \end{aligned}\]Here \(\mathrm{f}_{1}\) is the activity coefficient expressing the extent to which the thermodynamic properties of water in the aqueous solution are not ideal. For the pure solvent water at pressure \(\mathrm{p}\) (i.e. on the other side the of the semi-permeable membrane), \[\begin{aligned} &\mu_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\ &\quad \mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0)+\mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{p}\right] \end{aligned}\]But osmosis experiments explore an equilibrium characterized by equation (d). \[\mu_{1}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{\prime}\right)=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]Therefore using equations (b) and (c), \[\begin{array}{r} \mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right]+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)= \\ \mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right] \end{array}\]Or, \[\begin{aligned} &\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right] \\ &-\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \end{aligned}\]The terms \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right]\) and \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]\) describe the molar volumes of water at pressures \(\mathrm{p}\) and \(\mathrm{p}^{\prime}\); i.e. at pressure \(\mathrm{p}\) and (\(\mathrm{p}+\pi\)). We assume that both terms can be replaced by the molar volumes at average pressure \([(2 \, \mathrm{p}+\pi) / 2]\); namely \(\mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ;(2 \, \mathrm{p}+\pi) / 2]\). Therefore \[\pi \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ;(2 \, \mathrm{p}+\pi) / 2]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]In the event that \(\pi<<2 \, p\), \[\pi \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]If the thermodynamic properties of the solutions are ideal, \(\mathrm{f}_{1}\) equals unity. Then \[\pi^{\mathrm{id}} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]In the latter two equations \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\) is treated as a constant, independent of the thermodynamic properties of the solution. A further interesting development of equation (i) is possible for a solution prepared using \(\mathrm{n}_{1}\) moles of solvent water and \(\mathrm{n}_{j}\) moles of solute. Thus \[-\ln \left(\mathrm{x}_{1}\right)=\ln \left(\frac{1}{\mathrm{x}_{1}}\right)=\ln \left(\frac{\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)=\ln \left(1+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)\]But . \(\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}\right)<<1\). We expand the last term in equation (j). \[\ln \left(1+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}-\frac{1}{2} \,\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)^{2}+\frac{1}{3} \,\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)^{3}-\ldots . .\]If we retain only the first term; \[\pi^{\mathrm{id}} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=\mathrm{R} \, \mathrm{T} \,\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}\right)\]But for a dilute solution, the volume of the solution \(\mathrm{V}\) is given by \(\mathbf{n}_{1} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]\).Or, \[\pi^{\mathrm{id}} \, \mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]But concentration \[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\]Then \[\pi^{\mathrm{id}}=\mathrm{c}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]Agreement between \(\pi(\mathrm{obs})\) and \(\pi^{\mathrm{id}}\) for aqueous solutions containing neutral solutes (e.g. sucrose) confirms the validity of the thermodynamic analysis. The term ‘semi-permeable’ in the present context means that the membrane is only permeable to the solvent. Perhaps the optimum semipermeable membrane is the vapor phase. Historically, equation(o) owes much to the equation of state for an ideal gas; i.e. \(\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T}\). From an experimentally found proportionality between \(\pi\) and \(\mathrm{c}_{j}\), van’t Hoff showed that the proportionality constant can be approximated by \(\mathrm{R} \, \mathrm{T}\). \[\pi=\left[\mathrm{mol} \mathrm{m}^{-3}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]\]This page titled 1.14.50: Osmotic Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.51: Partial Molar Properties: General

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The laws of thermodynamics and the associated treatment of the thermodynamic properties of closed systems concentrate attention on macroscopic properties. Although we may define the composition of a closed system in terms of the amounts of each chemical substance in a system, general thermodynamic treatments direct our attention to macroscopic properties such as, volume \(\mathrm{V}\), Gibbs energy \(\mathrm{G}\), enthalpy \(\mathrm{H}\) and entropy \(\mathrm{S}\). We need to ‘tell’ these thermodynamic properties that a given system probably comprises different chemical substances. In this development the analysis is reasonably straightforward if we define the system under consideration by the ‘Gibbsian ‘set of independent variables; i.e. \(\mathrm{T}, \mathrm{~p}\) and amounts of each chemical substance. Thus, \[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \ldots \mathrm{n}_{\mathrm{i}}\right]\]In equation (a), the Gibbs energy is defined by intensive variables \(\mathrm{T}\) and \(\mathrm{p}\) together with extensive composition variables. In many cases the task of a chemist is to assay a system to determine the number and amounts of each chemical substance in the system.The analysis leads to the definition of the chemical potential for each substance \(\mathrm{j}\), \(\mu_{\mathrm{j}}\) in a closed system.Consider a solution comprising \(\mathrm{n}_{1}\) moles of solvent, liquid chemical substance 1, and \(\mathrm{n}_{\mathrm{j}}\) moles of solute, chemical substance \(\mathrm{j}\). We ask—what contributions are made by the solvent and by the solute to the volume of the solution at defined \(\mathrm{T}\) and \(\mathrm{p}\)? In fact we can only guess at these contributions. This is disappointing. The best we can do is to probe the sensitivity of the volume of a given solution to the addition of small amounts of either solute or solvent. This approach leads to a set of properties called partial molar volumes. Here we explore the definition of these properties. The starting point is the Gibbs energy of a solution. We develop the argument in a way which places the Gibbs energy at the centre from which all other thermodynamic variables develop. For a closed system containing i-chemical substances, \[\mathrm{G}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]The later equation signals that the total Gibbs energy is given by the sum of products of amounts and chemical potentials of each chemical substance in the system. For an aqueous solution containing \(\mathrm{n}_{\mathrm{j}}\) moles of solute \(\mathrm{j}\) and \(\mathrm{n}_{1}\) moles of solvent 1 ( water), \[G(a q)=n_{1} \, \mu_{1}(a q)+n_{j} \, \mu_{j}(a q)\]We do not have to attach to equation (c) the condition ‘at fixed \(\mathrm{T} and \mathrm{p}\)’. Similarly the volume of the solution is given by equation (d). \[V(a q)=n_{1} \, V_{1}(a q)+n_{j} \, V_{j}(a q)\]The same argument applies in the case of a system prepared using \(\mathrm{n}_{1}\) moles of water, \(\mathrm{n}_{\mathrm{X}}\) moles of solute \(\mathrm{X}\) and \(\mathrm{n}_{\mathrm{Y}}\) moles of solute \(\mathrm{Y}\). \[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{X}} \, \mu_{\mathrm{X}}(\mathrm{aq})+\mathrm{n}_{\mathrm{Y}} \, \mu_{\mathrm{Y}}(\mathrm{aq})\]Complications emerge however if solute \(\mathrm{X}\) and \(\mathrm{Y}\) are in chemical equilibrium; e.g. \(X(a q) \Leftrightarrow Y(a q)\). Then account must be taken of the fact that \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) depend on \(\mathrm{T}\) and \(\mathrm{p}\).Footnote I have on my desk a flask containing water (\(\ell ; 100 \mathrm{~cm}^{3}\)) and 20 small round steel balls, each having a volume of \(0.1 \mathrm{~cm}^{3}\). I add a steel ball to the flask and the volume of the system, water + steel ball, is \(100.1 \mathrm{~cm}^{3}\). I add one more steel ball and the volume of the system increases by \(0.1 \mathrm{~cm}^{3}\). So in this simple case I can equate directly the volume of the pure steel balls \(\mathrm{V}^{*}\)(balls) with the partial molar volume of the balls in the system, water + balls.I have on my desk an empty egg carton designed to hold six eggs. The volume of the carton is represented as \(\mathrm{V}(\mathrm{c})\) as judged by the volume occupied in a food store.. The volume of one egg is \(\mathrm{V}^{*}\)(egg), the superscript * indicating that we are discussing the property of pure eggs. I now ‘add’ one egg to the egg carton which does not change its volume ----again as judged by the volume occupied in a food store. In other words the partial molar volume of eggs in the egg carton \(\mathrm{V}(\mathrm{egg})\) is zero; \[V(\text { egg })=\left(\frac{\partial V(\text { system })}{\partial n(\text { egg })}\right)=\text { zero. Or, } V(\text { egg })-V^{*}(\text { egg })<0\]This page titled 1.14.51: Partial Molar Properties: General is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.52: Partial Molar Properties: Definitions

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A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. If the thermodynamic properties of the liquid mixture are ideal the volume of the mixture is given by the sum of products of amounts and molar volumes (at the same \(\mathrm{T}\) and \(\mathrm{p}\)); equation (a). \[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]If the thermodynamic properties of the mixture are not ideal, the volume of the (real) mixture is given by equation (b). \[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix})\]\(\mathrm{V}_{1}(\operatorname{mix})\) and \(\mathrm{V}_{1}(\operatorname{mix})\) are the partial molar volumes of chemical substances 1 and 2 defined by equations (c) and (d). \[V_{1}(m i x)=\left(\frac{\partial V}{\partial n_{1}}\right)_{T, p, n}\]\[V_{2}(\operatorname{mix})=\left(\frac{\partial V}{\partial n_{2}}\right)_{T, p, n}\]The similarities between equations (a) and (b) are obvious and indicate an important method for describing the extensive properties of a given system. This was the aim of G. N. Lewis who sought equations of the form shown in equation (b). In general terms, we identify an extensive property \(\mathrm{X}\) of a given system such that the variable can be written in the general form shown in equation (e). \[\mathrm{X}=\mathrm{n}_{1} \, \mathrm{X}_{1}+\mathrm{n}_{2} \, \mathrm{X}_{2}\]where \[\mathrm{X}_{1}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]\[X_{2}(\operatorname{mix})=\left(\frac{\partial X}{\partial n_{2}}\right)_{T_{, p, n}}\]Other than the composition variables, the conditions on the partial differentials in equations (f) and (g) are intensive properties;Partial molar properties can also be defined for different pairs of intensive thermal and non-thermal variables, other than \(\mathrm{T}\) and \(\mathrm{p}\). The concept of a partial property was extended to intensive properties such as isothermal and isentropic compressibilities.A further distinction between Lewisian and non-Lewisian partial molar properties has been proposed. G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259. J. C. R. Reis, J. Chem. Soc Faraday Trans.,2,1982,78,1575. J. C. R. Reis, J. Chem. Soc Faraday Trans.,1998,94,2385. J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Phys. Chem.Chem.Phys.,2001,3,1465.This page titled 1.14.52: Partial Molar Properties: Definitions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.53: Phase Rule

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According to the Gibbs-Duhem Equation, the properties of a single phase at equilibrium containing i chemical substances are related; we divide the Gibbs-Duhem Equation by the total amount in the system such that \(\mathrm{x}_{j}(\alpha)\) is the mole fraction of substance \(j\) in the \(\alpha\) phase. The Gibbs-Duhem Equation. requires that \[0=\mathrm{S}_{\mathrm{m}}(\alpha) \, \mathrm{dT}-\mathrm{V}_{\mathrm{m}}(\alpha) \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{x}_{\mathrm{j}}(\alpha) \, \mathrm{d} \mu_{\mathrm{j}}(\alpha)\]Within this phase, the definition of mole fraction means that over all \(\mathrm{i}\)-chemical substances, \[\sum_{j=1}^{j=i} x_{j}(\alpha)=1\]The number of independent intensive variables is \([\mathrm{P} \,(\mathrm{C}-1)+2]\) where \(\mathrm{C}\) is the number of independent chemical substances in phase \(\alpha\). The additional two variables refer to the intensive temperature and pressure. We consider the case where the closed system contains \(\mathrm{P}\) phases. Therefore we can set down \(\mathrm{P}\) equations of the form shown in equation (a). With reference to the chemical potential of substance \(j\), the overall equilibrium condition requires that the chemical potentials of this substance over all phases ( i.e. \(\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \alpha_{p}\)) are equal. \[\mu_{j}\left(\alpha_{1}\right)=\mu_{j}\left(\alpha_{2}\right)=\mu_{j}\left(\alpha_{3}\right)=\ldots \ldots \ldots . .=\mu_{j}\left(\alpha_{p}\right)\]Hence with reference to the intensive chemical potentials there are (\(\mathrm{P} - 1\)) constraints. Therefore the number of independent intensive variables for this system comprising \(\mathrm{i}\) chemical substances distributed through \(\mathrm{P}\) phases, namely \(\mathrm{F}\), equals \((\mathrm{C} −1) + 2 − (\mathrm{P} −1)\). Therefore \[\mathrm{P}+\mathrm{F}=\mathrm{C}+2\]The latter is the Phase Rule. This equation is possibly the most elegant and practical equation in chemistry.This page titled 1.14.53: Phase Rule is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.54: Poynting Relation

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A given closed system comprises chemical substance j in two homogeneous subsystems which are separated by an appropriate semipermeable diaphragm and which are at the same temperature but different pressures. The subsystems I and II are in thermodynamic equilibrium. Thus (cf. Topic 690), \[\mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)\]For subsystem I, \[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\left(\frac{\partial \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp}_{1}\]Or, \[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=-\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, \mathrm{dp}_{1}\]Here \(\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) and \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{\mathrm{l}}\right)\) are molar properties of chemical substance \(j\). Similarly, \[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)=-\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \, \mathrm{dp}_{2}\]The equality expressed in equation (a) is valid at all \(\mathrm{T}\) and \(\mathrm{p}\). Clearly this condition can only be satisfied if the following equation is satisfied. \[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)\]Then at constant temperature, \[V_{j}^{*}\left(I, T, p_{1}\right) \, d p_{1}=V_{j}^{*}\left(I I, T, p_{2}\right) \, d_{2}\]Hence, \[\frac{\mathrm{dp}_{1}}{\mathrm{dp}_{2}}=\frac{\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}_{1}, \mathrm{p}_{2}\right)}{\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)}\]The latter is the Poynting Equation. An interesting application of this equation concerns the case where system II is the vapor phase and system I is the liquid phase. The vapor phase is described as an ideal gas using equation (h) for one mole of chemical substance \(j\). \[\mathrm{p}_{2} \, \mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)=\mathrm{R} \, \mathrm{T}\]The liquid phase comprises one mole of liquid j for which \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) is the molar volume which is assumed to be a constant, independent of pressure.Hence from equations (g) and (h), \[\frac{\mathrm{dp}_{1}}{\mathrm{dp}_{2}}=\frac{1}{\mathrm{~V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)} \, \frac{\mathrm{R} \, \mathrm{T}}{\mathrm{p}_{2}}\]Or, \[\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \left(\mathrm{p}_{2}\right)=\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, d \mathrm{p}_{1}\]The assumption is made that, phase I being a liquid, \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) is independent of pressure. Then equation (j) is integrated between pressure limits \(\mathrm{p}_{2}\) and \({\mathrm{p}_{2}}^{\prime}\) and between \(\mathrm{p}_{1}\) and \({\mathrm{p}_{1}}^{\prime}\). Hence, \[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{2}^{\prime} / \mathrm{p}_{2}\right)=\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \,\left[\mathrm{p}_{1}^{\prime}-\mathrm{p}_{1}\right]\]An interesting application of equation (k) concerns the impact of an increase in pressure from \(\mathrm{p}_{1}\) and \({\mathrm{p}_{1}}^{\prime}\) on liquid \(j\). This increase might be produced for example by an increase in confining pressure of an inert gas insoluble in liquid \(j\). Equation (k) describes the increase in vapor pressure from \(\mathrm{p}_{2}\) to \({\mathrm{p}_{2}}^{\prime}\) of liquid \(j\). This pattern might seem intuitively somewhat unexpected. J. J. Vanderslice, H. W. Schamp Jr and E. A. Mason, Thermodynamics, Prentice Hall,Englewood Cliffs, N.J., 1966, page 106. Poynting, Phil. Mag.,1881,,12,32.This page titled 1.14.54: Poynting Relation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.55: Process

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In order to document the thermodynamics of processes a convention has been agreed. In general, the thermodynamic variable takes the following form. \[\Delta_{\text {proc }} \mathrm{X}^{0}\]HereIn recognition of the long tradition of using a ‘double-dagger’, a superscript \(\neq\) indicates activation as in the formation of ‘transition state from reactants.This page titled 1.14.55: Process is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.56: Properties: Equilibrium and Frozen

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A given closed system having Gibbs energy \(\mathrm{G}\) at temperature \(\mathrm{T}\), pressure \(\mathrm{p}\), molecular composition (organization \(\xi\)) and affinity for spontaneous change \(\mathrm{A}\) is described by equation (a). \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\]In the state defined by equation (a), there is an affinity for spontaneous chemical reaction \(\mathrm{A}\). Starting with the system in the state defined by equation (a) it is possible to change the pressure and perturb the system to a series of neighboring states for which affinity remains constant. The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\mathrm{A}\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\). Returning to the original state characterized by \(\mathrm{T}\), \(\mathrm{p}\) and \(\xi\), we imagine that it is possible to perturb the system by a change in pressure in such a way that the system remains at fixed extent of reaction, \(\xi\). The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\xi\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\). We explore these dependences of \(\mathrm{G}\) on pressure at fixed temperature and atThe procedure for relating \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\), and \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\) is a standard calculus operation. At fixed temperature, \[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}}\]This interesting equation shows that the differential dependence of Gibbs energy (at constant temperature) on pressure at constant affinity for spontaneous change does NOT equal the corresponding dependence at constant extent of chemical reaction. This inequality is not surprising. But our interest is drawn to the case where the system under discussion is, at fixed temperature and pressure, at thermodynamic equilibrium where \(\mathrm{A}\) is zero, \(\mathrm{d} \xi / \mathrm{dt}\) is zero, Gibbs energy is a minimum AND, significantly, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Hence \[\mathrm{V}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \mathrm{A}=0}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi^{\mathrm{eq}}}\]The dependence of \(\mathrm{G}\) on pressure for differential displacements at constant ‘\(\mathrm{A} = 0\)’ and \(\xi^{\mathrm{eq}}\) are identical. We confirm that the volume \(\mathrm{V}\) of a system is a ‘strong’ state variable. These comments seem trivial but the point is made if we go on to consider the volume of a system as a function of temperature at constant pressure. We use a calculus operation to derive equation (d). \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}}\]Again we are not surprised to discover that in general terms the differential dependence of \(\mathrm{V}\) on temperature at constant affinity does not equal the differential dependence of \(\mathrm{V}\) on temperature at constant composition/organization. Indeed, unlike the simplification we could use in connection with equation (b), {namely that at equilibrium \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero} we cannot assume that the volume of reaction, \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero at equilibrium. In other words for a closed system at thermodynamic equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) {when \(\mathrm{A}=0\), \(\xi = \xi^{\mathrm{eq}}\) and \(\mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}=0\)}, there are two thermal expansions, at constant \(\mathrm{A}\) and at constant \(\xi\)ξ.We consider a closed system in equilibrium state I defined by the set of variables,\(\left\{\mathrm{T}[\mathrm{I}], \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{I}]\right\}\). The equilibrium composition is \(\xi^{\mathrm{eq}}[\mathrm{I}]\) at zero affinity for spontaneous change. This system is perturbed to two nearby states at constant pressure.\[\Delta \mathrm{V}(\mathrm{A}=0)=\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}]\] At constant pressure we record the equilibrium thermal expansion; \[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left[\frac{\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}]}{\Delta \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}\] The equilibrium isobaric expansibility, \[\alpha_{\mathrm{p}}(\mathrm{A}=0)=\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0) / \mathrm{V}\] In order for the system to move from one equilibrium state, I with composition \(\xi^{\mathrm{eq}}[\mathrm{I}]\) to another equilibrium state, II with composition \(\xi^{\mathrm{eq}}[\mathrm{II}]\), the system changes by a change in chemical composition and/or molecular organization.Hence we define the ‘frozen’ isobaric expansion, \(\mathrm{E}_{\mathrm{p}} (\xi = \text { fixed})\). An alternative name is the instantaneous expansion because, practically, we would have to change the temperature at a such a high rate that there is no change in molecular composition or organisation in the system. \[\mathrm{E}_{\mathrm{p}}(\xi=\text { fixed })=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\]Further \[\alpha_{p}(\xi=\text { fixed })=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, \xi}\]Similar comments apply to isothermal compressibilities, \(\mathrm{K}_{\mathrm{T}}\); there are two limiting quantities \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) and \(\kappa_{\mathrm{T}}(\xi)\). In order to measure \(\kappa_{\mathrm{T}}(\xi)\) we have to change the pressure also in an infinitely short time.The entropy \(\mathrm{S}\) is given by the partial differential, \(-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \xi}\). At equilibrium where \(\mathrm{A}=0, \mathrm{~S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}=0}\). We carry over the argument described in the previous section but now concerned with a change in temperature. We consider the two pathways, constant \(\mathrm{A}\) and constant \(\xi\). \[\begin{aligned} &(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}}= \\ &\quad(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}, \xi}-(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \bar{\xi}} \,(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}} \end{aligned}\]But at equilibrium, \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero, and so \(\mathrm{S}(\mathrm{A}=0)\) equals \(\mathrm{S}\left(\xi^{\mathrm{eq}}\right)\). Then just as for volumes, the entropy of a system is not a property concerned with pathways between states; entropy is a strong function of state. Another important link involving Gibbs energy and temperature is provided by the Gibbs-Helmholtz equation. We explore the relationship between changes in \((\mathrm{G} / \mathrm{T})\) at constant affinity \(\mathrm{A}\) and at fixed \(\xi\), following perturbation by a change in temperature. \[\begin{aligned} &{[\partial(\mathrm{G} / \mathrm{T}) / \partial \mathrm{T}]_{\mathrm{p}, \mathrm{A}}=} \\ &{[\partial(\mathrm{G} / \mathrm{T}) / \partial \mathrm{T}]_{\mathrm{p}, \xi}-(1 / \mathrm{T}) \,(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \xi} \,(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}} \end{aligned}\]But at equilibrium, \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Then \(\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\). In other words, the variable enthalpy is another strong function of state. This is not the case for isobaric heat capacities. \[\begin{aligned} &(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}}= \\ &(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}-(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}} \,(\partial \mathrm{A} / \partial \mathrm{T})_{\mathrm{p}, \xi} \,(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}} \end{aligned}\]We cannot assume that the triple product term in the latter equation is zero. Hence, there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity \(C_{p}(A=0)\) and the frozen isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\xi \mathrm{eq})\). In other words, an isobaric heat capacity is not a strong function of state because it is concerned with a pathway between states. Unless otherwise stated, we use the symbol \(\mathrm{C}_{\mathrm{p}}\) to indicate an equilibrium transformation, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\).This page titled 1.14.56: Properties: Equilibrium and Frozen is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.57: Reversible Change

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In thermodynamics the term 'reversible' means that in such a system the affinity for spontaneous change \(\mathrm{A}\) is zero; we can in fact characterize the composition of the system by the symbol \(\xi^{\mathrm{eq}}\), indicating a time independent extent of chemical reaction. The composition of the system does not change because the affinity for spontaneous change is zero.For a reversible change the affinity for spontaneous change is zero at all stages. The composition is represented by \(\xi^{\mathrm{eq}}\), and the rate of change \(\mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}\) is zero, at defined \(\mathrm{T}\) and \(\mathrm{p}\). We represent the volume of the system using following equation. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]This equation means that the volume, a dependent variable, is unambiguously defined by the set of variables in the square brackets, [... ]. The pressure is changed from \(\mathrm{p}\) to \(\mathrm{p} + \Delta \mathrm{p}\), such that the new equilibrium composition is \(\xi + \Delta \xi\) where the affinity for spontaneous change is zero. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T},(\mathrm{p}+\Delta \mathrm{p}), \xi^{\mathrm{eq}}(\mathrm{p}+\Delta \mathrm{p}), \mathrm{A}=0\right]\]Under these circumstances the change from \(\mathrm{V}(\mathrm{p})\) to \(\mathrm{V}(\mathrm{p} + \Delta \mathrm{p})\) is from one equilibrium state where \(\mathrm{A} = 0\) to another equilibrium state where \(\mathrm{A}\) is also zero. Such an equilibrium transformation is, in thermodynamic terms, reversible. All changes under the constraint that \(\mathrm{A}\) remains at zero are reversible.This page titled 1.14.57: Reversible Change is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.58: Reversible Chemical Reactions

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Two important themes in thermodynamics concern the description of chemical equilibria and the kinetics of chemical reactions in closed systems at fixed temperature and pressure. These two themes are often linked in descriptions of chemical reactions. We comment on this link.A given aqueous solution (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), which is close to the standard pressure, \(\mathrm{p}^{0}\)) is prepared using chemical substance \(\mathrm{X}\). Spontaneous chemical reaction forms chemical substance \(\mathrm{Z}\) in the following chemical reaction. \[\mathrm{X}(\mathrm{aq}) \rightarrow \mathrm{Z}(\mathrm{aq})\]Experiment confirms that the extent of chemical reaction is given by equation (b). \[\frac{1}{\mathrm{~V}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}}=-\mathrm{k}_{1} \,[\mathrm{X}]\]In this system, \(\mathrm{c}_{\mathrm{x}}\{=[\mathrm{X}]\}=\mathrm{n}_{\mathrm{x}} / \mathrm{V}\). The common assumption is that for dilute solutions both \(\mathrm{k}_{1}\) and volume \(\mathrm{V}\) are independent of time. \[\mathrm{dc}_{\mathrm{x}} / \mathrm{dt}=-\mathrm{k}_{1} \, \mathrm{c}_{\mathrm{x}}\]Rate constant \(\mathrm{k}_{1}\) is expressed using the unit, \(\mathrm{s}^{-1}\). We consider a system prepared using chemical substance \(\mathrm{Z}\) which undergoes spontaneous chemical reaction to form chemical substance \(\mathrm{X}\). The analogue of equation (c) takes the following form.; \[\mathrm{dc}_{\mathrm{Z}} / \mathrm{dt}=-\mathrm{k}_{2} \, \mathrm{c}_{\mathrm{Z}}\]We assert that the chemical reaction described by equations (c) and (d) proceed until the properties of the system (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) are independent of time. In other words the system is in thermodynamic equilibrium with the surroundings with \(\mathbf{c}_{X}=\mathbf{c}_{X}^{e q}\) and \(\mathbf{c}_{Z}=\mathbf{c}_{Z}^{e q}\) where, macroscopically, \(\mathrm{dc}_{\mathrm{X}} / \mathrm{dt}\) and \(\mathrm{dc}_{\mathrm{Z}} / \mathrm{dt}\) are zero. Also \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{z}} \rightarrow 0 ; \mathrm{m}_{\mathrm{x}} \rightarrow 0\right) \gamma_{\mathrm{z}}=1\) and \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{x}} \rightarrow 0\right.\); and \(\left.\mathrm{m}_{\mathrm{z}}=0\right) \gamma_{\mathrm{x}}=1\) Thus \[\mathrm{k}_{1} \, \mathrm{c}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{k}_{2} \, \mathrm{c}_{\mathrm{Z}}^{\mathrm{eq}}\]From a thermodynamic point of view, at equilibrium (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) the affinity for spontaneous change \(\mathrm{A}\) is zero, and the system is at a minimum in Gibbs energy \(\mathrm{G}\). If the molalities of substances \(\mathrm{X}\) and \(\mathrm{Z}\) are \(\mathrm{m}_{\mathrm{X}}^{\mathrm{eq}}\) and \(\mathrm{m}_{\mathrm{Z}}^{\mathrm{eq}}\) respectively, the standard increase in Gibbs energy \(\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{~T}, \mathrm{p}, \mathrm{aq})\) is related to a (dimensionless) thermodynamic equilibrium constant \(\mathrm{K}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) using equation (f). \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln (\mathrm{K})=\mu_{\mathrm{Z}}^{0}(\mathrm{aq})-\mu_{\mathrm{x}}^{0}(\mathrm{aq})\]where, \[\mathrm{K}=\left(\mathrm{m}_{\mathrm{z}} \, \gamma_{\mathrm{z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} /\left(\mathrm{m}_{\mathrm{x}} \, \gamma_{\mathrm{x}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\]Here \(\gamma_{\mathrm{X}}\) and \(\gamma_{\mathrm{Z}}\) are the activity coefficients for solutes \(\mathrm{X}\) and \(\mathrm{Z}\). If the aqueous solution is quite dilute, we can assume that the thermodynamic properties of the solution are ideal. Moreover the ratio \(\left(m_{Z} / m_{X}\right)^{e q}\) is, based on the same approximation, equal to \(\left(c_{Z} / c_{X}\right)^{e q}\). In other words equations (e) and (g) can be written in the following forms. Law of Mass Action \[\left(c_{Z} / c_{X}\right)^{e q}=\left(k_{1} / k_{2}\right)^{e q}\]Thermodynamics \[\left(\mathrm{c}_{\mathrm{Z}} / \mathrm{c}_{\mathrm{X}}\right)^{\mathrm{eq}}=\mathrm{K}\]Within the context of the assumptions outlined above , we obtain by comparing equations (h) and (i) the following classic equation. \[\mathrm{K}=\mathrm{k}_{1} / \mathrm{k}_{2}\]Equation (j) is fascinating because the two sides of the equation have different origins, Law of Mass Action and the Laws of Thermodynamics. Indeed equation (j) is often used in an introduction to the concept of chemical equilibrium, the latter emerging as a ‘balance of rates of reaction’. In a wider context equation (j) is used in treatments of fast chemical reactions where a given closed system is only marginally displaced from equilibrium by transient changes in electric field, magnetic field, pressure or temperature. E. Caldin, Fast Reactions in Solution, Blackwell Scientific Publications, Oxford, 1964. M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press, London, 1973. The analysis takes a similar form in cases where the reaction stoichiometry is more complicated. Consider the case of an association reaction in aqueous solution. \[\mathrm{X}(\mathrm{aq})+\mathrm{Y}(\mathrm{aq}) \rightarrow \mathrm{Z}(\mathrm{aq})\]Law of Mass Action For the forward reaction \(\mathrm{dc}_{\mathrm{X}} / \mathrm{dt}=-\mathrm{k}_{1} \, \mathrm{c}_{\mathrm{X}} \, \mathrm{c}_{\mathrm{Y}}\) For the reverse reaction \[\mathrm{Z}(\mathrm{aq}) \rightarrow \mathrm{X}(\mathrm{aq})+\mathrm{Y}(\mathrm{aq})\]\[\mathrm{dc}_{\mathrm{Z}} / \mathrm{dt}=-\mathrm{k}_{2} \, \mathrm{c}_{\mathrm{z}}\]For a system where, macroscopically, \(\mathrm{dc}_{\mathrm{X}} / \mathrm{dt}=\mathrm{dc}_{\mathrm{Y}} / \mathrm{dt}=\mathrm{dc}_{\mathrm{Z}} / \mathrm{dt}=0\), \[\mathrm{k}_{1} \,\left(\mathrm{c}_{\mathrm{X}} \, \mathrm{c}_{\mathrm{Y}}\right)^{\mathrm{cq}}=\mathrm{k}_{2} \,\left(\mathrm{c}_{\mathrm{Z}}\right)^{\mathrm{eq}}\]Or, \[\mathrm{k}_{1} / \mathrm{k}_{2}=\left(\mathrm{c}_{\mathrm{Z}}\right)^{\mathrm{eq}} /\left(\mathrm{c}_{\mathrm{X}} \, \mathrm{c}_{\mathrm{Y}}\right)^{\mathrm{eq}}\]From a thermodynamic viewpoint, at equilibrium (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)), \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln (\mathrm{K})=\mu_{\mathrm{Z}}^{0}(\mathrm{aq})-\mu_{\mathrm{X}}^{0}(\mathrm{aq})-\mu_{\mathrm{Y}}^{0}(\mathrm{aq})\]where \[\mathrm{K}=\left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} /\left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} \,\left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\]In the limit that the solution is dilute, \(\left(\gamma_{\mathrm{Z}}\right)^{e q}=\left(\gamma_{\mathrm{x}}\right)^{\mathrm{eq}}=\left(\gamma_{\mathrm{Y}}\right)^{\mathrm{eq}}=1\). Then \[\mathrm{K}=\left(\mathrm{c}_{\mathrm{Z}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}} /\left(\mathrm{c}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}} \,\left(\mathrm{c}_{\mathrm{Y}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}\]Comparison of equations (e) and (h) allows identification of the ratio \(\mathrm{k}_{1} / \mathrm{k}_{2}\) with the equilibrium constant \(\mathrm{K}\).If at equilibrium \(\mathrm{dc}_{\mathrm{X}} / \mathrm{dt}=0\) and \(\mathbf{c}_{\mathrm{x}}^{\mathrm{cq}} \neq 0\) then according to equation (c), \(\mathrm{k}_{1}\) must be zero. The same problem arises from equation (d). To circumvent this objection, the Principle of Microscopic Reversibility states that at Equilibrium, the amount of chemical substance \(\mathrm{X}\) consumed by equation (a) and described by equation (c) equals the amount of chemical substance \(\mathrm{X}\) produced by the reverse reaction described by equation (d).This page titled 1.14.58: Reversible Chemical Reactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.59: Salting-In and Salting-Out

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A classic subject concerning the properties of salt solutions has centred for more than a century on the effects of an added salt on the solubility of an apolar (volatile) solute.In terms of the Phase Rule, a given closed system contains two phases, gas and liquid. The liquid phase is an aqueous salt solution. A volatile chemical substance is distributed between the vapour and liquid phases. Hence the number of phases \(\mathrm{P}\) equals 2; the number of components \(\mathrm{C}\) equals 3; i.e. water + salt + volatile chemical substance. Hence the number of degrees of freedom \(\mathrm{F}\) equals 3. If therefore we define the temperature, pressure and concentration of salt in the aqueous salt solution, the thermodynamic equilibrium is completely defined.Similarly if the closed system contains pure liquid \(j\) and an aqueous salt solution which also contains solute \(j\), the number of degrees of freedom is again 3. Then the equilibrium state is completely defined by specifying \(\mathrm{T}, \mathrm{~p}\) and the concentration (molality) of salt in solution. In this case the equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\) is defined by equation \ref{a}.\[\mu_{\mathrm{j}}^{*}(\ell)=\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\label{a}\]In the absence of salt, treating substance \(j\) as a solute in aqueous solution, equation \ref{b} describes this equilibrium in terms of the equilibrium composition of the solution assuming ambient pressure is close to the standard pressure.\[\mu_{\mathrm{j}}^{*}(\ell)=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)_{\mathrm{aq}}^{\mathrm{cq}}\label{b}\]A similar equilibrium is established but this time the aqueous solution contains a salt, molality \(\mathrm{m}_{\mathrm{s}}\). Equation \ref{b} takes the following form.\[\mu_{\mathrm{j}}^{*}(\ell)=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)_{\mathrm{s}}^{\mathrm{\alpha q}}\label{c}\]The subscript on the last term in equation \ref{c} indicates that the aqueous solution contains salt \(\mathrm{S}\) as well as apolar solute \(j\). According to equations \ref{b} and \ref{c}, the two solubilities of substance \(j\) are related.\[\left(m_{j} \, \gamma_{j}\right)_{a q}^{e q}=\left(m_{j} \, \gamma_{j}\right)_{s}^{e q}\label{d}\]Equation \ref{d} is thermodynamically correct. The change in solubility of chemical substance \(j\) on adding salt \(\mathrm{S}\), molality ms is compensated by a change in the activity coefficient of solute \(j\). The corresponding equation on the concentration scale has the form shown in equation \ref{e}.\[\left(c_{j} \, y_{j}\right)_{a q}^{e q}=\left(c_{j} \, y_{j}\right)_{s}^{e q}\label{e}\]The latter is the usual form of the equation. The analysis is readily repeated for the case where the chemical substance \(j\) is a volatile gas.At this stage a number of extra-thermodynamic assumptions are built into the analysis. To reduce the clutter of symbols we drop the designation ‘\(\mathrm{eq}\)’ taking this condition as implicit in all that follows. Further we assume that substance \(j\) is sparingly soluble so that in the aqueous solution \(j-j\) solute-solute interactions are unimportant. Therefore the properties of solute \(j\) are ideal; \(\left(\mathrm{y}_{\mathrm{j}}\right)_{\mathrm{aq}}=1\). Hence from equation \ref{e},\[\ln \left[\left(c_{j}\right)_{\mathrm{aq}} /\left(\mathrm{c}_{\mathrm{j}}\right)_{\mathrm{s}}\right]=\ln \left[\left(\mathrm{y}_{\mathrm{j}}\right)_{\mathrm{s}}\right]\label{f}\]In these terms \(\left(\mathrm{y}_{j}\right)_{\mathrm{s}}\) is the activity coefficient of solute \(j\) in the aqueous salt solutions where the concentration of salt is represented by \(\mathrm{c}_{\mathrm{s}}\). For dilute salt solutions the assumption is made that \(\ln \left[\left(\mathrm{y}_{\mathrm{j}}\right)_{\mathrm{s}}\right]\) is a linear function of \(\mathrm{c}_{\mathrm{s}}\).\[\ln \left[\left(\mathrm{y}_{\mathrm{j}}\right)_{\mathrm{s}}\right]=\mathrm{k} \, \mathrm{c}_{\mathrm{s}}\label{g}\]Combination of equations \ref{f} and \ref{g} yields the following equation.\[\ln \left[\left(c_{j}\right)_{a q} /\left(c_{j}\right)_{s}\right]=k \, c_{s}\label{h}\]Equation \ref{h} is one form of the Setchenow equation in which constant \(\mathrm{k}\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is characteristic of salt \(\mathrm{S}\) and solute \(j\). An alternative form starts by expressing \(\left(c_{j}\right)_{s}\) as \(\left(c_{j}\right)_{a q}-\delta c_{j}\) implying a reduction in the solubility of solute \(j\) when a salt is added; i.e. a salting-out. Hence,\[\delta \mathrm{c}_{\mathrm{j}} /\left(\mathrm{c}_{\mathrm{j}}\right)_{\mathrm{aq}}=\mathrm{k} \, \mathrm{c}_{\mathrm{s}}\label{i}\]This Setchenow Equation requires that \(\delta \mathrm{c}_{\mathrm{j}} /\left(\mathrm{c}_{\mathrm{j}}\right)_{\mathrm{aq}}\) is a linear function of \(\mathrm{c}_{\mathrm{s}}\). A positive \(\mathrm{k}\) describes a salting-out; a negative \(\mathrm{k}\) describes a salting-in. The phenomenon by which solubilities of gases in aqueous solutions are changed by adding a salt attracts enormous interest, both from practical and theoretical standpoints. Conway reviewed theoretical models which attempt to account quantitatively for the phenomenon. Considerable attention has been given to theories based on the relationship between the impact of the non-polar solute on the dielectric properties of the solvent and hence the chemical potential of the salt in solution.For the most part salting-out is the commonly observed pattern. Nevertheless there are some interesting cases where apolar solutes are salted-in by tetra-alkylammonium salts; benzene , methane and helium in \(\mathrm{Bu}_{4}\mathrm{N}^{+} \mathrm{~Br}^{-} (\mathrm{aq})\). It would appear that an added apolar solute is stabilized by interaction with the apolar alkyl groups of the cations. J.Setchenow, Z. Phys. Chem.,1889,4,117.\[\begin{aligned} &\ln \left[\frac{\left(c_{j}\right)_{a q}}{\left(c_{j}\right)_{s}}\right]=\ln \left[\frac{\left(c_{j}\right)_{a q}}{\left(c_{j}\right)_{a q}-\delta c_{j}}\right]=-\ln \left[\frac{\left(c_{j}\right)_{\mathrm{aq}}-\delta c_{j}}{\left(c_{j}\right)_{\mathrm{aq}}}\right] \\ &\quad-\ln \left[1-\frac{\delta c_{j}}{\left(c_{j}\right)_{\mathrm{aq}}}\right] \equiv \frac{\delta c_{j}}{\left(c_{j}\right)_{\mathrm{aq}}} \end{aligned}\] F. A. Long and W. F. McDevit, Chem. Rev.,1952,51,119. For a review of the definitions of units used in this subject area see H. L. Clever, J. Chem. Eng. Data, 1983,28,340. B. E. Conway, Pure Appl. Chem.,1985,57,263. P. Debye and J. MacAulay, Z. Phys. Chem.,1925,26,22. B. E. Conway, J. E. Desnoyers and A. C. Smith, Philos. Trans. R. Soc.,A 1964, 256A, 389. Benzene in \(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~Br}^{-} (\mathrm{aq})\);J. E. Desnoyers, G. E. Pelletier and C. Jolicoeur, Can. J. Chem.,1965,43,3232. Benzene in R4NBr(aq); H. E. Wirth and A. LoSurdo, J. Phys. Chem.,1968,72,751. RH in \(\mathrm{R}_{4}\mathrm{NBr}(\mathrm{aq})\);W.-Y. Wen and J. H. Hung, J.Phys.Chem.,1970,74,170. A. Feillolay and M. Lucas, J. Phys. Chem.,1972,76,3068. C. Treiner and A. K. Chattopadhyay, J. Chem. Soc Faraday Trans. 1, 1983,79,2915.This page titled 1.14.59: Salting-In and Salting-Out is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.6: Extent of Reaction - General

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The variable \(\xi\) describes in quite general terms the molecular composition/organisation. For a closed system at fixed \(\mathrm{T}\) and \(\mathrm{p}\), there is a composition/organisation \(\xi^{\mathrm{eq}}\) corresponding to a minimum in Gibbs energy where the affinity for spontaneous change is zero. In general terms there is an extent of reaction \(\xi\) corresponding to a given affinity \(\mathrm{A}\) at defined \(\mathrm{T}\) and \(\mathrm{p}\). In fact we can express \(\xi\) as a dependent variable defined by the independent variables \(\mathrm{T}\), \(\mathrm{p}\), and \(\mathrm{A}\). Thus \[\xi=\xi[\mathrm{T}, \mathrm{p}, \mathrm{A}]\]The general differential takes the following form. \[\mathrm{d} \xi=\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}\]The change in chemical composition occurs spontaneously. The change in composition is described in terms of the extent of chemical reaction, \(\xi\). In a given aqueous solution, the chemical reaction is: \[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})\]At each stage the extent of chemical reaction is represented by the symbol \(\xi\). \[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})\]At \(t = 0\) (i.e. as prepared) \[n(\text { ester })^{0} \quad n\left(\mathrm{OH}^{-}\right)^{0} \quad 0 \quad 0\]After extent of reaction \(\xi\) (at some time later) \[n(\text { ester })^{0} - \xi \quad n\left(\mathrm{OH}^{-}\right)^{0} -\xi \quad \xi \quad \xi\]As the reaction proceeds so \(\xi\) increases. [NB The zero superscript signals ‘at time zero’.] At a given stage of the reaction and time \(t\), (accepting \(\mathrm{dt}\) is positive), Rate of Reaction = \(\mathrm{d}\xi / \mathrm{dt}\)We now ask ‘why did chemical reaction proceed in this direction?’. The answer is ---- the chemical reaction was driven by the affinity for spontaneous change, symbol \(\mathrm{A}\). By identifying these two ideas, affinity for spontaneous change and the rate of reaction \(\mathrm{d}\xi / \mathrm{dt}\), we arrive at two important criteria for chemical equilibrium. Affinity for spontaneous change \(\mathrm{A} = 0\) Rate of change \(\mathrm{d}\xi / \mathrm{dt} = 0\)However we need to stand back a little and examine how we might advance generalizations concerning the direction of Spontaneous Chemical Reaction. What macroscopic property can be identified which accounts for the fact that alkaline hydrolysis of ethyl ethanoate is spontaneous? To make further progress we introduce two laws of thermodynamics. Actually these are not laws in the sense of being laid down by government or by religious doctrine. Rather these laws are AXIOMS. We explore these axioms in the context for which ξ refers to a change in composition resulting from chemical reaction.Footnotes For a discussion of the significance of extent of reaction \(\xi\), see: The usefulness of the concept of extent of chemical reaction \(\xi\) is further illustrated by the following examples.This page titled 1.14.6: Extent of Reaction - General is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.60: Second Law of Thermodynamics

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The Second Law introduces an extensive function of state, a property of a given system, called the entropy, symbol \(\mathrm{S}\).Spontaneous chemical reaction in a closed system is driven by the affinity for spontaneous change \(\mathrm{A}\) producing a change in chemical composition \(\xi\). The change in entropy \(\mathrm{dS}\) at temperature \(\mathrm{T}\) is given by Equation \ref{a}.\[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi \label{a}\]where\[\mathrm{A} \, \mathrm{d} \xi>0 \label{b}\]The latter inequality is the LAW. This inequality is the key to chemistry. In effect the law states that if there is an affinity for a given chemical reaction ( i.e. a driving ‘force’ for reaction) the chemical reaction will spontaneously proceed in that direction. This is the thermodynamic selection rule for which there are no exceptions.In the limit that a system undergoes a ‘reversible ‘ change, \(\mathrm{A}\) is zero; the system is at equilibrium with the surroundings. For a reversible change\[\mathrm{T} \, \mathrm{d} \mathrm{S}=\mathrm{q} \label{c}\]Often texts seek to answer the question ‘what is entropy?’ This is a fruitless task unless one draws attention to Equation \ref{c} which reminds us that the product \(\mathrm{T} \, \mathrm{dS}\) is in fact a thermal energy. Chemists are familiar with spontaneous chemical reactions and Equations \ref{a} and \ref{b} present no conceptual problems. Robert Park, Voodoo Science, Oxford,2000. From page 7; ‘The first law says you can’t win; the second law says you can’t even break even’. This comment is with respect to fraudulent claims of discoveries of perpetual motion machines.This page titled 1.14.60: Second Law of Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.61: Solubility Products

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A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient) contains an aqueous solution of a sparingly soluble salt \(\mathrm{MX}\); e.g. \(\mathrm{AgCl}\). The system also contains solid salt \(\mathrm{MX}\). When a soluble salt (e.g. \(\mathrm{KNO}_{3}\)) is added the solubility of salt \(\mathrm{MX}\) increases. This remarkable observation is readily accounted for. The equilibrium involving the sparingly soluble salt is represented as follows.We represent the salt \(\mathrm{MX}\) by the symbol \(j\). At equilibrium, \[\mu_{\mathrm{j}}^{\prime \prime}(\mathrm{s})=\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\]In terms of the solubility \(\mathrm{S}_{j}\) of the salt \(\mathrm{MX}\), a 1:1 salt, \[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{\mathrm{o}}\right)\]By definition \[\Delta_{\text {sol }} G^{0}=-R \, T \, \ln K_{s}=\mu_{j}^{0}(a q)-\mu_{j}^{*}(s)\]\(\mathrm{K}_{\mathrm{S}}\) is the solubility product, a characteristic property of salt \(\mathrm{MX}\) (at defined \(\mathrm{T}\) and \(\mathrm{p}\)). \[\mathrm{K}_{\mathrm{S}}=\left[\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]^{2}\]Or, \[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)-\ln \left(\gamma_{\pm}\right)\]According to the DHLL, \[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\]\(\mathrm{I}\) is the ionic strength of the solution which can be changed by adding a soluble salt. From equations (e) and (f), \[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)+\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\]The key point to note is the positive sign in equation (g) showing that the theory accounts for the observed salting–in of the sparingly soluble salt. Further a plot of \(\ln \left(S_{j} / m^{0}\right)\) against \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) is linear yielding an estimate for \(\mathrm{K}_{\mathrm{S}}\) from the intercept.This page titled 1.14.61: Solubility Products is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.62: Solubilities of Gases in Liquids

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Comparison of the solubilities of volatile chemical substance \(j\) in liquids \(\ell_{1}\) and \(\ell_{2}\) yields an estimate of the difference in reference chemical potentials, \(\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~T})\).This is a classic subject with two consequences.A closed system contains two phases, liquid and gaseous, at temperature \(\mathrm{T}\). The liquid is water; a sparingly soluble chemical substance \(j\) exists in both gas and liquid phases. A phase equilibrium is established for substance \(j\) in the two phases. In terms of the Phase Rule, there are two phases and two components. Hence there are two degrees of freedom. If the temperature and pressure are defined, the compositions of the two phases are fixed. In terms of chemical potentials with reference to substance \(j\) the following condition holds. \[\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}, \mathrm{p}, \mathrm{T}\right)=\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{p}_{\mathrm{j}}, \mathrm{T}\right)\]Here \(\mathrm{p}_{j}\) is the equilibrium partial pressure of substance \(j\) in the gas phase at pressure \(\mathrm{p}\) where pressure \(\mathrm{p}\) equals \(\left(\mathrm{p}_{j} + \mathrm{p}_{1}\right)^{\mathrm{eq}}\) where \(\mathrm{p}_{1}\) is the equilibrium partial pressure of water in the vapor phase. Equation (a) establishes the thermodynamic basis of the phenomenon discussed here. However historical and practical developments resulted in quite different approaches to the description of the solubilities of gases in liquids. The thermodynamic treatment is not straightforward if we recognize that the thermodynamic properties of the vapor (i.e. gas phase) and the solution are not ideal. When both the solubility and the partial pressure of the ‘solute’ in the gas phase are low, the assumption is often made that the thermodynamic properties of gas and solution are ideal. Then the analysis of solubility is reasonably straightforward. In a sophisticated analysis, account must be taken of the intermolecular interactions in the vapor phase and solute-solute interactions in solution. We review the basis of analyses where the thermodynamic properties of gas and liquid phases are ideal. Nevertheless equation (a) is the common starting point for the analysis.By definition, the Bunsen Coefficient \(\alpha\) is the volume of a gas at \(273.15 \mathrm{~K}\) and standard pressure \(\mathrm{p}^{0}\) which dissolves in unit volume of a solvent when the partial pressure of the gas equals \(\mathrm{p}^{0}\). Experiment yields the volume \(\mathrm{V}_{j}(\mathrm{g})\) of gas \(j\) at temperature \(\mathrm{T}\) and partial pressure \(\mathrm{p}_{j}\) absorbed by volume \(\mathrm{V}_{\mathrm{s}}\) of solvent at temperature \(\mathrm{T}\). The volume of gas at \(273.15 \mathrm{~K}\) and standard pressure \(\mathrm{p}^{0}\), \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)\) is given by equation (b) assuming that gas \(j\) has the properties of a perfect gas. \[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)=\left[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right) \, 273.15 \mathrm{~K} / \mathrm{p}^{0} \, \mathrm{T}\right]\]Hence experiment yields the ratio, \(\left[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right) / \mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})\right]\). By simple proportion we obtain the volume, \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)\) in the event that the gas \(j\) was at pressure \(\mathrm{p}^{0}\) above the liquid phase. Bunsen coefficient, \[\alpha=\frac{\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right)}{\mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})} \,\left(\frac{\mathrm{p}_{\mathrm{j}}}{\mathrm{p}^{0}}\right) \, \frac{273.15 \mathrm{~K}}{\mathrm{~T}}\]At ambient pressure, if the partial pressure of the solvent is negligibly small, \[\alpha=\frac{\mathrm{V}_{\mathrm{j}}(\mathrm{g} ; \mathrm{T} ; \mathrm{p})}{\mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})} \,\left(\frac{273.15 \mathrm{~K}}{\mathrm{~T}}\right)\]The assumption that substance \(j\) is a perfect gas can be debated but the correction is often less that 1%.A given closed system comprises gaseous and liquid phases, at temperature \(\mathrm{T}\). The system is at equilibrium such that equation (a) holds. We assume that the thermodynamic properties of the solution and the gas phase are ideal and that an equilibrium exists for chemical substance \(j\) between the two phases. The solution is at pressure \(\mathrm{p}^{0}\), standard pressure which is close to ambient pressure. \[\begin{aligned} \mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, & \ln \left(\mathrm{p}_{\mathrm{e}}^{\mathrm{eq}} / \mathrm{p}^{0}\right) \\ &=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \end{aligned}\]For a perfect gas, \[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{g})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]For a solution, \[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\]In these terms \(\mathrm{V}(\mathrm{aq})\) is the volume of solution which dissolves \(\mathrm{n}_{j}\) moles of chemical substance \(j\) from the gas phase. \[\begin{aligned} \mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{1}{\mathrm{p}^{0}}\right) \\ &=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})} \, \frac{1}{\mathrm{c}_{\mathrm{r}}}\right) \end{aligned}\]The Ostwald Coefficient \(\mathrm{L}\) is defined in terms of reference chemical potentials of substance \(j\) in solution and gas phase. \[\Delta_{s \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=-\mathrm{R} \, \mathrm{T} \, \ln (\mathrm{L})=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)\]From equation (h), \[\Delta_{\mathrm{s} \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{V}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{n}_{\mathrm{j}} \, \mathrm{p}^{0}}\right]\]\[\Delta_{s \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{V}(\mathrm{aq})}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{R} \, \mathrm{T} \, \mathrm{c}_{\mathrm{r}}}{\mathrm{p}^{0}}\right]\]Because \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{g})\) are expressed in the same units, the Oswald coefficient is dimensionless. The key assumption is that the thermodynamic properties of gas and solution phases are ideal.Ostwald coefficients can be defined in several ways. In the analysis set out above we refer to the volume of the solution containing solvent and solute \(j\). Another definition refers to the volume of pure liquid which dissolves a volume of gas \(\mathrm{V}_{j}\). \[\mathrm{L}^{0}=\left[\mathrm{V}_{\mathrm{g}} / \mathrm{V}^{*}(\ell)\right]^{\mathrm{eq}}\]A third definition refers to ratio of concentrations of substances \(j\) in liquid and gas phases. \[\mathrm{L}_{\mathrm{c}}=\left[\mathrm{c}_{\mathrm{j}}^{\mathrm{L}} / \mathrm{c}_{\mathrm{j}}^{\mathrm{V}}\right]^{\mathrm{eq}}\]In effect an Oswald coefficient describes an equilibrium distribution for a volatile solute between gas phase and solution. The Oswald coefficient is related to the (equilibrium) mole fraction of dissolved gas, \(x_{j}\) using equation (n) where \(\mathrm{p}_{j}\) is the partial pressure of chemical substance \(j\) and \(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of the solvent. \[\mathrm{x}_{2}=\left\{\left[\mathrm{R} \, \mathrm{T} / \mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)\right]+1\right\}^{-1}\]In a given closed system at temperature \(\mathrm{T}\), gas and solution phases are in equilibrium. The thermodynamic properties of both phases are ideal. Then according to Henry’s Law, the partial pressure of volatile solute \(j\) is a linear function of the concentration \(\mathrm{c}_{j}\) at fixed temperature. \[\mathrm{p}_{\mathrm{j}}(\operatorname{vap})=\mathrm{K}_{\mathrm{c}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\]\(\mathrm{K}_{\mathrm{c}}\) is the Henry’s Law constant (on the concentration scale), characteristic of solvent, solute and temperature. Similarly on the mole fraction scale, \[\mathrm{p}_{\mathrm{j}}(\operatorname{vap})=\mathrm{K}_{\mathrm{x}} \, \mathrm{x}_{\mathrm{j}}\]This subject, gas solubilities, is enormously important. We draw attention to some interesting reports concerning solubilities with particular reference to aqueous solutions and the environment.Footnote R. Battino and H. L. Clever, Chem.Rev.,1966,66,395. E. Wilhelm and R. Battino, Chem.Rev.,1973,73,1. E. Wilhelm, R. Battino and R. J. Wilcock, Chem.Rev.,1977,77, 219. R. Battino, Fluid Phase Equilib.,1984,15,231. E. Wilhelm, Pure Appl.Chem.,1985,57,303. E. Wilhelm, Fluid Phase Equilib.,1986,27,233 E. Wilhelm, Thermochim. Acta,1990,162,43. R.Fernandez-Prini and R. Crovetto, J. Phys. Chem. Ref.Data,1989,18,1231. R. Battino, T. R. Rettich and T. Tominaga, J. Phys. Chem. Ref. Data, 1984,13,563. T. R. Rettich, Y. P. Handa, R. Battino and E. Wilhelm, J. Phys. Chem.,1981,85,3230. \[\begin{aligned} &\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{V}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{n}_{\mathrm{j}} \, \mathrm{p}^{0}}=\frac{[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}{\left[\mathrm{m}^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]}{[\mathrm{mol}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]} \\ &=\frac{[\mathrm{N} \mathrm{m}]}{\left[\mathrm{m}^{3}\right]} \, \frac{1}{\left[\mathrm{~N} \mathrm{\textrm {m } ^ { - 2 } ]}\right.}= \end{aligned}\]For a perfect gas \(j\), \[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{g})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]\[V_{j}(g)=n_{j} \, R \, T / p_{j}\]For \(\mathrm{n}_{1}\) moles of liquid 1, density \(\rho_{1}^{*}(\ell)\), \[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)=\mathrm{n}_{1} \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell)\]By definition, at temperature \(\mathrm{T}\), \[\mathrm{L}=\mathrm{V}_{\mathrm{j}}(\mathrm{g}) / \mathrm{V}_{1}^{*}(\ell)\]\[\mathrm{L}=\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{p}_{\mathrm{j}}} \, \frac{\rho_{\mathrm{l}}^{*}(\ell)}{\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{l}}}\]Hence, \[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{L} \, \mathrm{p}_{\mathrm{j}}} \, \frac{\rho_{1}^{*}(\ell)}{\mathrm{M}_{1}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)}\]But mole fraction of solute \(j\) in solution, \[\mathrm{x}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}}=\frac{1}{\left(\mathrm{n}_{\mathrm{l}} / \mathrm{n}_{\mathrm{j}}\right)+1}\]From equations (f) and (g), \[\mathrm{x}_{\mathrm{j}}=\left[\left\{\mathrm{R} \, \mathrm{T} / \mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)\right\}+1\right]^{-1}\]This page titled 1.14.62: Solubilities of Gases in Liquids is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.63: Solubilities of Solids in Liquids

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This very large subject can be divided into two groups. The first group concerns the solubility of a given solid substance \(j\) in a given solvent, liquid \(\ell_{1}\). The second group involves comparison of the solubilities of a given solid in two liquids, \(\ell_{1}\) and \(\ell_{2}\).A closed system (at defined \(\mathrm{T}\) and \(\mathrm{p}\), the latter being close to the standard pressure) contains solid substance \(j\) in equilibrium with an aqueous solution containing solute \(j\). The system is characterized by the (equilibrium) solubility, \(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\). At equilibrium, \[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right]\]Then \[\Delta \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{*}(\mathrm{~s})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right]\]If the aqueous solution is dilute and the solubility is low, it can often be assumed that the properties of the solution are ideal. Hence, \[\Delta \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{\mathrm{*}}(\mathrm{s})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right]\]It should be noted that the sign of \(\Delta \mu_{j}^{0}\) depends on whether or not \(m_{j}^{e q}(a q)\) is larger or less than unity.We illustrate the second approach by considering a combination of the experiment described above and an experiment where the solvent is a binary aqueous mixture, mole fraction composition \(\mathrm{x}_{2}\). At equilibrium, \[\mu_{j}^{*}(s)=\mu_{j}^{0}\left(s \ln ; x_{2}\right)+R \, T \, \ln \left[m_{j}^{\mathrm{eq}}\left(s \ln ; x_{2}\right) \, \gamma_{j}^{\mathrm{eq}}\left(s \ln ; x_{2}\right) / m^{0}\right]\]\[\begin{aligned} \Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{x}_{2}\right)-\mu_{\mathrm{j}}^{0}(\mathrm{aq}) \\ =-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right] \end{aligned}\]If both solutions are dilute in substance \(j\), the ratio, \(\gamma_{j}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\) can be assumed to be close to unity. In fact this is a better approximation than assuming both activity coefficients are unity. Then \[\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right]\]In other words if the solubility of substance \(j\) increases on adding solvent component 2 then \(\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{\mathrm{c}}\) is negative. This stabilization is a consequence of a difference in solute-solvent interactions.This page titled 1.14.63: Solubilities of Solids in Liquids is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.64: Solutions: Solute and Solvent

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A given solution (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), where the latter is close to the standard pressure) is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{j}\) moles of a simple solute. The Gibbs energy \(\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equation (a). \[\begin{aligned} \mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]In the event that the thermodynamic properties of the solution are ideal, \[\begin{aligned} \mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]The excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) for the solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) is given by equation (c). \[\mathrm{G}^{\mathrm{E}}=\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)\]Therefore \[\mathrm{G}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Hence at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the dependence of \(\mathrm{G}^{\mathrm{E}}\) on \(\mathrm{m}_{j}\) is given by equation (e). \[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}=1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{j}_{\mathrm{j}}\]According to the Gibbs-Duhem equation, the chemicals potentials of solvent \(\mu_{1}(\mathrm{aq})\) and solute \(\mu_{j}(\mathrm{aq})\) are linked. At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0\]Then for a solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)), \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0\]In terms of the impact of adding \(\mathrm{dm}_{j}\) moles of solute, \[\left(1 / M_{1}\right) \, d \mu_{1}(a q) / d m_{j}+m_{j} \, d \mu_{j}(a q) / d m_{j}=0\]The Gibbs-Duhem relation describes moderation of the effects of added \(\mathrm{dm}_{j}\) moles of solute \(j\) on the changes in chemical potentials of solute and solute. We use the equation which relates these chemical potentials to the composition of the solution. For simple solutes (e.g. urea) at ambient pressure, equation (g) takes the following form. \[\begin{aligned} &{\left[1 / \mathrm{M}_{1}\right] \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]} \\ &\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right]=0\right. \end{aligned\]Hence, \[\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right]=0\right.\]The simple differential equation (j) is important in developing links between the thermodynamic properties of solutions, solvent and solute. The integrated form of this equation is important. From equation (j), \[\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=0\]Therefore, \[-\phi \, d m_{j}-m_{j} \, d \phi+d m_{j}+m_{j} \, d \ln \left(\gamma_{j}\right)=0\]Or, \[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1) \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi\]Or, with a slight re-arrangement, \[d \ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{d} \phi+(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]Hence we obtain an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the dependence of \((\phi - 1)\) on molality of solute bearing in mind that \(\ln \left(\gamma_{j}\right)\) equals zero and \(\phi\) equals 1 at \(\mathrm{m}_{j} = 0\). \[\ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1)+\int_{\mathrm{o}}^{\mathrm{m}_{\mathrm{j}}}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]In another approach we start again with equation (j). \[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Or, \[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}}\right)-\ln \left(\mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Or, \[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{dm}_{\mathrm{j}} \,+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Following integration from ‘\(\mathrm{m}_{j} =0\)’ to \(\mathrm{m}_{j}\), \[\phi \, \mathrm{m}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}}+\int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]\[\phi=1+\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]\[\phi-1=\left(1 / m_{j}\right) \, \int_{0}^{m_{j}} m_{j} \, d \ln \left(\gamma_{j}\right)\]In other words \((\phi - 1)\) is related to the integral of \(m_{j} \, d \ln \left(\gamma_{j}\right)\) between the limits ‘\(\mathrm{m}_{j} = 0\)’ and \(\mathrm{m}_{j}\). Equation (e) can be re-expressed as an equation of \(\ln \left(\gamma_{\mathrm{j}}\right)\). \[\ln \left(\gamma_{\mathrm{j}}\right)=-(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}+(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}{ }^{\mathrm{E}} / \mathrm{dm} \mathrm{j}_{\mathrm{j}}\]Hence from equation (r), \[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\phi \, \mathrm{dm} \mathrm{j}_{\mathrm{j}}=\mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Or, \[\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\phi-1-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0\]Or, \[-(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0\]Then with reference to equation (v), \[\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\]Combination of equations (z) and (d) yields an equation for \((1 - \phi)\) in terms of \(\mathrm{G}^{\mathrm{E}}\). Thus \[\mathrm{G}^{\mathrm{E}} / \mathrm{R} \, \mathrm{T}=\mathrm{m}_{\mathrm{j}} \,(1-\phi)+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{R} \, \mathrm{T}\right) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\]Or, \[(1-\phi)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}-\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]\]Or, \[(1-\phi)=-\left(\mathrm{m}_{\mathrm{j}} / \mathrm{R} \, \mathrm{T}\right) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\}\] \[\mathrm{G}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\] \[\ln \left(\gamma_{\mathrm{j}}\right)=\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]} \, \frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}=\] \[(1-\phi)=\left[\frac{}{\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}\right]=\]This page titled 1.14.64: Solutions: Solute and Solvent is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.65: Spontaneous Change: Isothermal and Isobaric

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.65%3A_Spontaneous_Change%3A_Isothermal_and_Isobaric

By definition, \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]\(\mathrm{G}, \mathrm{~H} \text { and } \mathrm{S}\) are extensive functions of state. At fixed \(\mathrm{T}\) and \(\mathrm{p}\), the dependences of these variables on extent of reaction, \(\xi\) are related. \[(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}-\mathrm{T} \,(\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\]For a spontaneous change \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\) where the affinity for spontaneous change \(\mathrm{A}\left[=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right]\) is positive. This can arise under two limiting circumstances.If for a given possible process, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}>0\), then the process is not spontaneous; there is no affinity for spontaneous change. If \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero at defined \(\mathrm{T}\) and \(\mathrm{p}\), the system is at equilibrium with the surroundings; the affinity for spontaneous change is zero. The chemical equilibrium is stable if \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is negative.This page titled 1.14.65: Spontaneous Change: Isothermal and Isobaric is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.66: Spontaneous Chemical Reaction

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.66%3A_Spontaneous_Chemical_Reaction

A closed reaction vessel at \(298.2 \mathrm{~K}\) and \(101325 \mathrm{~N m}^{-2}\) is filled with a solution having the initial composition, water (\(1.2 \mathrm{~mol}\)), \(\mathrm{NaOH}(\mathrm{aq}, 0.5 \mathrm{~mol})\), \(\mathrm{CH}_{3}.\mathrm{COOC}_{2}\mathrm{H}_{5}(\mathrm{aq}, 0.2 \mathrm{~mol})\). Experiment shows that the system spontaneously changes composition. We write the overall chemical reaction as: \[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}+\mathrm{OH}^{-} \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\]In this connection we state that the reaction is driven by the affinity for spontaneous reaction leading to a change in chemical composition characterized by the extent of reaction, \(\xi\).This page titled 1.14.66: Spontaneous Chemical Reaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.68: Surroundings and System

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.68%3A_Surroundings_and_System

The word ‘system’ describes that part of the universe which we have identified for the purpose of studying its chemical properties. The term "universe" in the latter sentence is somewhat pretentious (= implying ‘of enormous importance’ in a way that is doubtful). From the practical point of view, a chemist identifies the system as the contents of the reaction vessel (flask) under investigation.The rest of the universe comprises the surroundings. We as observers of the properties plus all our measuring equipment including spectrophotometers and calorimeters are part of the surroundings. As far as chemists are concerned the surroundings mean the laboratory (+ chemist!) surrounding the system.This page titled 1.14.68: Surroundings and System is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.69: Temperature of Maximum Density: Aqueous Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.24%3A_Misc/1.14.69%3A_Temperature_of_Maximum_Density%3A_Aqueous_Solutions

At ambient pressure, the molar volume of water (\(\ell\)) is a minimum near \(277 \mathrm{~K}\), the temperature of maximum density, \(\mathrm{TMD}\). The \(\mathrm{TMD}\) is sensitive to the concentration and nature of added solute. Generally added salts lower the \(\mathrm{TMD}\), the extent of lowering being often written \(\Delta \theta\). For dilute salt solutions, \(\mathrm{TMD}\) is a linear function of the molality of salt, \(\mathrm{m}_{j}\),\(\left(\partial \Delta \theta / \partial m_{\mathrm{j}}\right)\) being negative; Despretz Law. However considerable interest is generated by the observation that some organic solutes at low mole fractions raise the \(\mathrm{TMD}\); i.e. \(\Delta \theta > 0\); e.g. 2-methylpropan-2-ol.Although the phenomenon of a shift in \(\mathrm{TMD}\) is straightforward from an experimental standpoint, explanations distinguish between possible contributions to the shift in \(\mathrm{TMD}\). Most treatments identify two contributions to the shift in \(\mathrm{TMD}\), an ‘ideal’ shift and a contribution which reflects the fact that the thermodynamic properties of the aqueous system are not ideal.At the outset we assume that the molar volume of water at ambient pressure in the region of the \(\mathrm{TMD}\) is a quadratic function of the difference (\(\mathrm{T}-\mathrm{TMD}^{*}\)) where \(\mathrm{TMD}^{*}\) is the temperature of maximum density of water (\(\ell\)). At temperature \(\mathrm{T}\) the molar volume \(\mathrm{V}_{1}^{*}(\ell, \mathrm{T})\) is given by equation (a) where \(\chi_{1}\) is a dimensionless property of water (\(\ell\)). \[\mathrm{V}_{1}^{*}(\ell, \mathrm{T})=\mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\}\]We consider briefly three types of systems;For the non-aqueous component, the dependence of molar volume \(\mathrm{V}_{2}^{*}(\ell, \mathrm{T})\) on temperature is given by equation (b) where \(\chi_{2}\) is a dimensionless property of the non-aqueous component. \[\mathrm{V}_{2}^{*}(\ell, \mathrm{T})=\mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{2} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right\}\]\(\mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right)\) is the molar volume of non-aqueous component at the temperature \(\mathrm{TMD}^{*}\). The volume of a mixture prepared using \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles of the two liquids at temperature \(\theta\) under the no-mix condition is given by equation (c). \[\mathrm{V}(\mathrm{no}-\operatorname{mix} ; \theta)=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \theta)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \theta)\]Hence, \[\begin{aligned} \mathrm{V}(\mathrm{no}-\operatorname{mix} ; \theta) &=\mathrm{n}_{1} \,\left[\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\}\right] \\ &+\mathrm{n}_{2} \,\left[\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right\}\right] \end{aligned}\]At temperature \(\theta\), the volume of the real mixture \(\mathrm{V}(\operatorname{mix} ; \theta)\) is given by equation (e) where \(\mathrm{V}_{1}(\operatorname{mix} ; \theta)\) and \(\mathrm{V}_{2}(\operatorname{mix} ; \theta)\) are the partial molar volumes of the two components in the mixture at temperature \(\theta\). \[\mathrm{V}(\operatorname{mix} ; \theta)=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix} ; \theta)+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix} ; \theta)\]The volume of mixing at temperature \(\theta\) is given by equation (f). By definition, \[\Delta_{\text {mix }} V(\theta)=V(\operatorname{mix} ; \theta)-V(\text { no }-\operatorname{mix} ; \theta)\]Hence, \[\begin{aligned} \Delta_{\text {mix }} \mathrm{V}(\theta)=& \mathrm{n}_{1} \,\left\{\mathrm{V}_{1}(\operatorname{mix} ; \theta)-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right]\right\} \\ &+\mathrm{n}_{2} \,\left\{\mathrm{V}_{2}(\operatorname{mix} ; \theta)-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right]\right\} \end{aligned}\]But the molar volume of mixing at temperature \(\theta\), \[\Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta)=\Delta_{\text {mix }} \mathrm{V}(\theta) /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\]Hence, \[\begin{aligned} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\theta) &=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix} ; \theta)+\mathrm{x}_{2} \, \mathrm{V}_{2}(\operatorname{mix} ; \theta) \\ -\left\{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \,\left[1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right]\right\} \\ &-\left\{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right]\right\} \end{aligned}\]Then the differential of \(\Delta_{\mathrm{mix}} \mathrm{~V}_{\mathrm{m}}\) is given by equation (j). \[\begin{aligned} \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=& \mathrm{x}_{1} \, \mathrm{dV}(\operatorname{mix} ; \mathrm{T})+\mathrm{V}_{1}(\operatorname{mix} ; \mathrm{T}) \, \mathrm{dx}{ }_{1} \\ &+\mathrm{x}_{2} \, \mathrm{dV}_{2}(\operatorname{mix} ; \mathrm{T})+\mathrm{V}_{2}(\mathrm{mix} ; \mathrm{T}) \, \mathrm{dx} \mathrm{x}_{2} \\ &-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,(\mathrm{T}-\mathrm{TMD})^{2} \,[\mathrm{K}]^{-2}\right] \, \mathrm{dx} \\ &\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT} \\ &-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,(\mathrm{T}-\mathrm{TMD})^{2} \,[\mathrm{K}]^{-2}\right] \, \mathrm{dx}_{2} \\ &-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \mathrm{TMD}) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT} \end{aligned}\]But according to the Gibbs-Duhem Equation, at fixed pressure \[x_{1} \, d V_{1}(\operatorname{mix} ; T)+x_{2} \, d V_{2}(\operatorname{mix} ; T)=E_{p m}(\operatorname{mix} ; T) \, d T\]In equation (k), \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{T})\) is the molar isobaric expansion of the mixture. Equation (j) can therefore be reorganized into equation (l). \[\begin{aligned} &\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=\\ &\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{T}) \, \mathrm{dT}\\ &+\left\{\mathrm{V}_{1}(\operatorname{mix} ; \mathrm{T})-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} \,[\mathrm{K}]^{-2}\right\} \, \mathrm{dx} \mathrm{x}_{1}\right.\\ &+\left\{\mathrm{V}_{2}(\mathrm{mix} ; \mathrm{T})-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-1}\right] \, \mathrm{dx}\right.\\ &\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT}\\ &-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT} \end{aligned}\]\We note that \(\mathrm{dx}_{1} = −\mathrm{dx}_{2}\) and that the coefficients of \(\mathrm{dx}_{1}\) and \(\mathrm{dx}_{2}\) are in fact excess partial molar volumes at temperature \(\mathrm{T}\). Hence, \[\begin{aligned} \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=& \\ \mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{T}) \, & \mathrm{dT}+\left[\mathrm{V}_{2}^{\mathrm{E}}(\mathrm{T})-\mathrm{V}_{1}^{\mathrm{E}}(\mathrm{T})\right] \, \mathrm{dx} \mathrm{x}_{2} \\ &\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, 2 \, \chi_{1} \,(\mathrm{T}-\mathrm{TMD}) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT} \\ &-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \mathrm{TMD}) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT} \end{aligned}\]Or, \[\begin{aligned} \frac{\mathrm{d} \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\mathrm{T})}{\mathrm{dT}} &=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{T})+\left[\mathrm{V}_{2}^{\mathrm{E}}(\mathrm{T})-\mathrm{V}_{1}^{\mathrm{E}}(\mathrm{T})\right] \, \frac{\mathrm{dx}}{\mathrm{dT}} \\ &\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \\ &-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1} \end{aligned}\]We use equation (n) at temperature ‘\(\mathrm{T} = \theta\)’ and at fixed composition; i.e. \(\mathrm{dx}_{2} = 0\). Moreover, by definition at the \(\mathrm{TMD}\), \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \theta)\) is zero. Hence the dependence of \(\Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta)\) on temperature at fixed pressure and composition is given by equation (o). \[\begin{aligned} \left(\frac{\partial \Delta_{\text {mix }} V_{\mathrm{m}}(\mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{x}}=&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \\ &-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1} \end{aligned}\]We identify temperature \(\theta\) with the recorded \(\mathrm{TMD}\). Hence from equation (o) with \(\Delta \theta=\theta-\mathrm{TMD}^{*}\), \[\begin{array}{r} \Delta \theta=-\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}} \\ -\frac{\left[\partial \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta) / \partial \mathrm{T}\right]_{\mathrm{p}, \mathrm{x}}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}} \end{array}\]In other words, the shift in the \(\mathrm{TMD}\), \(\Delta \theta\), is made up of two contributions. For binary system having thermodynamic properties which are ideal, the second term on the r.h.s. of equation (p) is zero. The first term on the r.h.s. side of equation (p) predicts that \(\Delta \theta\) is negative in agreement with the Despretz rule. In summary therefore equation (p) can be written in the following simple form. \[\Delta \theta=\Delta \theta(\text { ideal })+\Delta \theta(\text { struct })\]The sign of \(\Delta \theta (\text{struct})\) is determined by the sign of \(\left[\partial \Delta_{\operatorname{mix}} V_{\mathrm{m}}(\theta) / \partial \mathrm{T}\right]\). If the latter term is negative, \(\Delta \theta (\text{struct})\) is positive and for some systems can be the dominant term. As noted above, this is the case at low mole fractions \(\mathrm{x}_{2}\) for 2-methylpropan-2-ol, a trend attributed to enhancement of water-water hydrogen bonding by the non-aqueous component.The volume of a solution at temperature \(\mathrm{TMD}\), prepared using \(1 \mathrm{~kg}\) of solvent water and mj moles of a simple neutral solute is given by equation (r). \[\mathrm{V}(\mathrm{aq} ; \mathrm{TMD})=\left(\mathrm{l} / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD})+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right)\]But at the \(\mathrm{TMD}\), \[\left(1 / \mathrm{M}_{1}\right) \,\left[\partial \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) / \partial \mathrm{T}\right]=-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right) / \partial \mathrm{T}\right]\]We use equation (a) to relate \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD})\) to \(\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right)\) with \(\Delta \mathrm{T}\) representing (\(\mathrm{TMD} - \mathrm{~TMD}^{*}\)). \[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, \chi_{1} \, 2 \, \Delta \mathrm{T} /[\mathrm{K}]^{2}=-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right) / \partial \mathrm{T}\right]\]We assume that for dilute real solutions \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of the molality of solute \(j\) and that the proportionality term is the pairwise volumetric interaction parameter \(\mathrm{v}_{jj}\). Thus, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}+\mathrm{v}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Then \[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \mathrm{dT}=\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}+\left[\partial \mathrm{v}_{\mathrm{ij}} / \partial \mathrm{T}\right]_{\mathrm{p}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Hence, \[\begin{aligned} &\left\{2 \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{1} / \mathrm{M}_{1} \,[\mathrm{K}]^{2}\right\} \, \Delta \mathrm{T}= \\ &\quad-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\partial \mathrm{T}\right]-\left[\partial \mathrm{v}_{\mathrm{ij}} / \partial \mathrm{T}\right] \,\left(\mathrm{m}^{0}\right)^{-1} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \end{aligned}\]We rewrite equation (w) as an equation for the change in \(\mathrm{TMD}\), \(\Delta \mathrm{T}\) using a quadratic in \(\mathrm{m}_{j}\). Thus, \[\Delta \mathrm{T}=\mathrm{q}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{q}_{2} \, \mathrm{m}_{\mathrm{j}}^{2}\]Consequently a plot of \(\Delta \mathrm{T}\) \(\mathrm{m}_{j}\) is linear having intercept \(\mathrm{q}_{1}\) and slope \(\mathrm{q}_{2}\). If the solution is ideal [i.e. \(\mathrm{v}_{jj}\) is zero] then \(\mathrm{q}_{2}\) in zero and \(\left[\Delta \mathrm{T} / \mathrm{m}_{\mathrm{j}}\right]\) is constant independent of \(\mathrm{m}_{j}\).The above analysis forms the basis for an analysis of the effects of salts on \(\mathrm{TMD}\) except that the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) is expressed using the following equation where \(\mathrm{S}_{\mathrm{V}}\) is the Debye-Huckel Limiting Law volumetric parameter. \[\phi\left(V_{j}\right)=\phi\left(V_{j}\right)^{\infty}+S_{v} \,\left(m_{j} / m^{0}\right)^{1 / 2}+b \,\left(m_{j} / m^{0}\right)\]The foregoing analysis has been extended to include consideration of isobaric expansions and limiting partial molar expansions. C. Wada and S.Umeda, Bull. Chem. Soc. Jpn,1962,35,646,1797. F. Franks and B. Watson, Trans. Faraday Soc.,1969,65,2339. The symbol [K] indicates the unit of temperature, kelvin. The term \(\left\{1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\}\) is dimensionless as required by equation (a). As noted \(\chi_{1}\) and \(\chi_{2}\) are dimensionless and characteristic properties of the two components. \[\begin{aligned} &\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}} \\ &=\frac{ \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \,[\mathrm{K}]^{-1}}{ \, \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \,[\mathrm{K}]^{-2}}=[\mathrm{K}] \\ &\frac{\left[\partial \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta) / \partial \theta\right]}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{ \, \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \,[\mathrm{K}]^{-2}}=[\mathrm{K}] \end{aligned}\] M. V. Kaulgud, J. Chem. Soc. Faraday Trans.1,1979,75,2246; 1990,86,911. J. R. Kuppers, J.Phys.Chem.,1974,78,1041. T. H. Lilley and S. Murphy, J.Chem. Thermodyn., 1973,5,467. T. Wakabayashi and K. Takazuimi, Bull. Chem. Soc. Jpn., 1982,55,2239. T. Wakabayashi and K. Takazuimi, Bull. Chem. Soc. Jpn., 1982,55,3073. For comments on salts in D2O, see A. J. Darnell and J. Greyson, J. Phys. Chem.,1968, 73,3032. G. Wada and M. Miura, Bull. Chem. Soc. Jpn., 1969,42,2498. J. R. Kuppers, J. Phys. Chem.,1975,79,2105. D. A. Armitage, M. J. Blandamer, K. W. Morcom and N. C. Treloar, Nature, 1968,219,718. J. E. Garrod and T. M. Herrington, J. Phys.Chem.,1970,74,363. T. M. Herrington and E. L. Mole, J. Chem. Soc. Faraday Trans.1,1982,78,213. D. D. Macdonald and J. B. Hyne, Can. J. Chem.,1976,54,3073. D. D. Macdonald, B. Dolan and J. B. Hyne, J. Solution Chem.,1976,5,405.This page titled 1.14.69: Temperature of Maximum Density: Aqueous Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.7: Extrathermodynamics - Background

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Essentially thermodynamics is used to analyze experimental data. In these terms, thermodynamics shows how properties of systems are related and how one can link measured properties with important thermodynamic variables. Nevertheless, there are cases where a pattern seems to emerge from measured variables which is not a consequence of the laws of thermodynamics. Furthermore, it is often discovered that the patterns can actually be accounted for if one or two additional postulates are made. These new postulates are therefore extra-thermodynamic and the analytical method is called extrathermodynamics. The analysis has merit in that the new postulates point to patterns which can be developed for other systems.The essence of the argument can be understood by considering the molar volume of pure ethanol at ambient pressure and \(298.2 \mathrm{~K}\). Clearly \(\mathrm{V}^{*}\left(\ce{C2H5OH} ; \ell ; 298.2 \mathrm{~K} ; \left.101325 \mathrm{~N} \mathrm{~m}^{-2}\right)\) is a properly defined thermodynamic variable. But as chemists we might be tempted to explore an extrathermodynamic postulate in which \(\mathrm{V}^{*}\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} ; \ell\right)\) can be subdivided into group contributions. Thus\[\mathrm{V}^{*}\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} ; \ell\right)=\mathrm{V}\left(\mathrm{CH}_{3}\right)+\mathrm{V}\left(\mathrm{CH}_{2}\right)+\mathrm{V}(\mathrm{OH}) \label{a}\]This equation cannot be justified on thermodynamic grounds. Nevertheless we might examine molar volumes of several (liquid) alcohols at the same \(\mathrm{T}\) and \(\mathrm{p}\) and come up with a self-consistent set of group volumes. For example,\[\left[\mathrm{V}^{*}\left(\mathrm{n}-\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH} ; \ell\right)\right]=\mathrm{V}\left(\mathrm{CH}_{3}\right)+2 * \mathrm{~V}\left(\mathrm{CH}_{2}\right)+\mathrm{V}(\mathrm{OH}) \label{b}\]Hence comparison of Equations \ref{a} and \ref{b} yields directly \(\mathrm{V}\left(\mathrm{CH}_{2}\right)\), the contribution of methylene groups to the molar volume of (liquid) alcohols at the same \(\mathrm{T}\) and \(\mathrm{p}\). Although such an analysis might be judged naïve, the general approach finds merit in several subject areas; e.g chemical equilibria and chemical kinetics.Footnotes J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, London, 1963. E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997.This page titled 1.14.7: Extrathermodynamics - Background is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.70: Solutions: Neutral Solutes: Inter-Solute Distances

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At its simplest a solution comprises one liquid component is vast excess, the solvent, and another, the solute, which is dispersed in the solvent. The solvent is more than just a useful medium in which to disperse the solute although one might argue that a key role is to inhibit associations of the solute molecules. In an even cursory examination of the properties of solutions, a key consideration is the distance between solute molecules. An interesting calculation offers insight into the dependence of solute-solute distances on solute concentration. For a simple non-ionic solute ( e.g. urea) in aqueous solution at concentration \(\mathrm{c}_{j} \mathrm{~mol dm}^{-3}\), the average solute-solute distance \(\mathrm{d}\) is given by equation (a) where \(\mathrm{N}_{\mathrm{A}}\) is the Avogadro number. \[d=\left(N_{A} \, c_{j}\right)^{-1 / 3}\]At \(\mathrm{c} = 10^{-2} \mathrm{~mol dm}^{-3}\), \(\mathrm{d} = 5.5 \mathrm{~nm}\). If the solute is a 1:1 salt where 1 mole of salt yields two moles of solute ions, \(\mathrm{d} = 4.4 \mathrm{~nm}\). With increase in solute concentration, the mean distance between solute molecules decreases.An interesting feature of aqueous solutions is worthy of comment. If a given water molecule is hydrogen bonded (indicating strong cohesion) to four nearest neighbour water molecules, that water molecule exists in a state of low density-high molar volume. In other words cohesion is linked to low density, a pattern contrary to that encountered in most systems. Nevertheless in reviewing the properties of aqueous solutions and water, one must be wary of overstressing the importance of hydrogen bonding. Indeed liquid water has a modest viscosity which is not the conclusion would draw from some models for liquid water which emphasize the role of water-water hydrogen bonding.Footnote R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd. Edn. Revised, 1965.This page titled 1.14.70: Solutions: Neutral Solutes: Inter-Solute Distances is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.71: Time and Thermodynamics (Timenote)

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We note two comments in monographs dealing with thermodynamics.One comment states that ‘… thermodynamics deals with systems at equilibrium, time is not a thermodynamic co-ordinate.’The reference here is in the context of systems at equilibrium.A stronger statement with a different view is made by McGlashan.Thus‘We shall be using time \(\mathrm{t}\) as one of our variables in this chapter. There are those who say that time has no place in thermodynamics. They are wrong.’Some history sets the scene.Once upon a time chemists used the calorie as a unit of energy. In fact there were three different units named calorie: thermochemical calorie, international calorie and 150 C calorie. In common they defined energy in terms of the amount of energy required to raise by one Kelvin, the temperature of one gram of pure liquid water under specified conditions of temperature and pressure. Time is not mentioned, directly or indirectly, in this definition. Then Joule showed there is an equivalence between heat and mechanical energy. It is just a small step to relate thermal energy to kinetic energy and, hence, to time. If a calorimetric definition of energy had been adopted, then its unit would be a base unit. In practice, this would be a regression to the situation before Joule determined the mechanical equivalent of heat.Wood and Battino and McGlashan are both right. Time is an important thermodynamic variable for formulating the conditions under which systems approach an equilibrium state. However, time is not used to describe the properties of these systems after equilibrium is attained.Footnotes S. E. Wood and R. Battino, Thermodynamics of Chemical Systems, Cambridge Univeristy Press, Cambridge,1990, page 2. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 102; the footnotes in this text are often provocative.This page titled 1.14.71: Time and Thermodynamics (Timenote) is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.72: Variables: Independent and Dependent

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A colleague has filled a flask with water and asks us by phone to estimate the volume of water in the flask. Clearly this is an impossible task but our colleague offers further information. In answer to the first question, our colleague informs us that there are 2 moles of water in the flask. Immediately we suggest that the volume of water is \(36 \mathrm{cm}^{3}\). Not good enough! Our colleague demands a more precise estimate. We know that the volume of water(\(\ell\)) depends on temperature and pressure and so request new this information. We are told that the temperature is \(298.2 \mathrm{~K}\) and the pressure is \(101325 \mathrm{~N m}^{-2}\). We summarize this information in the following form. \[\mathrm{V}=\mathrm{V}\left[298.2 \mathrm{~K} ; 101325 \mathrm{~N} \mathrm{~m} \mathrm{~m}^{-2} ;(\ell) ; 2 \text { moles }\right]\]Our colleague offers further information such as the vapor pressure and heat capacity of water(\(\ell\)) under these conditions. But we decline this offer on the grounds that no further information is required. We know that having defined the variables in the square brackets [.....], a unique volume \(\mathrm{V}\) is defined. We may not immediately know the actual volume but given a little time in a scientific library we will be in a position to report volume \(\mathrm{V}\).The variables in the square brackets are the INDEPENDENT VARIABLES. For a system containing one chemical substance we define the volume as follows. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}\right]\]The term independent means that within limits we can change \(\mathrm{T}\) independently of the pressure and \(\mathrm{n}_{1}\); change \(\mathrm{p}\) independently of \(\mathrm{T}\) and \(\mathrm{n}_{1}\); change \(\mathrm{n}_{1}\) independently of \(\mathrm{T}\) and \(\mathrm{p}\). There are some restrictions in our choice of independent variables. At least one variable must define the amount of all chemical substances in the system and one variable must define the 'hotness' of the system.The molar volume of liquid chemical substance 1 at the specified temperature and pressure, \(V_{1}^{*}(\ell)\) is obtained from equation (b) by fixing \(\mathrm{n}_{1}\) at 1 mol. Thus \[\mathrm{V}_{1}^{*}(\ell)=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}=1 \mathrm{~mol}\right]\]If the composition of a given closed system is specified in terms of the amounts of two chemical substances, 1 and 2, four independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) define the independent variable \(\mathrm{V}\). Thus \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\]Actually there is merit in writing equation (d) in terms of three intensive variables which in turn defines the molar volume \(\mathrm{V}_{\mathrm{m}}\) of the binary system at given mole fraction \(x_{1}=1-x_{2}\). Thus \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right]\]For a system containing i - chemical substances where the amounts can be independently varied, the dependent extensive variable \(\mathrm{V}\) is defined by equation (f). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots, \mathrm{n}_{\mathrm{i}}\right]\]Similarly the dependent intensive variable \(\mathrm{V}_{\mathrm{m}}\) is defined by equation (g). \[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}, \mathrm{x}_{2} \ldots \ldots \ldots, \mathrm{x}_{\mathrm{i}-1}\right]\] The phrase 'independent variable' is important. With reference to the properties of an aqueous solution containing ethanoic acid, the number of components for such a solution is 2, amount of water and amount of ethanoic acid. The actual amounts of ethanoic acid, water, ethanoate and hydrogen ions are determined by an equilibrium constant which is an intrinsic property of this system at given \(\mathrm{T}\) and \(\mathrm{p}\). From the point of the Phase Rule, the number of components equals 2. For the same reason when we consider the volume of a system containing only \(\mathrm{n}_{j}\) moles of water we disregard evidence that water partly self-dissociates into \(\mathrm{H}^{+} (\mathrm{aq})\) and \(\mathrm{OH}^{-} (\mathrm{aq})\). In terms of the Phase Rule, for two components (\(\mathrm{C} = 2\)) and one phase (\(\mathrm{P} = 1\)), the number of degrees of freedom \(\mathrm{F}\) equals 3. These degrees of freedom refer to a set of intensive variables. Hence, for a solution where substance 1 is the solvent and substance 2 is the solute, the system is defined by specifying the three (intensive) degrees of freedom, \(\mathrm{T}, \mathrm{~p}\) and, for example, solute molality.This page titled 1.14.72: Variables: Independent and Dependent is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.73: Variables: Gibbsian and Non-Gibbsian

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Experience shows that the thermodynamic state of a closed single phase system can be defined by a minimum set of independent variables where at least one variable is a measure of the ‘hotness’ of the system; e.g. temperature. The volume of an aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of urea is defined by the set of independent variables, \(\mathrm{T}, \mathrm{~p}, \mathrm{~n}_{1} \text { and } \mathrm{n}_{j}\). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]Having defined the parameters set out in the brackets [...] the volume of the system, the dependent variable, is uniquely defined. In fact we can replace \(\mathrm{V}\) in this equation by \(\mathrm{G}, \mathrm{~H} \text { and } \mathrm{~S}\) in order to define unique Gibbs energy, enthalpy and entropy respectively.The set of independent variables in equation (a) is called Gibbsian because the set comprises the intensive variables \(\mathrm{T}\) and \(\mathrm{p}\) together with the extensive composition variables. The general form of equation (a) defining the thermodynamic potential function, Gibbs energy \(\mathrm{G}\) is as follows where \(\xi\) is the extensive composition variable. \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\]Other sets of independent variables are used in conjunction of the thermodynamic potential functions, enthalpy \(\mathrm{H}\), energy \(\mathrm{U}\) and Helmholtz energy \(\mathrm{F}\). \[\mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\]\[\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi]\]\[\mathrm{H}=\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]\]In equations (c) and (d), \(\mathrm{V}\) is an extensive variable and in equations (d) and (e) S is an extensive variable. The sets of independent variables in equations (c), (d) and (e) are called non-Gibbsian.Footnote J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.This page titled 1.14.73: Variables: Gibbsian and Non-Gibbsian is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.74: Vaporization

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A given chemical substance \(j\) can exist in phases I and II. For phase I,\[\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{I})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I}) \label{a}\]\(\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I}), \mathrm{~H}_{\mathrm{j}}^{*}(\mathrm{I}) \text { and } \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})\) are the molar thermodynamic energy, enthalpy and volume respectively of chemical substance \(j\) in phase I at pressure \(\mathrm{p}\). Chemical substance \(j\) can also exist in phase II at the same pressure \(\mathrm{p}\).\[\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{II})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II}) \label{b}\]Equations \ref{a} and \ref{b} are quite general. In an important application we identify phase II as the vapor phase which we assume to have the properties of a perfect gas. Phase I is the liquid state. For the process `liquid → vapor' ( i.e. vaporization) at temperature \(\mathrm{T}\),\[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{p} \,\left[\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})-\mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]\]But at temperature \(\mathrm{T}\), \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})-\mathrm{V}_{\mathrm{j}}^{*}(\ell) \gg 0\) Also for one mole of a perfect gas, \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})=\mathrm{R} \, \mathrm{T}\).Hence,\[\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}\]\(\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{\mathrm{N}}(\mathrm{T})\) is obtained from the dependence of vapour pressure on temperature; see Clausius - Clapeyron Equation. Hence we obtain the molar thermodynamic energy of vaporisation.This page titled 1.14.74: Vaporization is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.75: Viscosities: Salt Solutions

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The (shear) viscosities of salt solutions have been extensively studied. A frequently cited paper (always worth reading in terms of the care taken in the experimental study) was published by Jones and Dole. The dependence of the viscosity of an aqueous salt solution \(\eta(\mathrm{aq})\) (at fixed temperature and pressure) on concentration of salt \(j\) is described by the Jones-Dole equation; equation (a) where \(\eta_{1}^{*}(\ell)\) is the viscosity of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\).\[\eta(\mathrm{aq}) / \eta_{1}^{*}(\ell)=1+\mathrm{A} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}}\]Equation (a) is re-expressed as follows. By definition;\[\psi=\left[\eta(\mathrm{aq})-\eta_{1}^{*}(\ell)\right] /\left[\eta_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}^{1 / 2}\right]\]Hence\[\psi=\mathrm{A}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}\]A plot of \(\psi\) against \(\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}\) has intercept \(\mathrm{A}\) and slope \(\mathrm{B}\), the Jones-Dole \(\mathrm{B}\)-viscosity coefficient. The \(\mathrm{A}\) coefficient describes the impact of charge-charge interactions on the viscosity of a solution, being generally positive and estimated using the Falkenhagen equation. The \(\mathrm{B}\) coefficient characterizes ion-solvent interactions at defined \(\mathrm{T}\) and \(\mathrm{p}\). For a 1:1 salt \(j\), the \(\mathrm{B}_{j}\) coefficient for salt \(j\) is expressed as the sum of ionic \(\mathrm{B}\) coefficients.\[\mathrm{B}_{\mathrm{j}}=\mathrm{B}_{+}+\mathrm{B}\]For example, \(\mathrm{B}_{j}\) for a series of salts with a common anion, the changes in \(\mathrm{B}_{j}\) reflect changes in \(\mathrm{B}_{+}\) for the cations. The pattern in \(\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)\) reflects the changes in \(\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)\) through the series from \(\mathrm{R}\) = methyl to \(\mathrm{R}\) = n-butyl. In fact the change in this case is indicative of the change in character from ‘structure breaking’ \(\mathrm{Me}_{4}\mathrm{N}^{+}\) to hydrophobic ‘structure forming’ \(\mathrm{Bu}_{4}\mathrm{N}^{+}\) ions. In broad terms a positive \(\mathrm{B}\) coefficient indicates a tendency for the solute to enhance water-water interactions and thus raise the viscosity whereas a negative coefficient indicates a tendency to induce disorder. Ionic B-viscosity coefficients are linked to the hydration properties of ions. An important link was suggested by Gurney. On the grounds that \(\mathrm{K}^{+}\) and \(\mathrm{Cl}^{-}\) ions are roughly the same size, Gurney argued that for aqueous solutions, \(\mathrm{B}\left(\mathrm{K}^{+}\right)=\mathrm{B}\left(\mathrm{Cl}^{-}\right)\). Hence one can estimate single-ion \(\mathrm{B}\)-viscosity coefficients. \(\mathrm{A}\) negative \(\mathrm{B}\)-ionic coefficient indicates that the ion is a ‘structure breaker’ and a positive \(\mathrm{B}\)-ionic viscosity coefficient indicates that the ion is a ‘structure former’.The terms ‘structure breaker’ and ‘structure former’ were extensively used in the decades from 1950 to 1990. However their popularity waned towards the end of the century as more precise descriptions of ionic hydration were sought. We confine attention to shear viscosities (i.e. resistance to shear). The related bulk viscosities (i.e. resistance to compression) are rarely discussed in the present context. Units; dynamic viscosity; traditional unit = poise, symbol \(\mathrm{P}\)\(\mathrm{P}=10^{-1} \text { Pas}\). But \(\mathrm{Pa} = \mathrm{~kg m}^{-1} \mathrm{~s}^{-2}\) Then \(\mathrm{P} = 10^{-1} \mathrm{~kg m}^{-1} \mathrm{~s}^{-1} SI unit; η = [kg m-1 s-1 ] G. Jones and M. Dole, J. Am. Chem. Soc., 1929,51,2950. Other equations have been suggested as alternatives to the Jones-Dole equation. See for example, H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolyte Solutions, Reinhold, New York, 3rd. edition, 1958,p.240. See for example; An alternative approach involves calculating the A coefficient using the Falkenhagen theory and hence writing the Jones-Dole equation as follows.\[\eta(\mathrm{aq}) / \eta_{\mathrm{I}}^{*}(\ell)-1-\mathrm{A} \,\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}=\mathrm{B} \, \mathrm{c}_{\mathrm{j}}\]In a plot of the LHS of this equation against \(\mathrm{c}_{j}\), the slope equals B. See for example, K. Tamaki, K. Suga and E. Tanihara, Bull. Chem. Soc. Jpn.,1987.60,1225. B. M. Lowe and G. A. Rubienski, Electrochem. Acta, 1974,19,393. E. R. Nightingale, J. Phys. Chem.,1959,63,1381;1962,66,894. See also, D. T. Burns, Electrochim Acta, 1965,10,985. R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.This page titled 1.14.75: Viscosities: Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.76: Work

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This term ‘work’ makes it first key appearance (at least in thermodynamics) in the context of the statement that if work is done on a closed system the thermodynamic energy of the system increases given that heat q is zero. This simple statement understates the complexity of the term ‘work’ in thermodynamics.In general terms work done on a closed thermally insulated system raises the energy of that system and is given by the product of intensive and capacity factors. Three examples make the point.The analysis is complicated by the fact that changes in a given system can take one of two limiting forms; e.g. frozen and equilibrium. In the case of surface tension, frozen ( plastic) surface tension describes the case where the intermolecular distances in the surface increase. The equilibrium case describes the case where molecules in the bulk phase and in the surface exchange to hold the change in the surface as a reversible (equilibrium) process.Footnote E. F. Caldin, Chemical Thermodynamics, Clarendon Press, Oxford, 1958.This page titled 1.14.76: Work is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.8: Extrathermodynamics - Equilbrium - Acid Strength

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In aqueous solution at ambient pressure and \(298.15 \mathrm{~K}\), benzoic acid exists in the form of a chemical equilibrium described in equation (a) \[\mathrm{PhCOOH}(\mathrm{aq}) \Leftrightarrow=\mathrm{H}^{+}(\mathrm{aq})+\mathrm{PhCOO}^{-}(\mathrm{aq})\]At defined \(\mathrm{T}\) and \(\mathrm{p}\), \[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{PhCOO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{PhCOOH} ; \mathrm{aq})\]In the case of a substituted benzoic acid, \(\mathrm{XC}_{6} \mathrm{H}_{4} \mathrm{COOH} \quad[=\mathrm{XPhCOOH}]\), the corresponding description of the chemical equilibrium takes the following form. \[\begin{aligned} &\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq}) \\ &=\mu^{0}(\mathrm{XPhCOO} ; ; \mathrm{aq})+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{XPhCOOH} ; \mathrm{aq}) \end{aligned}\]In aqueous solution at ambient pressure and \(298.15 \mathrm{~K}\), the properties of an aqueous solution containing phenol can be described in terms of the following equilibrium. \[\mathrm{PhOH}(\mathrm{aq}) \Leftrightarrow=\Longrightarrow \mathrm{H}^{+}(\mathrm{aq})+\mathrm{PhO}^{-}(\mathrm{aq})\]Then, (cf. equation (b)), \[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{PhO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{PhOH} ; \mathrm{aq})\]In the case of a substituted phenol \(\mathrm{XPhOH}\), the equation corresponding to equation (d) takes the following form. \[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{XPhO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{XPhOH} ; \mathrm{aq})\]In the following we compare situations where \(\mathrm{X}\) is common to the substituted phenol and benzoic acid including position in the aromatic ring. The interesting point to emerge is that for a range of substituents, \(\mathrm{X}\), the recorded dependence of \(\Delta_{\Delta_{r}} \mathrm{G}^{0}(\mathrm{XPhOH} ; \mathrm{aq})\) on \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})\) is linear. Such a pattern is not a requirement of thermodynamics. The challenge is to suggest a set of minimum relationships which account for this pattern.Consider the reference chemical potential for solute \(\mathrm{RX}\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\). As chemists we recognise that groups \(\mathrm{R}\) and \(\mathrm{X}\) do not make independent contributions to \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\). The postulate, Single Interaction Mechanism, recognises that the groups \(\mathrm{R}\) and \(\mathrm{X}\) interact such that \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\) is given by equation (g). \[\mu^{0}(\mathrm{RX} ; \mathrm{aq})=\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}, \mathrm{X})\]Here symbol \(\mathrm{R}\) identifies the substituent zone and \(\mathrm{X}\) identifies the reaction zone so that \(\mathrm{I}(\mathrm{R}, \mathrm{X})\) describes interaction between these two zones.The interaction variable \(\mathrm{I}(\mathrm{R}, \mathrm{X})\) is a function of scalar variables. Then \[\mu^{0}(\mathrm{RX} ; \mathrm{aq})=\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}) \, \mathrm{I}(\mathrm{X})\]Hence for benzoic acid \(\mathrm{PhCOOH}(\mathrm{aq})\), \[\mu^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}(\mathrm{COOH})+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH})\]Similarly, \[\mu^{0}(\mathrm{PhCOO} ; ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)\]Hence, \[\begin{gathered} \Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\ -\mu^{0}(\mathrm{Ph})-\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH}) \end{gathered}\]Or, \[\begin{gathered} \Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\ -\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH}) \end{gathered}\]A similar equation emerges describing the acid dissociation of the substituted acid. Thus, \[\begin{gathered} \Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\ -\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}(\mathrm{COOH}) \end{gathered}\]By definition, \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})-\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})\]Hence, \[\begin{gathered} \Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { acids })=\left[\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}(\mathrm{COOH})\right] \\ -\left[\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH})\right] \end{gathered}\]Or, \[\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { acids })=[\mathrm{I}(\mathrm{XPh})-\mathrm{I}(\mathrm{Ph})] \,\left[\mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{COOH})\right]\]Thus \(\Delta_{r} G^{0}\) is given by the product of two terms;We turn our attention to the acid strength of phenol and susbstituted phenols in aqueous solution at the same \(\mathrm{T}\) and \(\mathrm{p}\). A similar analysis to that set out above yields the following equation. \[G (phenols) [I(XPh) I(Ph)] [I(O ) I(OH)] 0 ∆∆r = − ⋅ − − (q)Comparison of equations (p) and (q) yields equation (r). \[\left.\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { phenols })=\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids }\right) \,\left\{\left[\mathrm{I}\left(\mathrm{O}^{-}\right)-\mathrm{I}(\mathrm{OH})\right] /\left[\mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{COOH})\right]\right\}\]The analysis rationalises the observation that \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (phenols) }\) is a linear function of \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids) }\). In other words we have not proved that such a linear function exists. Rather we have identified the minimum hypothesis required to account for the observation. In these terms the extrathermodynamic analysis has pointed to a reason for the recorded dependences of \(\Delta \Delta_{\mathrm{r}} G^{0} \text { (phenols) }\) on \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids) }\). The pattern is not a requirement of thermodynamics.Footnotes See for example, J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, London, 1963. E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997. The superscript ‘0’ is retained although the meaning here is somewhat obscure. It effectively reminds us that we are dealing with the properties of a solute in its solution reference state.This page titled 1.14.8: Extrathermodynamics - Equilbrium - Acid Strength is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.9: Extrathermodynamics - Solvent Polarity

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A solution comprises at least two chemical substances, solvent and solute. The amount of solvent far exceeds the amount of solute so in this sense a solute is dispersed through a solvent. Although the solvent molecules are in vast excess, our interest centres on the minor (solute) component because chemists attempt to understand how interactions between a solute molecule and surrounding solvent molecules control the properties of the solute molecule; e.g. control reactivity, solubility, and colour. Out of this interest in solute - solvent interactions emerges the concept of solvent polarity which attempts to characterise this interaction.The general concept of solvent polarity can be understood by considering developments in two subjects;We review briefly each of these subject areas, indicating how the concept of polarity and/or solvent polarity emerged. We show how the intuitive concept of solvent polarity in these subject areas developed in quantitative terms. We used the word intuitive and this usage can be understood in the following terms. Asked to prepare a solution of sodium chloride (table salt), a first year freshman student would choose water as the solvent rather than (liquid) benzene or ethanol because (the student would argue) water is more "polar" than either ethanol or benzene. Here the term "polar" is little more than laboratory jargon. We seek a quantitative measure of solvent polarity.In 1862 Bertholet and Pean de Saint Gilles noted that the rate of chemical reaction depends on the solvent. In 1890, Menschutkin confirmed that finding in a very detailed study. So for more than 100 years chemists have attempted to describe quantitively these solvent effects. Perhaps not suprisingly the first attempts concentrated on the dependendence of rate constants on the relative permittivity of solvents. Many authors sought correlations using the treatments described by Kirkwood. The quantity used in these correlations usually takes the form \(\left(\varepsilon_{\mathrm{r}}-1\right) /\left(2 \, \varepsilon_{\mathrm{r}}+1\right)\). But as many authors point out, this Kirkwood function is little better than the relative permittivity for describing interactions at the molecular level. Nevertheless the challenge remained to describe kinetic solvent effects. A particular important stage was the growth of interest in physical organic chemistry. Probably the ‘father’ of this subject was C.K.Ingold. In his classic monograph , Ingold actually used the pharase ‘solvent polarity’when commenting on the rates of reactions through a series of - 1 - solvents of diminishing polarity; water, ethanol, propanone, benzene. But Ingold did not offer a polarity scale. One of the reactions discussed by Ingold was the hydrolysis of \(\left(\mathrm{CH}_{3}\right)_{3}\mathrm{CCl}\). \[\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CCl}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{aq}) \rightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})\]This classic reaction (although the mechanism is still debated) formed the basis of a quantitative description of solvent polarity described by Winstein and coworkers. Using the rate of reaction described above they identified a reference solvent, a mixture formed by ethanol (80 vol%) and water(20 vol%). If the rate constant for this reaction is \(\mathrm{k}^{0}\) in this solvent mixture and the rate constant is \(\mathrm{k}\) in a new solvent, the Y-value of this new solvent is given by \[\mathrm{Y}=\log \left(\mathrm{k} / \mathrm{k}^{0}\right)\]In effect \(\mathrm{Y}\) measures the ionising power of the solvent – the extent to which the solvent favours charge separation within the neutral solute. Hence by measuring the rate constant for the above reaction in a given solvent, the polarity of the solvent is obtained as shown by its Y-value.This kinetic approach to the determination of solvent polarities has attracted attention, particularly in the context probing reaction mechanisms. The Y-value approach can be rationalised using an extrathermodynamic analysis. Nevetheless application of the solvent polarity scale based on Y-values is limited. The range of solvents for which Y-values can be measured is restricted.A feature of many dye molecules is the sensitivity of their colour to the solvent. This fact was exploited by Brooker and coworkers who used two dyes to define \(\chi_{\mathrm{B}}\) and \(\chi_{\mathrm{R}}\) values. These scales have not found wide application. A polarity which has attracted attention was suggested by Kosower. The scale is based on the uv/visible spectra of N-methyl pyridinium iodide. The low energy absorption band in the spectra characterises the charge transfer from iodide to the pyridinium ring. Kosower examined correlations between Z – and Y- values and between Z-values and other solvent sensitive partaneters . The consensus is that Z provides a reasonable sastifactory measure of solvent polarity.There can be little doubt that chemists find the concept of solvent polarity intuitively attractive . Granted the need there is an associated demand for a convenient, readily available method for measuring solvent polarity. Reichardt synthesised a betaine dye which - 2 - is particularly solvent sensitive as shown by the dependence on solvent of an intramolecular charge transfer band. Reichardt expresses the energy of the energy band maximum of the absorption band in kilocalories per mol which defines the \(\mathrm{E}_{\mathrm{T}}\) value for a given solvent. Solutions of the dye in methanol are red, violet in ethanol and green in propanone. So one has a striking visual indicator of solvent polarity.Foonotes In a solution which is defined as ideal in a thermodynamic sense there are no (solute molecule) \(\leftrightarrow\) (solute molecule) interactions. Hence the solute molecules are effectively infinitely far apart. Some indication of the ratio of solute to solvent molecules is indicated by the following rough calculation. Dilute aqueous solutions used in a study of chemical kinetics have concentrations of approx. \(10^{-3} \mathrm{mol dm}^{-3}\). In \(1 \mathrm{dm}^{3}\) of water there are \(55.5\) moles of water so the ratio of solute to solvent molecules is around \(55000\). Although the term is not used by chemists it may be helpful to imagine each solute molecule bathed in solvent molecules, implying a limitless expanse of solvent molecules around each solute molecule. We confine attention to the properties of solvents (e.g. polarities) at ambient pressure and at \(298.2 \mathrm{~K}\); i.e. 25 Celsius which is just above conventional room temperature. J.G.Kirkwood, J. Chem. Phys.,1934,2,351. Amis discusses treatments of kinetic data based on solvent permittivities; E. S. Amis, Solvent Effects on Reaction Rates and mechanisms, Academic Press, New York, 1966. See comments by N. S. Isaacs, Physical Organic Chemistry, Longmans, London,1987. See also comments concerning attempts to identify a single solvent property which accounts for solvent effects on rates of chemical reactions; J. B. F. N. Engberts, in Water-A Comprehensive Treatise, ed. F.Franks, Plenum Press, New York, 1979,Volume 6, chapter 4. C.K.Ingold, Structure and Mechanism in Organic Chemistry, G. Bell, London, 1953; see page 347. See also A. Streitweiser, Solvolytic Displacement Reactions, McGraw-Hill, New York, 1962 Y-values have been used in the context of kinetics of reactions of inorganic solutes; M.J.Blandamer, J. Burgess, and S. Hamshere, Transit. Metals Chem.,1979,4, 291. J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions,Wiley, New York, 1963; Dover Publications, New York, 1989. L. G. S. Brooker, A. C. Craig, D.W. Heseltine, P.W.Jenkins and L. L. Lincoln, J. Am. Chem. Soc.,1965,87,2443. E. Kosower, J.Am. Chem. Soc.1958,80,3253. E. Kosower, Physical Organic Chemistry, Wiley, New York, 1968. C.Reichardt, Chem. Rev.,1994,94,2319; Chem. Soc. Rev., 1992,147; Solvents and Solvent Effects in Organic Chemistry, VCH, Weinheim, 2nd. edn.,1988.This page titled 1.14.9: Extrathermodynamics - Solvent Polarity is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.15.1: Heat Capacities: Isobaric: Neutral Solutes

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An aqueous solution molality \(\mathrm{m}_{j}\), at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), contains a simple neutral solute, \(j\). The chemical potential of the solute is given by equation (a). \[\begin{aligned} &\mu_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp} \end{aligned}\]\[\mathrm{H}_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]where \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]\[\begin{aligned} &\mathrm{C}_{\mathrm{pj}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}^{2}\right]_{\mathrm{p}} \end{aligned}\]Here \[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) / \partial \mathrm{T}\right]_{\mathrm{p}}\]For the solvent, the chemical potential is given by equation (f). \[\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{1}^{0}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{l}}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp}\]\[\mathrm{H}_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Hence, \[\begin{aligned} &\mathrm{C}_{\mathrm{pl} 1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}} \end{aligned}\]Where \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]However, \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p}_{1}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}} \text { (aq) }\]Hence, from equations (d) and (h), \[\begin{aligned} &\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\mathrm{n}_{1} \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\ &\left.+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right] \\ &+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right. \\ &\left.-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right] \end{aligned}\]We rearrange the latter equation to describe the isobaric heat capacity of a solution prepared using \(1 \mathrm{~kg}\) of water. \[\begin{aligned} &\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right. \\ &\left.\quad+2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}\right] \end{aligned}\]The term inside the [….] brackets is the apparent molar isobaric heat capacity for the solute. Thus, \[\begin{array}{r} \phi\left(C_{p j}\right)=C_{p j}^{\infty}(a q ; T ; p)-2 \, R \, T \,\left(\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right)_{p} \\ -R \, T^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{j}\right)}{\partial T^{2}}\right)_{p} \\ +2 \, R \, T \,\left(\frac{\partial \phi}{\partial T}\right)_{p}+R \, T^{2} \,\left(\frac{\partial^{2} \phi}{\partial T^{2}}\right)_{p} \end{array}\]Hence, \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]Then, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]Equation (m) shows that \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\) is a complicated property of a solution and that ‘the devil is in the detail’. A simplification in the algebra emerges if we define a set of J-properties which are excess properties. Thus for a given solution prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) mole of solute \(j\), \[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{J}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{J}_{\mathrm{j}}(\mathrm{aq})\]where \[J(a q)=C_{p}(a q)-C_{p}(a q ; i d)\]\[\mathrm{J}_{1}(\mathrm{aq})=\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})-\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)\]\[\mathrm{J}_{\mathrm{j}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]\[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=\phi\left(\mathrm{C}_{\mathrm{p}_{\mathrm{j}}}\right)-\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}\]But \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})-\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{id})=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}\right]\]Then \[\mathrm{J}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right)\]An extensive literature reports the partial molar heat capacities of solutes in aqueous solution.Further \({\mathrm{C}_{\mathrm{pj}}}^{\infty}(\mathrm{aq})\) for a range of related solutes can be analysed to yield group contributions; e.g. at \(298.15 \mathrm{~K}\) the contribution of a methyl group, \(\mathrm{CH}_{3}\) to \({\mathrm{C}_{\mathrm{pj}}}^{\infty}\) for an aliphatic solute is \(178 \mathrm{~J K}^{-1} \mathrm{~mol}^{-1}\). Granted that \({\mathrm{C}_{\mathrm{pj}}}^{\infty}(\mathrm{aq})\) has been obtained for solute \(j\) and that the molar heat capacity of pure liquid \(j\), \(\mathrm{C}_{\mathrm{pj}}^{*}(\ell)\) is known , the isobaric heat capacity of solution \(\Delta_{s \ln } \mathrm{C}_{\mathrm{pj}}^{0}\) is obtained. \[\Delta_{s \ln } C_{p j}^{0}=C_{p j}^{\infty}(a q)-C_{p j}^{*}(\ell)\] \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{w}_{1}=1.0 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]\[\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\]\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,\left[\mathrm{K}^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right.\]\[\mathrm{m}_{\mathrm{j}} \, 2 \, \mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{K}^{-1}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\right.\]\[\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial^{2} \phi}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-2}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]\] Sucrose(aq); an early determination; F. T. Gucker and F. D. Ayres, J. Am. Chem.Soc.,1937,59,447. For partial molar isobaric heat capacities of neutral solutes in aqueous solution see, ROH(aq); D. Mirejovsky and E. M. Arnett, J. Am. Chem.Soc.,1983,105,112. Ph-X; group additivity; G. Perron and J. E. Desnoyers, Fluid Phase Equilib, 1979,2,239. Amides(aq); R. Skold, J. Suurkuus and I. Wadso, J.Chem. Thermodyn., 1976,8,1075. \(\Delta_{s \ln } \mathrm{C}_{\mathrm{pj}}^{0}\) for gases(aq); G. Olofsson, A. A. Oshodj, E. pj Qvarstrom and I. Wadso, J. Chem. Thermodyn., 1984,16,1041.This page titled 1.15.1: Heat Capacities: Isobaric: Neutral Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.15.2: Heat Capacities: Solutions: Solutes: Interaction Parameters

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.15%3A_Heat_Capacities/1.15.2%3A_Heat_Capacities%3A_Solutions%3A_Solutes%3A_Interaction_Parameters

We describe an excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) for a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) ion terms of solute-solute enthalpic interaction parameters. \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]The corresponding excess isobaric heat capacity is defined by equation (b). \[C_{p}^{E}\left(a q ; w_{1}=1 k g\right)=c_{p i j} \,\left(m_{j} / m^{0}\right)^{2}\]where \[\mathrm{c}_{\mathrm{pij}}=\left(\frac{\partial \mathrm{h}_{\mathrm{ij}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Here \(\mathrm{c}_{\mathrm{pjj}}\) is a pairwise solute-solute interaction isobaric heat capacity. From \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]then, \[\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})=\mathrm{C}_{\mathrm{pl}^{2}}^{*}(\ell)-\mathrm{M}_{\mathrm{l}} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]From \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}^{\prime}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]then, \[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]Footnote For a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\]Hence \[\begin{gathered} \mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{C}_{\mathrm{pl}}^{*}(\ell)-\mathrm{M}_{1} \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \\ +\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+2 \, \mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right] \end{gathered}\]Then, \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{c}_{\mathrm{pji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]Since, \[\begin{gathered} \mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl} 1}^{*}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \\ \mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{c}_{\mathrm{pjj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \end{gathered}\]This page titled 1.15.2: Heat Capacities: Solutions: Solutes: Interaction Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.15.3: Heat Capacities: Isobaric: Solutions: Unit Volume

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.15%3A_Heat_Capacities/1.15.3%3A_Heat_Capacities%3A_Isobaric%3A_Solutions%3A_Unit_Volume

A given aqueous solution was prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of solute \(j\). Then, \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]For the solution, by definition, the isobaric heat capacity per unit volume, (or heat capacitance) \[\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\]Similarly for the solvent at the same temperature and pressure, \[\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\]With reference to equations (b) and (c), the four experimentally determined quantities are \(\sigma(\mathrm{aq}), \sigma_{1}^{*}(\mathrm{aq}), \rho(\mathrm{aq}) \text { and } \rho_{1}^{*}(\ell)\). The latter two quantities are the densities of the solution and solvent respectively.Hence \(\sigma(\mathrm{aq})\) is related to the concentration of the solution, \(\mathrm{c}_{j}\). \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}}\]The latter equation relates \(\sigma(\mathrm{aq})\) to the property for the pure solvent, \(\sigma_{1}^{*}(\ell)\) and to the concentration of solute, \(\mathrm{c}_{\mathrm{j}}\). Equation (d) relates \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\) to the measured quantities \(\sigma(\mathrm{aq})\) and \(\sigma_{1}^{*}(\ell)\) together with the apparent molar volume \(\phi \left(\mathrm{V}_{\mathrm{j}}\right)\). Thus \(\sigma(\mathrm{aq})\) and \(\phi \left(\mathrm{V}_{\mathrm{j}}\right)\) for a given solution yields together with \(\sigma_{1}^{*}(\ell)\), the apparent molar isobaric heat capacity of the solute, \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\). In cases where the composition of the solution is expressed using molalities, equation (e) is the equation for \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\). \[\begin{aligned} \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=& {\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] } \\ &+\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq}) \end{aligned}\] From equations (a) and (b), \[\sigma(\mathrm{aq})=\left[\mathrm{n}_{1} / \mathrm{V}(\mathrm{aq})\right] \,\left[\mathrm{V}_{1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\left[\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]The term \(\left[\mathrm{V}_{1}^{*}(\ell) / \mathrm{v}_{1}^{*}(\ell)\right]\) has been introduced with the definition of \(\sigma_{1}^{*}(\ell)\) in mind.But, \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]or, \[\mathrm{n}_{\mathrm{l}} \, \mathrm{V}_{1}^{*}(\ell)=\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Further concentration, \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\) Then, \[\sigma(\mathrm{aq})=\left[\mathrm{V}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,[\mathrm{V}(\mathrm{aq})]^{-1} \, \sigma_{1}^{*}(\ell)+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]Hence, \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell) \,\left[1-\mathrm{c}_{\mathrm{j}}\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\mathrm{c}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]or \[\sigma(\mathrm{aq})=\sigma_{1}^{*}(\ell)+\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell)\right] \, \mathrm{c}_{\mathrm{j}}\] \[\begin{aligned} &\phi\left(\mathrm{C}_{\mathrm{pj}}\right) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\frac{\mathrm{mol}}{\mathrm{m}^{3}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right] \\ &\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{JK}^{-1} \mathrm{~m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right] \end{aligned}\] From, \[\mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq})\]But, \[\mathrm{V}(\mathrm{aq})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) / \rho(\mathrm{aq})\]Then, \[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma(\mathrm{aq})\]or (dividing by \(\mathrm{n}_{\mathrm{j}}\)), \[\left[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho(\mathrm{aq}) \, \mathrm{n}_{\mathrm{j}}}\right]+\left[\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]=\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})}\]But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \rho_{1}^{*}(\ell)\) Then, \[\left[\frac{1}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}\right]-\left[\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \sigma(\mathrm{aq})}\right]+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)}{\sigma(\mathrm{aq})}\]As an equation for \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\); \[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\frac{\sigma(\mathrm{aq})}{\rho(\mathrm{aq}) \, \mathrm{m}_{\mathrm{j}}}-\frac{\sigma_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\frac{\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq})}{\rho(\mathrm{aq})}\]Hence \[\begin{gathered} \phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right] \\ +\mathrm{M}_{\mathrm{j}} \, \sigma(\mathrm{aq}) / \rho(\mathrm{aq}) \end{gathered}\] \[\begin{aligned} &{\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell) \, \sigma(\mathrm{aq})-\rho(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)\right]=} \\ &{\left[\frac{\mathrm{kg}}{\mathrm{mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\frac{\mathrm{kg}}{\mathrm{m}^{3}}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]} \end{aligned}\]This page titled 1.15.3: Heat Capacities: Isobaric: Solutions: Unit Volume is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.15.4: Heat Capacities: Isobaric: Salt Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.15%3A_Heat_Capacities/1.15.4%3A_Heat_Capacities%3A_Isobaric%3A_Salt_Solutions

The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) of an aqueous salt solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of a 1:1 salt is related to \(\mathrm{m}_{j}\) using the DHLL. Because \({\mathrm{C}_{\mathrm{p}}}^{\mathrm{E}}\) is the isobaric temperature dependence of \(\mathrm{H}^{\mathrm{E}}\), then \({\mathrm{C}_{\mathrm{p}}}^{\mathrm{E}}\) for this aqueous solution is given by equation (a). \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-(4 / 3) \, \mathrm{R} \, \mathrm{m}_{\mathrm{j}}^{(3 / 2)} \,\left(\mathrm{m}^{0}\right)^{-1 / 2}\left[2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]\]\[\mathrm{S}_{\mathrm{Cp}_{\mathrm{p}}}=2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right)\]\(\mathrm{S}_{\mathrm{Cp}}\) is the DHLL factor in the equation for the isobaric heat capacity. \[C_{p}^{E}\left(a q ; w_{1}=1 \mathrm{~kg}\right)=-(4 / 3) \, R \, S_{C p} \, m_{j}^{(3 / 2)} \,\left(m^{0}\right)^{-1 / 2}\]Using equation (c), \[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=-(4 / 3) \, \mathrm{R} \, \mathrm{S}_{\mathrm{Cp}_{\mathrm{p}}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]We could perhaps have anticipated that according to DHLL, \(\phi \left(\mathrm{J}_{j}}\right)\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}\). An extensive literature describes the limiting partial molar isobaric heat capacities of ions in aqueous solution. One of the earliest investigations of the isobaric heat capacities of salt solutions was made by Randall and Ramage and later by Randall and Taylor. The groups lead by Hepler and by Desnoyers have made significant contributions in this area. However no agreement has been reached on a scale of absolute values. Hepler reported relative estimates based on \(\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq} ; 298 \mathrm{~K}\right)\) equal to zero. Perhaps most attention has been directed at salts formed by alkylammonium cation and hydrophobic anions; e.g. amino acids, phenylcarboxylates, t-butylcarboxlates and cryptates. Data for \(\mathrm{R}_{4} \mathrm{~N}^{+} \mathrm{Br}^{-}(\mathrm{aq})\) show that \(\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\) increases with increase in hydrophobic character of the R-group. French and Criss argue in favour of a scale which sets \(\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right)\) at \(– 68 \mathrm{~J K}^{-1} \mathrm{~mol}^{-1}\). An attempt has identified the various contributions to \(\mathrm{C}_{\mathrm{p}}^{\infty}(\text { ion; aq })\). Certainly trends in \(\mathrm{C}_{\mathrm{p}}^{\infty}(\text { ion; aq })\) point to characteristic features associated with the properties of ions in aqueous solution. Nevertheless, interpretation is not straightforward. \[\mathrm{S}_{\mathrm{Cp}}=2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right)_{\mathrm{p}}= \,[\mathrm{K}] \,\left[\mathrm{K}^{-1}\right]+[\mathrm{K}]^{2} \,\left[\mathrm{K}^{-1}\right] \,[\mathrm{K}]^{-1}=\] \[\phi\left(\mathrm{J}_{\mathrm{j}}\right)= \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \, \,=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\] M. Randall and W.D.Ramage, J.Am.Chem.Soc.,1927,49,93 M.Randall and M.D.Taylor, J. Am. Chem. Soc.,1941,45,959. I. K. Hovey, L. G. Hepler and P. R. Tremaine, J. Phys. Chem.,1988, 92, 1323; Thermochim. Acta, 1988, 126, 245; J. Chem. Thermodyn.,1988, 20, 595. J. J. Spitzer, I. V. Oloffson, P. P. Singh and L. G. Hepler, Thermochim. Acta,1979, 28, 155. J.-L. Fortier, P.-A. Leduc and J. E. Desnoyers, J. Solution Chem, 1974, 3, 323. B. Chawla and J. C. Ahluwalia, J. Phys. Chem.,1972 76, 2582. E. M. Arnett and J. J. Campion, J. Am. Chem. Soc.,1970, 92, 7097. J. C. Ahluwalia, C. Ostiguy, G. Perron and J. E. Desnoyers, Can. J. Chem., 1977, 55, 3364, and 3368. M. Lucas and H. Le Bail, J. Phys. Chem., 1976, 80, 2620. N. Morel-Desrosiers and J. P. Morel, J. Phys. Chem., 1985, 89, 1541. R. N. French and C. M. Criss, J. Solution Chem.,1982, 11, 625. C. Shin, I. Worsley and C.M.Criss, J. Solution Chem.,1976, 5 , 867. For further details of heat capacities of salt solutions see—This page titled 1.15.4: Heat Capacities: Isobaric: Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.15.5: Heat Capacities: Isochoric: Liquid Mixtures: Ideal

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For an ideal binary liquid mixture the molar isobaric heat capacity is given by the mole fraction weighted sum of the isobaric heat capacities of the pure liquid components. \[\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\]Both \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\) and \(\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\) can be measured so that \(\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) can be calculated for a given mixture as a function of mole fraction composition. Further \[\Delta_{\operatorname{mix}} \mathrm{C}_{\mathrm{p}}(\mathrm{id})=0\]The isochoric heat capacity of the corresponding ideal mixture is related to the isobaric heat capacity using equation (c). \[\mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\text { mix } ; \mathrm{id})=\mathrm{C}_{\mathrm{pm}}(\text { mix } ; \mathrm{id})-\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\text { mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\text { mix } ; \mathrm{id})}\]Equations (a) and (c) provide an equation for \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) in terms of the isochoric heat capacities of the pure liquid components. \[\begin{aligned} &\mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\operatorname{mix} ; \mathrm{id})=\\ &\mathrm{x}_{1} \,\left[\mathrm{C}_{\mathrm{V} 1}^{*}(\ell)+\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\mathrm{x}_{2} \,\left[\mathrm{C}_{\mathrm{V} 2}^{*}(\ell)+\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right]\\ &-\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})} \end{aligned}\]In terms of forming an ideal binary liquid mixture from two pure components, \[\begin{gathered} \Delta_{\operatorname{mix}} \mathrm{C}_{\mathrm{V}_{\mathrm{m}}}(\mathrm{id})=\mathrm{x}_{1} \,\left[\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right] \\ -\frac{\mathrm{T} \,\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{~K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})} \end{gathered}\]The equations become more complicated as we switch conditions from the intensive variables, \(\mathrm{T}\) and \(\mathrm{p}\), to extensive variables such as entropy and volume. The equations become even more complicated when we turn to a description of real mixtures.Footnote Consider a closed system subjected to a change in temperature, the system remaining at equilibrium where the affinity for spontaneous change is zero. Then \[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0} \quad \text { and } \quad \mathrm{C}_{\mathrm{V}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{T}}\right)_{\mathrm{V}, \mathrm{A}=0}\]In the following we drop the condition ‘\(\mathrm{A}=0\)’ and take it as implicit in the following analysis. [A similar set of equations can be written for the condition ‘at fixed ξ’.] Then \(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{U}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\) but by definition, \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\) Then \[C_{p}-C_{V}=\left(\frac{\partial H}{\partial T}\right)_{p}-\left(\frac{\partial H}{\partial T}\right)_{v}+V \,\left(\frac{\partial p}{\partial T}\right)_{v}\]Using a calculus operation, \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{v}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}}\) Then, \[\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\left[\mathrm{V}-\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right] \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]By definition \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\); then \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]A Maxwell equation requires that \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Then, \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{V}-\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Hence, \[\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]A calculus operation requires that \(\left(\frac{\partial p}{\partial T}\right)_{V} \,\left(\frac{\partial T}{\partial V}\right)_{p} \,\left(\frac{\partial V}{\partial p}\right)_{T}=-1\) Then \[C_{p}-C_{V}=-T \,\left[\left(\frac{\partial V}{\partial T}\right)_{p}\right]^{2} \,\left[\left(\frac{\partial V}{\partial p}\right)_{T}\right]^{-1}\]This page titled 1.15.5: Heat Capacities: Isochoric: Liquid Mixtures: Ideal is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.16.1: Ion Association

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The term ‘ strong electrolyte’ has a long and honourable history in the development of an understanding of the properties of salt solutions. This term describes salt solutions where each ion contributes to the properties almost independently of all other ions in a given solution. The word ‘almost’ signals that the properties of a given salt solution are determined in part by charge –charge interactions between ions through the solvent separating ions in solution. Otherwise the ions can be regarded as free. Such is the case for aqueous salt solutions at ambient temperatures and pressures prepared using 1:1 salts such as \(\mathrm{Na}^{+} \mathrm{Cl}^{-}\), \(\mathrm{Et}_{4}\mathrm{N}^{+} \mathrm{Br}^{-}\) …However with decrease in relative permittivity of the solvent, the properties of salt solutions indicate that not all the ions can be regarded as free; a fraction of the ions are associated. For dilute salts solutions in apolar solvents such as propanone a fraction of the salt is described as being present as ion pairs formed by association of cations and anions. With further decrease in the permittivity of the solvent higher clusters are envisaged; e.g. triple ions, quadruple ions…. Here we concentrate attention on ion pair formation building on the model proposed by N. Bjerrum.The analysis identifies a given \(j\) ion in a salt solution as the reference ion such that at distant \(\mathrm{r}\) from this ion the electric potential equals \(\psi_{j}\) whereby the potential energy of ion \(\mathrm{i}\) with charge number \(\mathrm{z}_{\mathrm{i}}\) equals \(\mathrm{z}_{i} \, e \, \psi_{j}. The solvent is a structureless continuum and each ion is a hard non-polarisable sphere characterised by its charge, \(\mathrm{z}_{j} \, e\), and radius \(\mathrm{r}_{j}\).If the bulk number concentration of \(\mathrm{i}\) ions is \(\mathrm{p}_{\mathrm{i}}\), the average local concentration of \(\mathrm{i}\) ions' \(\mathrm{p}_{\mathrm{i}}\) is given by equation (a). \[\mathrm{p}_{\mathrm{i}}^{\prime}=\mathrm{n}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right)\]The number of \(\mathrm{i}\)-ions, \(\mathrm{dn}_{\mathrm{i}}\) in a shell thickness \(\mathrm{dr}\) distance \(\mathrm{r}\) from the reference \(j\) ion is given by equation (b). \[\mathrm{p}_{\mathrm{j}}=\mathrm{n}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]At small \(\mathrm{r}\), the electric potential arising from the \(j\) ion is dominant. Hence, \[\psi_{\mathrm{j}}=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\]Hence, \[\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e}}{\mathrm{k} \, \mathrm{T}} \, \frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]Or, \[\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e}^{2}}{4 \, \pi \, \mathrm{k} \, \mathrm{T} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]Using equation (e), the number of ions in a shell, thickness \(\mathrm{dr}\) and distance \(\mathrm{r}\) from the \(j\) ion, at temperature \(\mathrm{T}\) in a solvent having relative permittivity \(\varepsilon_{\mathrm{r}}\) is obtained for ions with charge numbers \(\mathrm{z}_{\mathrm{i}}\) and \(\mathrm{z}_{j}\).For two ions having the same sign \(\mathrm{dn}_{\mathrm{i}}\) increases with increase in \(\mathrm{r}\), a pattern intuitively predicted. However for ions of opposite sign an interesting pattern emerges in which \(\mathrm{dn}_{\mathrm{i}}\) decreases with increase in \(\mathrm{r}\), passes through a minimum and then increases. In other words there exists a distance \(\mathrm{q}\) at which there is a minimum in the probability of finding a counterion. Thus \[\mathrm{q}=\frac{\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}}\right| \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\]For a given salt, \(\mathrm{q}\) increases with decrease in \(\varepsilon_{\mathrm{r}}\) at fixed \(\mathrm{T}\). Bjerrum suggested that the term ‘ion pair’ describes two counter ions where their distance apart is less than \(\mathrm{q}\). In other words the proportion of a given salt in solution in the form of ions pairs increases with decrease in \(\varepsilon_{\mathrm{r}}\). The interplay between solvent permittivity and ion size \(\mathrm{a}_{j}\) as determined by the sum of cation and anion radii is important. For a fixed \(\mathrm{a}_{j}\), the fraction of ions present as ion pairs increases with decrease in relative permittivity of the solvent. Thus high \(\varepsilon_{\mathrm{r}}\) favours description of a salt as present as only ‘free’ cations and anions. The properties of such a real solutions might therefore be described using the Debye-Huckel Limiting Law. By way of contrast as \(\varepsilon_{\mathrm{r}}\) decreases the extent of ion pair formation increases with decrease in ion size.The fraction of salt in solution \(\theta\) in the form of ion pairs is given by the integral of equation (e) within the limits \(\mathrm{a}\) and \(\mathrm{q}\) where \(\mathrm{a}\) is the distance of closest approach of cation and anion. Thus \[\theta=4 \, \pi \, p_{i} \, \int_{a}^{q} \exp \left(-\frac{z_{+} \, z_{-} \, e^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, k \, T \, r}\right) \, r^{2} \, d r\]Hence, for a solution where the concentration of salt \(\mathrm{c}_{j}\) expressed using the unit, \(\mathrm{mol dm}^{-3}\), \(\theta\) is given by equation (h). \[\theta=\frac{4 \, \pi \, N}{10^{3}} \,\left(\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}\right|}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\right)^{3} \, \mathrm{Q}(\mathrm{b})\]where \[Q(b)=\int_{2}^{b} x^{-4} \, e^{x} \, d x\]with \[\mathrm{b}=\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T} \, \mathrm{a}}\]and \[\mathrm{x}=-\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T} \, \mathrm{r}}\]The integral \(\mathrm{Q}(\mathrm{b})\) has been tabulated as a function of \(\mathrm{b}\). According to equation (h), \(\theta\) increases with increase in \(\mathrm{b}\); i.e. with increase in \(\mathrm{a}\) and decrease in \(\varepsilon_{\mathrm{r}}\).The analysis leading to equation (h) is based on concentrations of salts in solution. Therefore the equilibrium between ions and ion pairs is described using concentration units. Here we consider the case of a 1:1 salt (e.g \(\mathrm{Na}^{+} \mathrm{Cl}^{-}\)) in the form of the following equilibrium describing the dissociation of ion pairs. [A common convention in this subject is to consider ‘dissociation’.] For a 1:1 salt \(j\) in solution the chemical potential \(\mu_{j}(s \ln )\) is given by equation (l). \[\mu_{\mathrm{j}}(\mathrm{s} \ln )=\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\pm} / \mathrm{c}_{\mathrm{r}}\right)\]The mean ionic activity coefficient (concentration scale) is defined by equation (m) \[\operatorname{limit}\left(c_{j} \rightarrow 0\right) y_{\pm}=1.0 \text { at all T and } p\]The thermodynamic properties of the neutral (dipolar) ion pair are treated as ideal. Then, \[\mu_{\mathrm{ip}}(\mathrm{s} \ln )=\mu_{\mathrm{ip}}^{0}(\mathrm{~s} \ln )+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{ip}} / \mathrm{c}_{\mathrm{r}}\right)\]The equilibrium between ‘free’ ions (i.e. salt \(j\)) and ion pairs is described by the following equation. \[\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s} \ln ) \Leftarrow \Rightarrow \mathrm{M}^{+}(\mathrm{s} \ln )+\mathrm{X}^{-}(\mathrm{s} \ln )\]Then, \[\mu_{i p}(s \ln )=\mu_{j}(s \ln )\]Hence the ion pair dissociation constants \(\mathrm{K}_{\mathrm{D}}\) is given by equation (q). \[\Delta_{\text {diss }} G^{0}=-R \, T \, \ln \left(K_{D}\right)\]where \[\Delta_{\text {diss }} G^{0}=\mu_{j}^{0}(s \ln )-\mu_{\mathrm{ip}}^{0}(\mathrm{~s} \ln )\]Hence, \[K_{D}=\frac{\left(c_{j} \, y_{\pm} / c_{r}\right)^{2}}{\left(c_{i p} / c_{r}\right)}\]But \(\mathrm{c}_{\mathrm{j}}=\theta \, \mathrm{c}_{\mathrm{s}}\) and \(\mathrm{c}_{\mathrm{ip}}=(1-\theta) \, \mathrm{c}_{\mathrm{s}}\) where \(\mathrm{c}_{\mathrm{s}}\) is the total concentration of salt \(\mathrm{M}^{+} \mathrm{X}^{-}\). Then, \[\mathrm{K}_{\mathrm{D}}=\frac{\theta^{2} \, \mathrm{y}_{\pm}^{2} \, \mathrm{c}_{\mathrm{s}}}{(1-\theta) \, \mathrm{c}_{\mathrm{r}}}\]\(\mathrm{K}_{\mathrm{D}}\) is dimensionless. The long-established convention in this subject defines a quantity \(\mathrm{K}_{\mathrm{D}}^{\prime}\). Thus \[\mathrm{K}_{\mathrm{D}}^{\prime}=\frac{\theta^{2} \, \mathrm{y}_{\pm}^{2} \, \mathrm{c}_{\mathrm{s}}}{(1-\theta)}\]For very dilute solutions, the assumption is made that \(\theta = 1\) and \(\mathrm{y}_{\pm} = 1\). Hence using equation (h), \[\begin{aligned} &\frac{1}{\mathrm{~K}_{\mathrm{D}}^{\prime}} \cong \frac{1-\theta}{\mathrm{C}_{\mathrm{S}}} \\ &\theta=\frac{4 \, \pi \, N}{10^{3}} \,\left(\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\right)^{3} \, \mathrm{Q}(\mathrm{b}) \end{aligned}\]The molar conductances of salt solutions at fixed \(\mathrm{T}\) and \(\mathrm{p}\) can be precisely measured. To a first approximation the molar conductance of a given solution offers a method of counting the number of free ions. For salt solutions in solvents of low permittivity the molar conductance offers a direct method for assessing the fraction of salt present as free ions and hence the fraction present as ion pairs. Hence electrical conductivities of salt solutions in solvents of low relative permittivity have been extensively studied in order to probe the phenomenon of ion pair formation.The classic study was reported by Fuoss and Kraus in 1933 who studied the electrical conductivities of tetra-iso-amylammonium nitrate in dioxan + water mixtures at \(298.15 \mathrm{~K}\) over the range \(2.2 \leq \varepsilon_{\mathrm{r}} \leq 78.6\). The dependence of measured dissociation constants followed the pattern required by Bjerrum’s theory. Following the publication of the study by Fuoss and Kraus, many papers were published confirming the general validity of the Bjerrum ion-pair model. We note below a few examples of these studies which lead in turn to developments of the theory. For example in solvents of very low relative pemitivities triple ions are formed of the ++- and +-- type. Many experimental techniques have been used to support the Bjerrum model; e.g. cryoscopic studies, electric permittivities of solutions and Wien effects.Following the Bjerrum model, other models were suggested and developed. Denison and Ramsey suggested that the term ‘ion pair’ describes ions in contact, all other ions being free. Sadek and Fuoss proposed that association of free ions to form contact ion ion pairs involved formation of solvent separated ion pairs, although they later withdrew the proposal. Gilkerson modified equations describing ion-pair formation to include parameters describing ion-solvent interaction. In 1957 Fuoss restricted the definition of the term ‘ion pair’ to ions in contact. The dipolar nature of an ion pair was confirmed by dielectric relaxation studies. In the development of theories of ion pair formation Hammett notes the models of ion pair formation which involve charged spheres in a continuous dielectric may only be relevant under especially favourable circumstances.The initial proposal by Bjerrum concerning ion pair formation has had an enormous impact in many branches of chemistry including mechanistic organic chemistry. Spectroscopic studies identified ion-pairs in solution using charge transfer to solvent spectra. Electron spin resonance identified the presence of ion pairs in solution. Particularly interesting are those solutions where the counterion hops between two sites in an organic radical anion.Returning to the context of thermodynamics, the Bjerrum model of ion association has been extended to descriptions of partial molar volumes, apparent molar heat capacities and compressibilities of salts in non-aqueous solutions including cyanomethane.Nevertheless the debate concerning ion association in solution has continued particularly with the development of statistical thermodynamic treatments of salt solutions. Grunwald comments on the debate. To some extent the question arises as to the extent to which formation of ions pairs is either assumed from the outset or emerges from a given theoretical model for a salt solution. N. Bjerrum, K. Danske Vidensk Selskab, 1926,7, No. 9. For more recent accounts see— \[\begin{gathered} \frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}=\frac{ \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A} \mathrm{s}^{-1}\right]}{[\mathrm{J}]}= \\ \mathrm{p}_{\mathrm{i}}=\left[\mathrm{m}^{-3}\right] \quad \mathrm{p}_{\mathrm{i}}^{\prime}=\left[\mathrm{m}^{-3}\right] \end{gathered}\] \[\mathrm{p}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \Psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}=\frac{1}{\left[\mathrm{~m}^{3}\right]} \, \, \,\left[\mathrm{m}^{3}\right]=\] \[\psi_{\mathrm{j}}=\frac{ \,[\mathrm{C}]}{ \, \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \, \,[\mathrm{m}]}=\frac{[\mathrm{A} \mathrm{s}]}{\left[\mathrm{As} \mathrm{V}^{-1}\right]}=[\mathrm{V}]\] \[\begin{aligned} &\left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e}^{2}}{4 \, \pi \, \mathrm{k} \, \mathrm{T} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \\ &=\frac{ \, \,[\mathrm{C}]^{2}}{ \, \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{Fm}^{-1}\right] \, \,[\mathrm{m}]}=\frac{[\mathrm{As}]^{2}}{[\mathrm{~J}] \,[\mathrm{F}]} \\ &=\frac{[\mathrm{As}]^{2}}{[\mathrm{~J}] \,\left[\mathrm{As} \mathrm{A} \mathrm{s} \mathrm{J}^{-1}\right]}= \end{aligned}\]Hence, \(\mathrm{p}_{\mathrm{j}}=\left[\mathrm{mol} \mathrm{m}^{-3}\right] \, \, \, \,\left[\mathrm{m}^{2}\right] \,[\mathrm{m}]=\) \[\mathrm{q}=\frac{ \, \,[\mathrm{C}]^{2}}{ \, \,\left[\mathrm{Fm}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \, \mathrm{s}]^{2}}{\left[\mathrm{~A}^{2} \mathrm{~s}^{2} \mathrm{~J}^{-1} \mathrm{~m}^{-1}\right] \,[\mathrm{J}]}=[\mathrm{m}]\] Distance \(\mathrm{q}\) corresponds to the distance where \(\frac{\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{q}}=2 \, \mathrm{k} \, \mathrm{T}\) We stress the distinction between association of cation \(\mathrm{M}^{+}\) and anion \(\mathrm{X}^{-}\) to form an ion pair and association in solution of \(\mathrm{H}^{+}\) and \(\mathrm{CH}_{3}\mathrm{COO}^{-}\) ions to form undissociated ethanoic acid. In the later case the cohesion is discussed in quantum mechanical terms. R. M. Fuoss and C. A. Kraus, J. Am. Chem. Soc.,1933,55,1019. The liquid mixture dioxan + water is notable for being completely miscible and ambient \(\mathrm{T}\) and \(\mathrm{p}\), the relative permitivities having a remarkable range. No other water + organic liquid offers such a range. R. M. Fuoss and C. A. Kraus, J. Am. Chem. Soc.,1933,55,2387. R. M. Fuoss, Chem. Rev.,1935,17,227. F. M. Batson and C. A. Kraus, J. Am. Chem. Soc.,1934,56,2017. G. S. Hooper and C. A. Kraus, J. Am. Chem. Soc.,1934,56,2265. R. M. Fuoss, J.Am.Chem.Soc.,1934,56,1031. D. J. Mead and R. M. Fuoss, J. Am. Chem.Soc.,1939,61,2047. J. T. Denison and J. B. Ramsey, J.Chem.Phys.,1950,18,770. H. Sadek and R. M. Fuoss, J. Am. Chem.Soc.,1954,76,5897,5902,5905. H. Sadek and R. M. Fuoss, J. Am. Chem. Soc.,1959,81,4511. W. Gilkerson, J. Chem. Phys.,1956,25,1199. R. M. Fuoss, J. Am. Chem. Soc.,1957,79,3304. A. H. Sharbaugh, H. C. Eckstrom and C. A. Kraus, J. Chem, Phys.,1947.15,54. G. Williams, J. Phys.Chem.,1959,63,532. L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd. edition, 1970. See for example, S. Winstein, P. E.Klinedinst and E. Clippinger, J Am. Chem. Soc., 1961,83,4986. E. A. Moelwyn-Hughes, Chemical Statics and Kinetics of Solutions, Academic Press, London, 1971,p.405. For further references see M. J. Blandamer, and M. F. Fox., Chem. Revs., 1970.70, 59. T. A. Claxton, J. Oakes and M. C. R. Symons, Trans. Faraday Soc., 1968, 64, 596. J.-F. Cote, J. E. Desnoyers and J-C. Justice, J Solution Chem.,1996,25,113. J.-F. Cote and J. E. Desnoyers, J. Solution Chem..,1999,28,395. E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997,chapter 12.This page titled 1.16.1: Ion Association is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.16.2: Ionic Mobilities: Aqueous Solutions

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A classic subject in physical chemistry concerns the electric conductivities of salt solutions, most interest being centred on the aqueous salt solutions. Although the electric conductivities of these systems, transport properties, do not come under the heading ‘thermodynamic properties’, these conductivities have played a major part in the task of understanding the thermodynamic properties of salt solutions.Generally however interest in the electric conductivities of salt solutions has waned as spectroscopic properties in all its form have moved to a dominant position in physical chemistry. Nevertheless the contributions made by research into the electric properties of salt solutions have been and remain enormously important.At this point there is merit in commenting on the technique, mass spectrometry. In this important experimental technique, ions are produced in an ion source and then subjected to an electric field gradient, where (usually) cations are accelerated. The ions pass through a magnetic field, the path of a given ion depending on the charge and mass of the ion.Descriptions of the electrical conductivities of salt solutions start out from a quite different basis. To understand the point we consider a reasonably concentrated aqueous solution of sodium chloride; i.e. \(0.1 \mathrm{~mol dm}^{-3} \equiv 0.1 \mathrm{~mol}\) salt in water, mass \(1 \mathrm{~kg} \equiv 0.1 \mathrm{~mol}\) salt in \((1.0/0.018) \mathrm{~mol} \text{ water } \equiv 0.1 \mathrm{~mol Na}^{+} \text { ions } + 0.1 \mathrm{~mol Cl}^{-} \text { ions } + 55.6 \mathrm{~mol} \text { water}(\ell)\). In other words, for every sodium ion there are 556 molecules of water in this aqueous solution. The contrast with the mass spectrometer experiment could not be more dramatic. Further in conventional experiments studying the electric conductivities of salt solutions, the effect of a modest electric potential gradient is simply to bias the otherwise Brownian motion of the ion in a direction depending on the sign of the charge on a given ion. As each ion makes its way through the solution it is jostled and impeded by the large number of solvent molecules. Nevertheless in theoretical treatments of the electric conductivities of salt solutions the theory envisages a slow direct progress through the solution, in the case of, for example, a cation down the electric potential gradient. The key experimental fact is that the electric properties of salt solutions at low electric currents and low electric potential gradients obey the phenomenological law, Ohm’s Law. Deviations from this law are observed for example at high electrical field gradients; e.g. Wien Effects.The key term in the context of the electric conductivities of a salt solution, concentration of salt \(\mathrm{c}_{j}\) is the molar conductivity \(\Lambda\) defined by equation (a) where \(\kappa\) is the electrolytic conductivity. \[\Lambda=\kappa / \mathrm{c}_{j}\]For a salt solution prepared using a 1:1 salt , the molar conductivity can be expressed as the sum of ionic conductivities , \(\lambda_{+}\) and \(\lambda_{-}\). Thus \[\Lambda=\lambda_{+}+\lambda_{-}\]Using equation (a), the electrolytic conductivity \(\kappa\) is related to the ionic conductivities using equation (c) \[\kappa=\mathrm{c}_{\mathrm{j}} \,\left(\lambda_{+}+\lambda_{-}\right)\]The electric mobility of a given ion, \(\mathrm{u}_{j}\) is related to the mobility \(v_{j}\) using equation (d). \[\mathrm{u}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} / \mathrm{E}\] \[\begin{aligned} \kappa &=(\text { electric current density) } / \text { (electric field strength }) \\ &=[\mathrm{j}] /[\mathrm{E}] \\ &=\left[\mathrm{A} \mathrm{m}^{-2}\right] /\left[\mathrm{V} \mathrm{m}^{-1}\right]=\left[\mathrm{S} \mathrm{m}^{-1}\right] \end{aligned}\] \(\Lambda=\left[\mathrm{S} \mathrm{m}^{-1}\right] /\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{S} \mathrm{} \mathrm{m}^{2} \mathrm{~mol}^{-1}\right]\) \(\mathrm{u}_{\mathrm{j}}=\left[\mathrm{m} \mathrm{s}^{-1}\right] /\left[\mathrm{V} \mathrm{m}^{-1}\right]=\left[\mathrm{m}^{2} \mathrm{~s}^{-1} \mathrm{~V}^{-1}\right]\)This page titled 1.16.2: Ionic Mobilities: Aqueous Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.16.3: Ionic Strength: Ional Concentration

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The ionic strength of a salt solution I containing \(\mathrm{i}\)-ionic substances is defined by equation (a); \(\mathrm{m}_{j}\) is the molality of ionic substance-\(j\), charge number \(\mathrm{z}_{j}\). \[\mathrm{I}=(1 / 2) \, \sum_{\mathrm{j}=\mathrm{i}}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \, \mathrm{z}_{\mathrm{j}}^{2}\]The sum is taken over all \(\mathrm{i}\)-ionic substances in the solution. The situation is slightly complicated by the fact some authors use the term ‘ionic strength’ where the concentration \(\mathrm{c}_{j}\) (expressed using the unit, \(\mathrm{mol dm}^{-3}\)) replaces \(\mathrm{m}_{j}\). The substitution is reasonably satisfactory for dilute salt solutions at ambient \(\mathrm{T}\) and \(\mathrm{p}\) where the mass of water, volume \(1 \mathrm{~dm}^{3}\), is approx. \(1 \mathrm{~kg}\).The ional concentration of a salt solution \(\Gamma\) is defined by equation (b) where \(\mathrm{c}_{j}\) is expressed using the unit, \(\mathrm{mol dm}^{-3}\). \[\Gamma=\sum_{j=i}^{j=i} c_{j} \, z_{j}^{2}\] An aqueous solution contains \(\mathrm{K}_{2}\mathrm{SO}_{4}\) (\(0.1 \mathrm{~mol}\)) in \(1 \mathrm{~kg}\) of water(\(\ell\)).\(\mathrm{m}\left(\mathrm{K}_{2} \mathrm{SO}_{4}\right)=0.1 \mathrm{~mol} \mathrm{~kg}^{-1} ; \mathrm{m}\left(\mathrm{K}^{+}\right)=0.2 \mathrm{~mol} \mathrm{~kg}^{-1}\)and \(\mathrm{m}\left(\mathrm{SO}_{4}{ }^{2-}\right)=0.1 \mathrm{~mol} \mathrm{~kg}^{-1}\)Hence \(\mathrm{I}=(1 / 2) \,\left[(0.2)+\left(2^{2} \mathrm{X} 0.1\right)\right]=0.3 \mathrm{~mol} \mathrm{~kg}^{-1}\) H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, , Reinhold, New York, 2nd. revised edn.1950, p.33.This page titled 1.16.3: Ionic Strength: Ional Concentration is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.16.4: Ion-Water Interactions

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The seminal paper in this subject was published in 1933 by Bernal and Fowler. These authors drew attention to the possible impact of ions on water - water interactions in aqueous solutions beyond nearest-neighbour water molecules. The paper is also notable for the fact that the authors in their examination of the properties of ice, water and salt solutions did not use the term ‘hydrogen bond’. At that time there was much debate concerning the nature of this interaction bearing in mind that the ‘valency’ of hydrogen is unity. Rather Bernal and Fowler concluded that ‘the unique properties of water are due to a structure of an extended complex characterized by tetrahedral co-ordination’.Verwey drew attention to the importance of the interactions between an ion and near-neighbor water molecules in aqueous solution, these water molecules often being described as ‘electrostricted ‘ by strong ion- solvent dipole interactions. Solvent water plays an important role in the control of partial molar entropies of ions in aqueous solutions and transfer entropies of ions from aqueous to non-aqueous solvents.A major landmark was a paper published by Frank and Evans who developed the concept that solutes, polar and apolar, have an important impact on water - water interactions in aqueous solutions. In the development of models for ionic hydration, a distinction is drawn between hydrophilic and hydrophobic ions. Hydrophilic ions (alkali metal cations and halide anions) have strong attractive interactions with neighbouring dipolar water molecules. Neutron scattering data reveal important information concerning the arrangement of water molecules contiguous to ions.For example in the case of chloride ions, the \(\mathrm{Cl} – \mathrm{~H} - 0\) configuration is essentially linear. Nevertheless, there is clear evidence, albeit often secondary, that strong water - ion interactions have an impact on water - water interactions beyond the immediate hydration sheath. Viscosity data indicate that ions, such as iodide and potassium, have a structure breaking effect. The cospheres, for these ions, are drawn, showing two parts; an inner zone A and an outer zone B.In zone A, ion - water dipole interactions are strong, leading to the general description ionichydration. An indication of the structure of hydrated ions in solution emerges from X-ray crystallographic studies. In the case of \(\mathrm{KF}.4\mathrm{H}_{2}\mathrm{O}\), the structure comprises \(\mathrm{K}^{+} \left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\) and \(\mathrm{F}^{-} \left(\mathrm{H}_{2}\mathrm{O}\right)_{6}\) octahedra; the \(\mathrm{K}^{+} -\mathrm{O}\) distance is \(0.279 \mathrm{~nm}\). Kebarle showed that mass spectrometry could be used to study ion - water interactions and, interestingly, step-wise hydration in the gas phase.If a given ion in aqueous solution is indeed surrounded by two zones identified as zones A band B, the expectation is that ion - ion interactions in solution will reflect the impact of these structural features.In the context of the impact of zone B, the suggestion was that with increase in size of ions so zone B should increase. Hence the expectation was that, for example, the partial molar isobaric heat capacity of tetra-n-butylammonium bromide in aqueous solution would be large in magnitude and negative in sign. Such not the case; the sign is positive. A link was therefore established between the hydration characteristic of tetra-alkylammoniun ions and the structures of the corresponding salt hydrates; e.g. tetra-iso-amylammonium fluoride hydrate, \((\text{iso}-\mathrm{Am})_{4}\mathrm{N}^{+} \mathrm{~F}^{-} 38 \mathrm{~H}_{2}\mathrm{O}\). Generally, therefore, tetra-alkylammonium ions of \(\mathrm{C}_{4} - \mathrm{~C}_{9}\) carboxylates and tri-alkylsulphonium ions are often identified as hydrophobic where the interaction between these ions and neighboring water molecules is weak.. Interestingly, constricting the alkyl chains to form azoniaspiroalkane cations diminishes the hydrophobic character. The impact of replacing a hydrophobic terminal group in \(\mathrm{R}_{4}\mathrm{N}^{+}\) ions by a hydrophilic group on the properties of aqueous solutions is dramatic and offers an interesting insight into the role of ion – water interactions. In contrast Finney and co-workers report that neutron diffraction data for aqueous solutions, containing \(\mathrm{Me}_{4}\mathrm{N}^{+} \mathrm{~Cl}^{-}\), show no evidence for increased ice-like structure compared to pure water. Nevertheless thermodynamic and transport properties generally point to the conclusion that the ion \(\mathrm{Me}_{4} \mathrm{~N}^{+}\) does not promote near-neighbor water-water hydrogen bonding. J. D. Bernal and R. H. Fowler, J.Chem.Phys.,1933,1,515. See for example, P. A. Kollman and L. C. Allen, Chem. Rev.,1972, 72,283. E. J. Verwey, Rec. Trav. Chim.,1942,61,127. See also F. Vaslow, J. Phys.Chem.,1973,67,2773. C. M. Criss, J. Phys. Chem.,1974,78,1000. K. K. Kundu, Pure Appl. Chem.,1994,66,411. H. S. Frank and M. W. Evans, J. Chem. Phys.,1945,13,507. J. E. Enderby, Chem.Soc.Rev.,1995,24,159. G. W. Neilson and J. E. Enderby, Adv.Inorg.Chem.,1989,34,195; and references therein. J. E. Enderby and G.W. Neilson in Water A Comprehensive Treatise; ed.F. Franks, Plenum Press, New York, 1973,volume 6, chapter 1. H. S. Frank and W.-Y. Wen, Discuss Faraday Soc.,1957,24,756. W.-Y.Wen,in Ions and Molecules in Solution; ed. N. Tanaka, H. Otaki and R. Tamamushi, Elsevier, Studies in Physical and Theoretical Chemistry, Amsterdam, 1983, p.45. G. Beurskens and G. A. Jeffrey, J.Chem.Phys.,1964,41,917 and 924. P. Kebarle in Modern Aspects of Electrochemistry, ed.B.E.Conway and J. O’M. Bockris, 1974,9,1; and references therein H. S. Frank, J. Phys Chem.,1963,67,1554. E. M. Arnett and J. J. Campion, J.Am.Chem.Soc.,1970,92,7097. K. Tamaki, S. Yoshikawa and M. Kushida, Bull. Chem. Soc. Jpn., 1975,48,3018. G. A. Jeffrey, Prog. Inorg. Chem., 1967,8,43; and references therein D. Feil and G. A. Jeffrey, J.Chem.Phys.,1961,35,1863. C. Shin, I. Worsley and C.M. Criss, J. Solution Chem.,1976,5,867. S. Lindebaum, J. Phys. Chem.,1971,75,3733; and references therein P.-A. Leduc and J. E. Desnoyers, Can. J. Chem.,1973,51,2993. S. Lindenbaum, J.Chem.Thermodyn.,1971,3,625. A. H. Narten and S. Lindenbaum., J. Chem. Phys.,1969,51,1108. R. H. Boyd, J. Chem.Phys.,1969,51,1470. O. D. Bonner and C. F. Jumper, Infrared Physics, 1973,13,233. R. L. Kay, Adv. Chem. 1968, 73,1 T. S. Sarma and J. C. Ahluwalia, J. Phys. Chem.,1970,74,3547. T. S. Sarma, R. K. Mohanty and J. C. Ahluwalia, Trans. Faraday Soc.,1969, 65,2333. D. A. Johnson and J. F. Martin, J. Chem. Soc. Dalton, Trans.,1973,1585. T. S. Sunder, B. Chawla and J. C. Ahluwalia, J. Phys. Chem.,1974,78, 738. E. M. Arnett, M. Ho and L. L. Schaleyer, J.Am. Chem. Soc.,1970,93,77039. A. LoSurdo, W.-Y. Wen, C. Jolicoeur and J.-L.Fortier,J.Phys.Chem.,1977, 81, 1813;and references therein. W.-Y. Wen and S. Saito, J.Phys.Chem.,1965,69,3569. G.P. Cunningham, D.F. Evans and R.L. Kay, J.Phys.Chem.,1966,70,3998. J. Turner, A.K. Soper and J.L. Finney, Mol.Phvs.,1992,77,411. R. L. Kay, D. F. Evans and M.A. Matesich, in Solute-Solute Interactions, ed. J. F. Coetzee and C. D. Ritchie, Marcel Dekker, New York, 1976,voume.2, chapter 10. R. L. Kay, Water A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York, 1973, volume 3, chapter 4.This page titled 1.16.4: Ion-Water Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.1: Isentropic

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The term ‘adiabatic’ means that for a closed system no heat passes between system and surroundings; \(\mathrm{q} = 0\). The term ‘isentropic’ introduces the further constraint that the system remains at equilibrium with the surroundings; i.e. the affinity for spontaneous change is zero. From the Second Law, \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi \quad \text { where } \mathrm{A} \, \mathrm{d} \xi \geq 0\]The isentropic condition means that both \(\mathrm{A}\) and \(\mathrm{q}\) are zero. Hence \(\mathrm{dS}\) is zero, indicating that the entropy of the system remains constant. In other words, ‘isentropic’ describes an adiabatic change along an equilibrium and therefore reversible pathway.This page titled 1.17.1: Isentropic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.2: Isentropic Thermal Pressure Coefficient

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The volume of a given closed system is defined by the following set of independent variables where \(\xi\) is the general composition variable. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}} ; \mathrm{A}=0\right]\]We have rather over-defined the system. The aim is to identify the composition variable at equilibrium and the condition that the affinity for spontaneous change is zero. The dependent variable entropy for this system is defined in analogous fashion; equation (b). \[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}} ; \mathrm{A}=0\right]\]The system is perturbed by a change in temperature along a path for which the affinity for spontaneous change is zero. Moreover the entropy of the system remains the same as that given in equation (b). In order to hold the latter condition the equilibrium pressure must change. In the state defined by the independent variables \(\left[\mathrm{T}, \mathrm{p}, \xi^{e q} ; \mathrm{A}=0\right]\) the (equilibrium) isentropic differential dependence of pressure \(\mathrm{p}\) on temperature is the isentropic thermal pressure coefficient, \(\beta_{\mathrm{S}}\); equation.(c). \[\beta_{\mathrm{s}}=(\partial \mathrm{p} / \partial \mathrm{T})_{\mathrm{s}}\]Further \[\beta_{\mathrm{S}}=(\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}}\]Also, \[\beta_{\mathrm{s}}=\sigma /\left(\mathrm{T} \, \alpha_{\mathrm{p}}\right)\]Here \(\sigma\) is the isobaric heat capacity for unit volume (heat capacitance) of the system, \(\mathrm{C}_{\mathrm{p}} / \mathrm{~V}\). The three isentropic properties \(\alpha_{\mathrm{S}}\), \(\kappa_{\mathrm{S}}\) and \(\beta_{\mathrm{S}}\) are related using equation (f);. \[\beta_{\mathrm{s}}=-\alpha_{\mathrm{s}} / \kappa_{\mathrm{s}}\]With reference to the (equilibrium) thermal expansivity, \(\alpha_{\mathrm{S}}\), we envisage that the temperature is changed to produce a change in volume along a path for which the entropy remains the same as in equation (b) and the affinity for spontaneous change remains at zero. \[\alpha_{\mathrm{s}}(\mathrm{A}=0)=\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s} ; \mathrm{A}=0}\]In analogous fashion, \(\kappa_{\mathrm{S}}\) is a measure of the change in volume produced by a change in pressure. \[\kappa_{\mathrm{S}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S} ; \mathrm{A}=0}\] From \[\left[\frac{\partial}{\partial \mathrm{p}}\left(\frac{\partial \mathrm{H}}{\partial \mathrm{S}}\right)_{\mathrm{p}}\right]_{\mathrm{s}}=\left[\frac{\partial}{\partial \mathrm{S}}\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{s}}\right]_{\mathrm{p}}\]But at equilibrium where \(\mathrm{A}=0\), \(T=\left(\frac{\partial H}{\partial S}\right)_{P}\) and \(\mathrm{V}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{s}}\)Then \[\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}}\].From \(\beta_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}}\), Using the above Maxwell Relation, \[\beta_{\mathrm{S}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\] From the definition, \[\beta_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}}\]Using a calculus operation \[\beta_{\mathrm{s}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{T}}\]From the Gibbs - Helmholtz Equation, \[\left(\frac{\partial S}{\partial T}\right)_{p}=\frac{C_{p}}{T}\]From a Maxwell equation, \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\). Then \[\beta_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \, \frac{1}{\mathrm{E}_{\mathrm{p}}}\]But \[E_{p}=V \, \alpha_{p}\]Then, \[\beta_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{V}} \, \frac{1}{\mathrm{~T} \, \alpha_{\mathrm{p}}}\]Or, \[\beta_{\mathrm{s}}=\sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\] From the definition, \(\beta_{\mathrm{s}}=\left(\frac{\partial p}{\partial T}\right)_{\mathrm{s}}\), then, \(\beta_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}\) Then, \[\beta_{\mathrm{S}}=-\mathrm{E}_{\mathrm{S}} / \mathrm{K}_{\mathrm{s}}=-\left(\mathrm{E}_{\mathrm{S}} / \mathrm{V}\right) /\left(\mathrm{K}_{\mathrm{s}} / \mathrm{V}\right)=-\alpha_{\mathrm{S}} / \kappa_{\mathrm{S}}\]Also from and, \[\mathrm{E}_{\mathrm{s}} / \mathrm{K}_{\mathrm{s}}=-\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{V}} \, \frac{1}{\mathrm{~T} \, \alpha_{\mathrm{p}}}\]Then \[\alpha_{\mathrm{s}} / \kappa_{\mathrm{s}}=-\sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\]This page titled 1.17.2: Isentropic Thermal Pressure Coefficient is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.3: Iso-Variables

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Isobaric: A given system is held at constant pressure.Isothermal: A given system is held at constant temperature.Isochoric: A given closed system is held at constant volume.Isentropic: This condition, linked to the adiabatic constraint, requires that during a reversible change the entropy of a system remains constant in a particular thermodynamic process; e.g. compression. Adiabatic + Reversible = Isentropic. We can find isentropic processes which are irreversible. In this case they are not adiabatic.Isolated System: The boundary insulates a given system from the surroundings . This is not really an iso-variable in the thermodynamic sense.Isoperibol: In the vast majority of calorimetric experiments, the surroundings and the reaction vessel (the system+ container) are at constant temperature. When the experiment is initiated the composition of the closed system changes resulting from, for example, chemical reaction, mixing of liquids….. The temperature of the closed system changes albeit by a small amount because the processes taking place in the calorimeter are either exo- or endo-thermic. A sensitive detector is used to measure the change in temperature of the system. In pedantic terms the system is not constrained to be ‘isothermal’. So the calorimeter being used in such an experiment is an isoperibol calorimeter.This page titled 1.17.3: Iso-Variables is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.4: Isochoric Properties

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A given closed system is characterised by a given intensive variable \(\mathrm{X}\). In this section we have in mind an intensive property such as the relative permittivity of a liquid. The variable \(\mathrm{X}\) may also refer to an equilibrium constant and related parameters such as the enthalpy of reaction, \(\Delta_{\mathrm{r}}\mathrm{H}(\mathrm{T},\mathrm{p})\). In all cases we assert that the closed system is at thermodynamic equilibrium where the affinity for spontaneous change is zero. Thus we may define \(\mathrm{X}\) for a given system in terms of the temperature and pressure. \[\mathrm{X}=\mathrm{X}[\mathrm{T}, \mathrm{p}]\]The molar volume of the system is defined in analogous fashion. \[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}]\]Then \[\mathrm{dV}_{\mathrm{m}}=\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp}\]In other words the dependence of molar volume on \(\mathrm{T}\) and \(\mathrm{p}\) is characterised by the partial derivatives \(\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) and \(\left(\frac{\partial V_{m}}{\partial p}\right)_{T}\).With equation (b) and (c) in mind we return the intensive property \(\mathrm{X}\) described in equation (a). The dependence of \(\mathrm{X}\) on \(\mathrm{T}\) and \(\mathrm{p}\) is similarly characterized by the two partial derivatives, \(\left(\frac{\partial X}{\partial T}\right)_{p}\) and \(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\). A calculus operation yields an equation for the partial derivative \(\left(\frac{\partial \mathrm{X}}{\partial T}\right)_{\mathrm{V}(\mathrm{m})}\). Thus \[\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})}\]The property \(\left(\frac{\partial \mathrm{X}}{\partial T}\right)_{\mathrm{V}(\mathrm{m})}\) is the isochoric differential dependence of \(\mathrm{X}\) on \(\mathrm{T}\). Now (cf. equation (c)) volume \(\mathrm{V}_{\mathrm{m}\) depends on \(\mathrm{T}\). Hence to hold \(\mathrm{V}_{\mathrm{m}}\) constant, the pressure has to change. In fact equation (c) is used to find the required change in pressure for a given change in \(\mathrm{T}\); equation (e). \[\mathrm{dp}=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}_{\mathrm{m}}}\right)_{\mathrm{T}} \, \mathrm{dT}\]In other words the required change in pressure is determined by the equation of state for the system and is characteristic of the system, \(\mathrm{T}\) and \(\mathrm{p}\). For a given change in temperature, \(\delta \mathrm{T}(\exp )\) there is a defined change in pressure, \(\delta \mathrm{p}(\operatorname{def})\). The isochoric condition takes the following form granted that in the experiment we decide to change the temperature by an amount \(\delta \mathrm{T}\). \[\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}]=\mathrm{V}_{\mathrm{m}}[\mathrm{T}+\delta \mathrm{T}(\exp ) ; \mathrm{p}+\delta \mathrm{p}(\operatorname{def})]\]We now return to the property \(\mathrm{X}\) defined in equation (a). We consider the property \(\mathrm{X}\) at the two conditions highlighted in equation (f); \[\mathrm{X}[\mathrm{T}, \mathrm{p}] ; \quad \mathrm{X}[\mathrm{T}+\delta \mathrm{T}(\exp ) ; \mathrm{p}+\delta \mathrm{p}(\operatorname{def})]\]The term \(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})[\mathrm{T}, \mathrm{p}]}\) defines an isochoric dependence of \(\mathrm{X}\) on \(\mathrm{T}\) at pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\). At each temperature the isochoric dependence of \(\mathrm{X}\) on \(\mathrm{T}\) reflects the dependence of \(\mathrm{V}_{\mathrm{m}}\) on \(\mathrm{T}\).The analysis outlined above is repeated but in terms of the isochoric dependence of \(\mathrm{X}\) on pressure. In order that the volume of a system does not change when the pressure is changed by \(\delta \mathrm{p}(\exp )\), the temperature must be changed by an amount \(\delta \mathrm{T}(\operatorname{def})\) determined by the equation of state for the system. \[\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}]=\mathrm{V}_{\mathrm{m}}[\mathrm{T}+\delta \mathrm{T}(\operatorname{def}) ; \mathrm{p}+\delta \mathrm{p}(\exp )]\]We compare property \(\mathrm{X}\) under the isochoric condition given in equation (h); \[\mathrm{X}[\mathrm{T}, \mathrm{p}] ; \quad \mathrm{X}[\mathrm{T}+\delta \mathrm{T}(\operatorname{def}) ; \mathrm{p}+\delta \mathrm{p}(\exp )]\]\(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{v}_{(\mathrm{m})[\mathrm{T}, \mathrm{p}]}}\) describes the isochoric dependence of \(\mathrm{X}\) on pressure.We have carefully examined the concept of an isochoric dependence of a given variable on either \(\mathrm{T}\) or \(\mathrm{p}. The reason for this care emerges from the observation that the literature describes a number of isochoric parameters. In some cases the analysis is recognized as extrathermodynamic. In other cases a patina of thermodynamics is introduced into an analysis leading to further debate.This page titled 1.17.4: Isochoric Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.5: Isochoric Thermal Pressure Coefficient

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The equilibrium volume of a given closed system is defined by the following set of independent variables where \(\xi\) is the general composition variable. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}} ; \mathrm{A}=0\right]\]We have rather over-defined the system. The aim is to identify the composition variable at equilibrium and under the condition that the affinity for spontaneous change is zero. The system is perturbed by a change in temperature but we require that the system travels a path where the volume remains constant (and at equilibrium). The pressure must be changed in order to satisfy these conditions. By definition the isochoric differential dependence of pressure on temperature defines the isochoric thermal pressure coefficient. \[\beta_{V}=\left(\frac{\partial p}{\partial T}\right)_{v}\]Three interesting equations follow. \[\beta_{\mathrm{V}}=\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}\]\[\beta_{\mathrm{V}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}}\]\[\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}}\] From equation (a) \[\beta_{\mathrm{V}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\]Or, \[\beta_{\mathrm{v}}=\mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}=\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}\] Using a Maxwell relationship \[\beta_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}=-\left(\frac{\partial S}{\partial T}\right)_{V} \,\left(\frac{\partial T}{\partial V}\right)_{S}\]But \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}} / \mathrm{T}\) Then \[\beta_{\mathrm{v}}=-\mathrm{C}_{\mathrm{v}} / \mathrm{T} \, \mathrm{E}_{\mathrm{s}}=-\mathrm{C}_{\mathrm{v}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}}\] From and, \[\mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{E}_{\mathrm{S}}\]Or, \[\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{S}}\]This page titled 1.17.5: Isochoric Thermal Pressure Coefficient is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.6: Isotonic Method; Isopiestic Method

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The isotonic method is ‘beautifully simple’. The technique, described as both Isotonic and Isopiestic, leads to osmotic coefficients for solvents and activity coefficients for solutes in solution, generally aqueous solutions. Authors reporting their results describe apparatus and procedures which often differ marginally from those of other authors. Scatchard and coworkers describe how six small platinum cups, volume approx. \(15 \mathrm{~cm}^{3}\), are held in a gold-plated copper block, the cups being fitted with hinged lids. The cups and copper block, filled with solutions (see below) are held in a partially evacuated thermostatted chamber. The copper block is rocked gently. Over a period of time the cups are removed, weighed and replaced.The experiment ends when the masses of solutions in the cups are constant.The development of the isopiestic method can be traced to the experiments reported in 1917 by Bousfield (who used the word, iso-piestic). A closed system was set up containing several solid salts in separate sample cells together with a little water(\(\ell\)) in a separate sample cell. A little more water(\(\ell\)) was added to the separate sample cell and the sample cells containing salts reweighed over a period of many days. The uptake of water by the salts was monitored, eventually forming salt solutions. A quantity \(\mathrm{h}\), the number of moles of water taken up by a mole of salt was calculated; e.g. \(\mathrm{h } = 12.43(\mathrm{KCl}), 14.23 (\mathrm{NaCl}) \text { and } 17.18(\mathrm{LiCl})\). The system is isopiestic, meaning that all samples have equal vapour pressure. [One cannot help but feel sorry for Bousfield after reading the Discussion after the paper was presented at a meeting. The critics clearly did not appreciate what Bousfield was attempting to do.] Modern techniques developed from this approach.To illustrate the technique, consider the case where just two cups, \(\mathrm{A}\) and \(\mathrm{B}\), are used containing aqueous solutions of two salts, \(\mathrm{i}\) and \(\mathrm{j}\). Spontaneous transfer of solvent water occurs through the vapour phase until eventually (often after many hours) equilibrium is attained and no change in mass occurs. At equilibrium the chemical potentials of water in the two dishes are equal. Thus, \[\mu_{\mathrm{j}}^{\mathrm{eq}}(\operatorname{dish} \mathrm{A}, \mathrm{T}, \mathrm{p})=\mu_{\mathrm{i}}^{\mathrm{cq}}(\operatorname{dish} \mathrm{B}, \mathrm{T}, \mathrm{p})\]Granted that the masses of salts used to prepare the solutions in the two cups are accurately known, the mass of cups at equilibrium yields the equilibrium molalities. In most studies one dish (e.g. dish \(\mathrm{A}\)) holds a standard [e.g. \(\mathrm{KCl}(\mathrm{aq})\)] for which the dependence of practical osmotic coefficient on composition is accurately known.If, for example, the two cups contain aqueous salt solutions, equation (a) is rewritten as follows granted that the pressure is close to ambient. \[\left[\mu_{1}^{*}(\ell)-\phi_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]_{\mathrm{A}}=\left[\mu_{1}^{*}(\ell)-\phi_{\mathrm{i}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]_{\mathrm{B}}\]Here \(\mathrm{m}_{\mathrm{i}}\) and \(\mathrm{m}_{\mathrm{j}}\) are the equilibrium molalities, where the word ‘equilibrium’ refers to the solvent water. Hence \[\left(\phi_{\mathrm{j}} \, \mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}=\left(\phi_{\mathrm{i}} \, \mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}}\]The isopiestic ratio \(\mathrm{R}_{\text{iso}}\) is defined by equation (d). \[\mathrm{R}_{\text {iso }}=\left(\mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}} /\left(\mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}\]Hence, \[\phi_{\mathrm{B}}=\phi_{\mathrm{A}} / \mathrm{R}_{\mathrm{iso}}\]Therefore \(\phi_{\mathrm{B}}\) is obtained from the experimentally determined \(\mathrm{R}_{\text{iso}}\) and a known (i.e. previously published standard) \(\phi_{\mathrm{B}}\).In general terms, an ‘isopiestic experiment’ is based around the properties of the solvent water in a given solution. But the aim of the experiment is to gain information about the activity coefficient of the solute. The calculation therefore relies on the Gibbs - Duhem Equation . According to the Gibbs – Duhem equation the dependence of chemical potentials of salt and solvent are linked . If a given solution comprises \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{\mathrm{j}}\) moles of solute, then \[\mathrm{n}_{1} \,\left(\mathrm{d} \mu_{1} / \mathrm{dn} \mathrm{n}_{\mathrm{j}}\right)+\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{d} \mu_{\mathrm{j}} / \mathrm{dn}_{\mathrm{j}}\right)=0\]For a solution molality \(\mathrm{m}_{\mathrm{j}}\) in a solvent, molar mass \(\mathrm{M}_{1}\) \[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{\mathrm{1}} / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}} / \mathrm{dm}_{\mathrm{j}}=0\]Then if pressure \(\mathrm{p}\) is close to the standard pressure, \[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] / \mathrm{dm}_{\mathrm{j}}=0\]Or, \[-\phi-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+1+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}=0\]Thus \[-\left(\phi / \mathrm{m}_{\mathrm{j}}\right)-\mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\left(1 / \mathrm{m}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0\]No further progress can be made until we have determined in a series of experiments the dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\). The dependence of \(\phi_{\mathrm{B}}\) on molality \(\mathrm{m}_{\mathrm{B}}\) is obtained after many experiments. In a common procedure the dependence is fitted to a polynomial in mj such that integration yields the activity coefficient for the solute \(\gamma_{\mathrm{j}}\). Suppose for example we find that for a given system \(\phi\) is a linear function of molality \(\mathrm{m}_{\mathrm{j}}\). Thus \[\phi=1+\mathrm{a} \, \mathrm{m}_{\mathrm{j}}\]Or, \[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}=\mathrm{a}\]Hence, \[\ln \left(\gamma_{\mathrm{j}}\right)=2 \, \mathrm{a} \, \mathrm{m}_{\mathrm{j}}\]We note how the analysis relies on the fact the solute and solvent ‘communicate with each other’. G. Scatchard, W.J.Hamer and S.E.Wood, J.Am.Chem.Soc.,1938,60,3061. W. R. Bousfield, Trans. Farady Soc.,1917,13,401. J. A.Rard and R. F. Platford, Activity Coefficients in Electrolyte Solutions, ed. K. S. Pitzer, CRC Press, Boca Raton, 2nd edition, 1991. In fact one can regard the phenomenon as osmosis, the vapour phase being a perfect semi-permeable membrane. The system is partially evacuated so that equilibrium vapour pressure is reasonably rapidly attained.This page titled 1.17.6: Isotonic Method; Isopiestic Method is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.17.7: Isopiestic: Aqueous Salt Solutions

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An extensive literature reports applications of the isopiestic technique to the determination of osmotic coefficients and ionic activity coefficients for salt solutions. In effect the technique probes the role of ion-ion interactions in determining the properties of real salt solutions.Several approaches have been reported for analyzing isopiestic results. A common method starts with the isopiestic ratio \(\mathrm{R}_{\text{iso}}\). For solutions in dishes \(\mathrm{A}\) and \(\mathrm{B}\) at equilibrium, the isopiestic equilibrium conditions is given by equation (a). \[\left(\phi_{j} \, V_{j} \, m_{j}\right)_{A}=\left(\phi_{i} \, V_{i} \, m_{i}\right)_{B}\]The isopiestic ratio, \[\mathrm{R}_{\text {iso }}=\left(\mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}} /\left(\mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}\]An important task formulates an equation relating the osmotic coefficient for a given salt solution and the mean ionic coefficient \(\gamma_{\pm}\)If the salt solution contains a single salt, then according to the Gibbs-Duhem Equation, \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)=-m_{j} \, d \mu_{j}(a q)\]Hence (where pressure \(\mathrm{p}\) is close to the standard pressure) \[\begin{aligned} \left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\mathrm{l})\right.&\left.-\left(\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{v} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right)\right]=\\ &-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\left(\mathrm{v} \, \mathrm{Q} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]\right. \end{aligned}\]Then, \[\left.\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right)\right]=-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\left(\ln \left(\mathrm{m}_{\mathrm{j}}\right)+\ln \left(\gamma_{\pm}\right)\right]\right.\]Or, \[\left.-\phi \, \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}[\phi)\right]=-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d}\left(\mathrm{m}_{\mathrm{j}}\right) / \mathrm{m}_{\mathrm{j}}+\mathrm{d} \ln \left(\gamma_{\pm}\right)\right]\]Equation (f) is integrated between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\). Then, \[\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m(j)}(\phi-1) \, d \ln \left(m_{j}\right)\]And, \[\phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{\pm}\right)\]Hence the dependences of both \(\gamma_{\pm}\) and \(\phi\) are obtained for salt solutions and of both \(\gamma_{\mathrm{j}} and \(\phi\) for solutions containing neutral solutes.An important challenge at this stage is to express the experimentally determined dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\). Having expressed this dependence quantitively, the dependence of \(\gamma_{\pm}\) on \(\mathrm{m}_{\mathrm{j}}\) is obtained using equation (g). The integration can be done graphically or numerically using a computer- based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\). \[\phi=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{j}} \mathrm{A}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{r}(\mathrm{i})}\]The parameter \(\mathrm{r}(\mathrm{i})\) increases in quarter powers. Then, \[\ln \left(\gamma_{\pm}\right)=-\mathrm{S}_{\mathrm{\gamma}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{j}} \mathrm{A}_{\mathrm{i}} \,\left(\frac{\mathrm{r}_{\mathrm{i}}+1}{\mathrm{r}_{\mathrm{i}}}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{r}(\mathrm{i})}\]In more recent accounts, Pitzer’s equations have been used to represent the dependence of \(\phi\) on ionic strength.If the isopiestic experiments are repeated at several temperatures, the relative partial molar enthalpy of the solvent \(\mathrm{L}_{1}(\mathrm{aq})\) is obtained.In summary a large scientific literature reports thermodynamic data for aqueous solutions containing salts and mixed salt systems. G. Scatchard, W. J. Hamer and S. E. Wood, J.Am.Chem.Soc.,1938,60,3061. For reviews and further data compilations see R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn. (revised), 1965, p. 34. A. K. Covington and R. A. Matheson, J. Solution Chem., 1977, 6, 263; \(\mathrm{NH}_{4} \mathrm{~CNS}(\mathrm{aq})\). J. A. Rard and D. J. Miller, J. Chem.Eng. Data 1982,27,169; CsCl(aq) and \(\mathrm{SrCl}_{2}(\mathrm{aq})\). J. A. Rard, J.Chem.Eng.Data,1987,32,92. \(\mathrm{La}\left(\mathrm{NO}_{3}\right)_{3}(\mathrm{aq}) \text { and } \mathrm{Eu}\left(\mathrm{NO}_{3}\right)_{3}(\mathrm{aq})\). J. B. Maskill and R. G. Bates, J.Solution Chem.,1986,15,418 Tris(aq). L.M. Mukherjee and R. G. Bates, J.Solution Chem.,1985,14,255; \(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{Br}^{-} \left(\mathrm{D}_{2}\mathrm{O}\right)\). S. Lindenbaum, L. Leifer, G. E. Boyd and J. W. Chase, J. Phys. Chem., 1970,74, 761; \(\mathrm{R}_{4}\mathrm{NX}(\mathrm{aq})\)This page titled 1.17.7: Isopiestic: Aqueous Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.18.1: Liquid Mixtures: Regular Mixtures

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A given binary liquid mixture is prepared using liquid-1 and liquid –2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The chemical potentials, \(\mu_{1}\left(\operatorname{mix} ; x_{1}\right)\) and \(\mu_{2}\left(\operatorname{mix} ; x_{2}\right)\) are related to the mole fraction composition, \(x_{1}\) and \(x_{2} (= 1 - x_{1})\) using equations (a) and (b) where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\); \[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]\[\mu_{2}\left(\operatorname{mix} ; x_{2}\right)=\mu_{2}^{*}(\ell)+R \, T \, \ln \left(x_{2} \, f_{2}\right)\]Here for both \(\mathrm{i} = 1,2\) at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(x_{i} \rightarrow 1\right) f_{i}=1\]The term “regular mixture” describes a liquid mixture for which the rational activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are given by equations (d) and (e) where the property \(\mathrm{w}\) is independent of temperature and liquid mixture composition. \[\ln \left(f_{1}\right)=(w / R \, T) \, x_{2}^{2}\]\[\ln \left(f_{2}\right)=(w / R \, T) \,\left(1-x_{2}\right)^{2}\]Then, for example, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \ln \left(f_{1}\right)=0 ; f_{1}=1\]Similarly, \[\operatorname{limit}\left(x_{2} \rightarrow 1\right) \ln \left(f_{2}\right)=0 ; f_{2}=1\]Interest in regular liquid mixtures stems from the observation that the properties of such real (as opposed to ideal) mixtures are simply described. Of course the term “real” only means that the dependences of rational activity coefficients on mole fraction composition are defined by equations (d) and (e) and that the thermodynamic properties of the liquid mixture are not ideal.For example, according to equation (d), \[d \ln \left(f_{1}\right) / d T=-\left(w / R \, T^{2}\right) \, x_{2}^{2}\]With reference to the dependence of the properties of binary liquid mixtures on temperature (at fixed pressure), equation (a) yields equation (i). \[\frac{\mathrm{d}\left[\mu_{1}(\mathrm{mix}) / \mathrm{T}\right]}{\mathrm{dT}}=\frac{\mathrm{d}\left[\mu_{1}^{*}(\ell) / \mathrm{T}\right]}{\mathrm{dT}}+\mathrm{R} \,\left[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dT}}\right]\]From the Gibbs - Helmholtz equation, \[-\frac{\mathrm{H}_{1}(\mathrm{mix})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{1}^{*}(\ell)}{\mathrm{T}^{2}}-\mathrm{R} \,\left(\frac{\mathrm{w}}{\mathrm{R} \, \mathrm{T}^{2}}\right) \, \mathrm{x}_{2}^{2}\]Hence \[\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)+\mathrm{w} \, \mathrm{x}_{2}^{2}\]Here \[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{mix})=\mathrm{H}_{1}^{*}(\ell)\]According to equation (k), \(\mathrm{H}_{\mathrm{l}}(\mathrm{mix})\) is a quadratic function of the mole fraction composition.Further, \[\mathrm{S}_{1}(\operatorname{mix})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\]Hence entropic properties of regular mixtures do not deviate from the properties of an ideal liquid mixture. In terms of excess properties for regular mixtures, \(S_{m}^{E}=0\) and therefore \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\). Equations (d) and (e) can be written as explicit equations for \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) respectively. \[f_{1}=\exp \left[\left(\frac{w}{R \, T}\right) \, x_{2}^{2}\right]\]\[f_{2}=\exp \left[\left(\frac{w}{R \, T}\right) \,\left(1-x_{2}\right)^{2}\right]\]The partial pressures of the two chemical substances are given by Raoult’s Law. \[\mathrm{p}_{1}=\mathrm{p}_{1}^{*} \, \mathrm{x}_{1} \, \exp \left[\left(\frac{\mathrm{W}}{\mathrm{R} \, \mathrm{T}}\right) \, \mathrm{x}_{2}^{2}\right]\]\[\mathrm{p}_{2}=\mathrm{p}_{2}^{*} \, \mathrm{x}_{2} \, \exp \left[\left(\frac{\mathrm{w}}{\mathrm{R} \, \mathrm{T}}\right) \,\left(1-\mathrm{x}_{2}\right)^{2}\right]\]For both liquid components, deviations from ideal thermodynamic properties increase with increase in the magnitude of \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\). If \(\mathrm{w} >0\), the deviations are called positive whereas if \(\mathrm{w} <0\) the deviations are called negative. In the event that \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\) equals 2, the plots of \(\mathrm{p}_{1}\) and \(\mathrm{p}_{2}\) against mole fraction composition are horizontal when \(x_{1} = x_{2} = 0.5\). But in the event that \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\) equals 3, a range of binary liquid mixtures exist having intermediate mole fraction compositions and are unstable. These mixtures separate into two liquid mixtures, one rich in component 1 and the other rich in component 2. J. Hildebrand, J Am. Chem. Soc.,1929, 51,69. Accounts of this class of mixtures are given in references-. E. A. Guggenheim, Thermodynamics, North Holland Publishing Company, Amsterdam, 1950, chapter 5; note that Guggenheim uses the symbol \(x\) to represent the mole fraction composition of a binary liquid mixture \(x_{2}\); see page 173. E. A. Guggenheim, Mixtures, Clarendon Press, Oxford, 1952, chapter IV. M.L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, chapter 16. G. N. Lewis and M. L. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961, chapter 21. \[\mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\]Then, \(\mathrm{w}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\), a molar energy. From, \(\mu_{1}(\operatorname{mix})=\mathrm{H}_{1}(\operatorname{mix})-\mathrm{T} \, \mathrm{S}_{1}(\operatorname{mix})\)\[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{w} \, \mathrm{x}_{2}^{2}=\mathrm{H}_{1}^{*}(\ell)+\mathrm{w} \, \mathrm{x}_{2}^{2}-\mathrm{T} \, \mathrm{S}_{1}(\mathrm{mix})\]Or, \[\mu_{1}^{*}(\ell)-\mathrm{H}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=-\mathrm{T} \, \mathrm{S}_{1}(\operatorname{mix})\]Or, \[-\mathrm{T} \, \mathrm{S}_{1}^{\prime \prime}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=-\mathrm{T} \, \mathrm{S}_{1}(\mathrm{mix})\] From equation (p) with \((w / R \, T)=2\), \[\begin{aligned} &\mathrm{p}_{1}=\mathrm{p}_{1}^{*} \, \mathrm{x}_{1} \, \exp \left(2 \, \mathrm{x}_{2}^{2}\right)\\ &p_{1}=p_{1}^{*} \, x_{1} \, \exp \left[2 \,\left(1-x_{1}\right)^{2}\right]\\ &\frac{\mathrm{d}\left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right)}{\mathrm{dx}_{1}}=\exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right]-\mathrm{x}_{1} \, 4 \,\left(1-\mathrm{x}_{1}\right) \, \exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right]\\ &=\exp \left[2 \,\left(1-x_{1}\right)^{2}\right] \,\left[1-4 \, x_{1} \,\left(1-x_{1}\right)\right]\\ &=\exp \left[2 \,\left(1-x_{1}\right)^{2}\right] \,\left[1-4 \, x_{1}+4 \, x_{1}^{2}\right]\\ &\frac{\mathrm{d}\left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right)}{\mathrm{dx_{1 }}}=\exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right] \,\left(1-2 \, \mathrm{x}_{1}\right)^{2} \\ &\frac{\mathrm{d}\left(\mathrm{p}_{2} / \mathrm{p}_{2}^{*}\right)}{\mathrm{dx}_{2}}=\exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right]+\mathrm{x}_{2} \,(-4) \,\left(1-\mathrm{x}_{2}\right) \, \exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right] \\ &\frac{\mathrm{d}\left(\mathrm{p}_{2} / \mathrm{p}_{2}^{*}\right)}{\mathrm{dx_{2 }}}=\exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right] \,\left(1-2 \, \mathrm{x}_{2}\right)^{2} \end{aligned}\]This page titled 1.18.1: Liquid Mixtures: Regular Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.18.2: Liquid Mixtures: General Equations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.18%3A_Liquid_Mixtures/1.18.2%3A_Liquid_Mixtures%3A_General_Equations

A given binary liquid mixture is prepared using liquid-1 and liquid –2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The chemical potentials, \(\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)\) and \(\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)\) are related to the mole fraction composition, \(x_{1}\) and \(x_{2} (= 1 - x_{1})\) using equations (a) and (c) where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\); \[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]where \[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1\]\[\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\]where \[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{f}_{2}=1\]A general equation for activity coefficient \(\mathrm{f}_{1}\) takes the following form. \[\ln \left(f_{1}\right)=\sum_{k=1}^{k=\infty} \alpha_{k} \, x_{s}^{\lambda(k)}\]Equation (e) satisfies the condition, \[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \ln \left(f_{1}\right)=0 ; f_{1}=1\]The parameter \(\alpha_{\mathrm{k}}\) is characteristic of the mixture, temperature and pressure. The property \(\lambda_{\mathrm{k}}\) is a real number. In the limit that the liquid mixture is dilute in chemical substance liquid-2, equation (e) simplifies to equation (g). \[\ln \left(f_{1}\right)=\alpha \, x_{2}^{\lambda}\]In general terms, \[x_{1} \, d \ln \left(f_{1}\right)+x_{2} \, d \ln \left(f_{2}\right)=0\]We combine equations (e) and (h) with \(\lambda_{k} \geq 2\). \[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1} / \mathrm{f}_{2}\right)}{\mathrm{dx}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}}=\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \alpha_{\mathrm{k}} \, \lambda_{\mathrm{k}} \, \mathrm{x}_{2}^{\lambda(\mathrm{k})-2}\]Equation (i) is integrated to yield equation (j) where \(\mathrm{I}\) is the constant of integration. \[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k} \, x_{2}^{\lambda(k)-1}}{\lambda_{k}-1}-I\]Hence \[\begin{aligned} \ln \left(f_{2}\right) &=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k}}{\lambda_{k}-1} \,\left(x_{2}^{\lambda(k)-1}-1\right)-\sum_{k=1}^{k=\infty} \alpha_{k} \\ &=\ln \left(f_{1}\right)-\sum_{k=1}^{k=\infty} \alpha_{k} \,\left[\frac{\lambda_{k}}{\lambda_{k}-1} \,\left(x_{2}^{\lambda-(k-1)}-1\right)-1\right] \end{aligned}\]In other words, granted that \(\ln \left(f_{1}\right)\) is known as a function of \(x_{2}\), then \(\ln \left(f_{2}\right)\) can be calculated. I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1954. For a binary liquid mixture , the Gibbs-Duhem equation relates activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\). Thus, \[-S \, d T+V \, d p+n_{1} \, d \mu_{1}+n_{2} \, d \mu_{2}=0\]At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}=0\)Divide by \(\left(n_{1}+n_{2}\right)\); \(x_{1} \, d \mu_{1}+x_{2} \, d \mu_{2}=0\)\[\begin{gathered} \mathrm{x}_{1} \, \mathrm{d}\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{x}_{2} \, \mathrm{d}\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]=0 \\ \mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)=0 \\ \mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{f}_{2}\right)=0 \end{gathered}\]But \[\mathrm{x}_{1} \, \mathrm{d} \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \mathrm{d} \ln \left(\mathrm{x}_{2}\right)=\left(\mathrm{x}_{1} / \mathrm{x}_{1}\right) \, \mathrm{dx} \mathrm{x}_{1}+\left(\mathrm{x}_{2} / \mathrm{x}_{2}\right) \, \mathrm{dx} \mathrm{x}_{2}\]Also \(\mathrm{x}_{1}+\mathrm{x}_{2}=1\) so that \(\mathrm{dx}_{1}+\mathrm{dx}_{2}=0\) From equation (h) for a binary liquid mixture at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\begin{array}{r} \left(1-x_{2}\right) \, \frac{d \ln \left(f_{1}\right)}{d x_{2}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{2}}=0 \\ \frac{d \ln \left(f_{1}\right)}{d x_{2}}-x_{2} \, \frac{d \ln \left(f_{1}\right)}{d x_{2}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{2}}=0 \end{array}\]We divide by \(x_{2}\) and rearrange the equation. \[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}}-\frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx} \mathrm{x}_{2}}\]Or, \[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1} / \mathrm{f}_{2}\right)}{\mathrm{dx} \mathrm{x}_{2}}=\frac{1}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx_{2 }}}\] From equations (e) and (j), \[\ln \left(f_{2}\right)=\sum_{k=1}^{k=\infty} \alpha_{k} \, x_{2}^{\lambda(k)}-\sum_{k=1}^{k=\infty} \frac{\alpha_{k} \, \lambda_{k} \, x_{2}^{\lambda(k)-1}}{\lambda_{k}-1}-I\]But at \(x_{2} = 1, f_{2} = 1\). Then, \[0=\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \alpha_{\mathrm{k}}-\sum_{\mathrm{k}=1}^{\mathrm{k}=\infty} \frac{\alpha_{\mathrm{k}} \, \lambda_{\mathrm{k}}}{\lambda_{\mathrm{k}}-1}-\mathrm{I}\] J. N. Bronsted and P. Colmart, Z. Phys. Chem.,1934,A168, 381 ( as quoted in reference 1).This page titled 1.18.2: Liquid Mixtures: General Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.18.3: Liquid Mixtures: Series Functions for Activity Coefficients

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.18%3A_Liquid_Mixtures/1.18.3%3A_Liquid_Mixtures%3A_Series_Functions_for_Activity_Coefficients

A given binary liquid mixture is prepared using liquid-1 and liquid-2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The chemical potentials, \(\mu_{1}\left(\operatorname{mix} ; x_{1}\right)\) and \(\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)\) are related to the mole fraction composition, \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\left(=1-\mathrm{x}_{1}\right)\) using equations (a) and (c) where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\); \[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+R \, T \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]where \[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1\]\[\mu_{2}\left(\operatorname{mix} ; \mathrm{x}_{2}\right)=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\]where \[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{f}_{2}=1\]A quite general approach to understanding the properties of binary liquid mixtures expresses, for example, \(\ln \left(f_{1}\right)\) as a series function in terms of mole fraction \(x_{2}\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Using only three terms we obtain equation (e). \[\ln \left(f_{1}\right)=\alpha_{2} \, x_{2}^{2}+\alpha_{3} \, x_{2}^{3}+\alpha_{4} \, x_{2}^{4}\]As required, \[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \ln \left(f_{1}\right)=0 ; f_{1}=1\]Hence, \[\begin{aligned} \ln \left(f_{2}\right) &=\left[\alpha_{2}+(3 / 2) \, \alpha_{3}+2 \, \alpha_{4}\right] \, x_{1}^{2} \\ &-\left[\alpha_{3}+(8 / 3) \, \alpha_{4}\right] \, x_{1}^{3}+\alpha_{4} \, x_{1}^{4} \end{aligned}\]As required, \(\operatorname{limit}\left(x_{1} \rightarrow 0\right) \ln \left(f_{2}\right)=0 ; f_{2}=1\) From, \(\ln \left(f_{1}\right)=\alpha_{2} \, x_{2}^{2}+\alpha_{3} \, x_{2}^{3}+\alpha_{4} \, x_{2}^{4}\)\[\ln \left(f_{1}\right)=\alpha_{2} \,\left(1-x_{1}\right)^{2}+\alpha_{3} \,\left(1-x_{1}\right)^{3}+\alpha_{4} \,\left(1-x_{1}\right)^{4}\]Then \[\frac{\mathrm{d} \ln \left(f_{1}\right)}{d x_{1}}=-2 \, \alpha_{2} \,\left(1-x_{1}\right)-3 \, \alpha_{3} \,\left(1-x_{1}\right)^{2}-4 \, \alpha_{4} \,\left(1-x_{1}\right)^{3}\]But from the Gibbs-Duhem equation (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) \[x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}}=0\]Or, \[\frac{\mathrm{d} \ln \left(f_{2}\right)}{\mathrm{dx}_{1}}=-\frac{\mathrm{x}_{1}}{\mathrm{x}_{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dx}_{1}}\]Or, \[\frac{d \ln \left(f_{2}\right)}{d x_{1}}=-\frac{x_{1}}{\left(1-x_{1}\right)} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}\]Then, \[\frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{1}}=2 \, \alpha_{2} \, \mathrm{x}_{1}+3 \, \mathrm{x}_{1} \, \alpha_{3} \,\left(1-\mathrm{x}_{1}\right)+4 \, \alpha_{4} \, \mathrm{x}_{1} \,\left(1-\mathrm{x}_{1}\right)^{2}\]\[\frac{\mathrm{d} \ln \left(f_{2}\right)}{d x_{1}}=2 \, \alpha_{2} \, x_{1}+3 \, x_{1} \, \alpha_{3}-3 \, \alpha_{3} \, x_{1}^{2}+4 \, \alpha_{4} \, x_{1}-8 \, \alpha_{4} \, x_{1}^{2}+4 \, \alpha_{4} \, x_{1}^{3}\]Or, \[\frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{1}}=\left[2 \, \alpha_{2}+3 \, \alpha_{3}+4 \, \alpha_{4}\right] \, \mathrm{x}_{1}-\left[3 \, \alpha_{3}+8 \, \alpha_{4}\right] \, \mathrm{x}_{1}^{2}+4 \, \alpha_{4} \, \mathrm{x}_{1}^{3}\]The latter equation is integrated. \[\ln \left(f_{2}\right)=\left[\alpha_{2}+(3 / 2) \, \alpha_{3}+2 \, \alpha_{4}\right] \, x_{1}^{2}-\left[\alpha_{3}+(8 / 3) \, \alpha_{4}\right] \, x_{1}^{3}+\alpha_{4} \, x_{1}^{4}\]This page titled 1.18.3: Liquid Mixtures: Series Functions for Activity Coefficients is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.18.4: Liquid Mixtures: Binary: Less Common Properties

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For an ideal liquid mixture containing i-liquid components four important molar properties are related to the corresponding properties of the pure liquid components using the following equations. \[\mathrm{C}_{\mathrm{Vm}_{\mathrm{m}}}(\operatorname{mix} ; \mathrm{id})=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{E}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{C}_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)}\right] \,\left[\beta_{\mathrm{v}}(\operatorname{mix} ; \mathrm{id})-\beta_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)\right] \, \mathrm{C}_{\mathrm{V}_{\mathrm{i}}}^{*}(\ell)\right\}\]\[\begin{aligned} &\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id}) \\ &=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{C}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{E}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\left[\beta_{\mathrm{v}}(\mathrm{mix} ; \mathrm{id})\right]^{-1}-\left[\beta_{\mathrm{vi}_{\mathrm{i}}}^{*}(\ell)\right]^{-1}\right] \, \mathrm{E}_{\mathrm{Si}}^{*}(\ell)\right\} \end{aligned}\]\[\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{K}_{\mathrm{pi}}^{*}(\ell)}{\mathrm{E}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\beta_{\mathrm{s}}(\operatorname{mix} ; \mathrm{id})-\beta_{\mathrm{Si}}^{*}(\ell)\right] \, \mathrm{E}_{\mathrm{Si}}^{*}(\ell)\right\}\]\[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\text { mix; id }) \\ &\quad=\sum_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} \,\left\{1-\left[\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{K}_{\mathrm{Si}}^{*}(\ell)}\right] \,\left[\left[\beta_{\mathrm{s}}(\text { mix; id) }]^{-1}-\left[\beta_{\mathrm{si}}^{*}(\ell)\right]^{-1}\right] \, \mathrm{K}_{\mathrm{Si}}^{*}(\ell)\right\}\right. \end{aligned}\]With reference to these four equations, interesting features emerge. If \(\mathrm{V}_{1}^{*}(\ell)\) and \(\mathrm{V}_{2}^{*}(\ell)\) for the two components of a binary liquid mixture having ideal thermodynamic properties are linearly related at different temperatures and pressures then at fixed liquid mixture composition, \[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{1}^{*}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{2}^{*}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{v}_{(\operatorname{mix} ; \mathrm{d})}}\]Or, \[\beta_{\mathrm{v}_{1}}^{*}(\ell)=\beta_{\mathrm{v}_{2}}^{*}(\ell)=\beta_{\mathrm{v}}(\mathrm{mix} ; \mathrm{id})\]Under these conditions the two properties described in equations (a) and (b) are given by the mole fraction weighted sums of the properties of the pure liquids. The internal pressure pint is given by \(\left[\mathrm{T} \, \beta_{\mathrm{V}}-1\right]\). Hence the same condition holds with respect to the two properties defined by equations (a) and (b) if the internal pressures are equal. In practice liquids have different internal pressures. However this difference is often small for chemically similar liquids.An interesting feature emerges if the molar entropies of the two liquids are linearly related over a range of temperatures and pressures. Thus, \[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\theta)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}_{2}^{*}(\theta)}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\text { mid; } ; \mathrm{d})}\]Or, \[\beta_{\mathrm{s} 1}^{*}(\ell)=\beta_{\mathrm{S} 2}^{*}(\ell)=\beta_{\mathrm{s}}(\operatorname{mix} ; \mathrm{id})\]Therefore a liquid mixture where the components have identical isentropic thermal pressure coefficients, \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) and \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) are given by the mole fraction weighted sums of the properties of the pure components.In the case of an ideal binary liquid mixture the following three equations relate the isochoric heat capacities, isentropic compressions and isentropic expansions to the properties of the component pure liquids. \[\begin{aligned} &\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id}) \\ &=\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{v} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{V} 2}^{*}(\ell) \\ &+\mathrm{T} \,\left\{\left[\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 1}^{*}(\ell)}\right]+\left[\frac{\mathrm{x}_{2} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{~K}_{\mathrm{T} 2}^{*}(\ell)}\right]-\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}\right]\right\} \end{aligned}\]\[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{S} 2}^{*}(\ell) \\ &+\mathrm{T} \,\left\{\left[\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(1 \ell)}\right]+\left[\frac{\mathrm{x}_{2} \,\left[\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}\right]-\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}\right]\right\} \end{aligned}\]\[\begin{aligned} &\mathrm{E}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{S} 2}^{*}(\ell)\\ &+\mathrm{T} \,\left\{\begin{array}{l} {\left[\frac{\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}\right]+\left[\frac{\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell) \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}\right]} \\ -\left[\frac{\left[\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right] \,\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\right]}{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}\right] \end{array}\right\} \end{aligned}\]Inspection shows that in each case the condition for simple additivity requires that the sum inside the brackets {……} vanishes. In the case of \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) a sufficient condition ( and most probably also necessary) is that \(\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]\) and that \(\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\left[=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})\right]\) at a given \(\mathrm{T}\) and \(\mathrm{p}\). In the case of \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) the required conditions are that \(\mathrm{E}_{\mathrm{pl}}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\text { mix; } \mathrm{id})\right]\) and that \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]\) at a given \(\mathrm{T}\) and \(\mathrm{p}\). But since \(\mathrm{E}_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}\) and \(\mathrm{C}_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\), the condition can be restated as follows. Although molar entropies of liquid 1 and 2 may differ, they should have identical isobaric dependences on temperature and isothermal dependence on pressure. In the case of \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) the three conditions are that \(\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]\), \(\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\left[=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix} ; \mathrm{id})\right]\), and \(\mathrm{C}_{\mathrm{pl}}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 2}^{*}(\ell)\left[=\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]\).If we extend the foregoing analysis to the variables isentropic compressibilities \(\kappa_{\mathrm{S}}\) and isentropic expansibilities \(\alpha_{\mathrm{S}}\) we find that because \(\kappa_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})=\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id}) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\), the condition described above for \(\mathrm{K}_{\mathrm{Sm}}(\text { mix; } \mathrm{id})\) requires that \(\kappa_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})\) is given by the volume weighted sum of \(\kappa_{\mathrm{sl}}^{*}(\ell)\) and \(\kappa_{S 2}^{*}(\ell)\). Similarly we find that the three conditions described above in the context of \(\mathrm{E}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) is necessary in order that \(\alpha_{S}(\operatorname{mix} ; \mathrm{id})\) is given by the volume weighted sum of \(\alpha_{\mathrm{s} 1}^{*}(\ell)\) and \(\alpha_{\mathrm{s} 2}^{*}(\ell)\).The conditions described above are expressed in thermodynamic terms but we note that in no case can the properties of real pure liquids comply with these conditions. Nevertheless they provide useful pointers in the task of understanding the properties of real liquid mixtures. Even for a mixture prepared using \(\mathrm{H}_{2}\mathrm{O}(\ell)\) and \(\mathrm{D}_{2}\mathrm{O}(\ell)\), \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\) would depart from mole fraction additivity. Only for mixtures of ideal gases would the condition hold for \(\mathrm{C}_{\mathrm{Vm}}(\operatorname{mix} ; \mathrm{id})\). Indeed for a monatomic gas the energy is entirely translational and \(\mathrm{C}_{\mathrm{pm}}=(5 / 2) \ldot \mathrm{R}\).Then \(\left(\partial \mathrm{S}_{\mathrm{m}} / \partial \mathrm{T}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{pm}} / \mathrm{T}=(5 / 2) \, \mathrm{R} / \mathrm{T}\) for both pure gases and the mixture, a consequence of the Sackur-Tetrode equation for the molar entropy of ideal gases.Footnote G. Douheret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, Chem. Phys. Chem. Phys.,2001,2,148.This page titled 1.18.4: Liquid Mixtures: Binary: Less Common Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.18.5: Liquid Mixtures: Binary: Pseudo-Excess Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.18%3A_Liquid_Mixtures/1.18.5%3A_Liquid_Mixtures%3A_Binary%3A_Pseudo-Excess_Properties

At defined \(\mathrm{T}\) and \(\mathrm{p}\), a thermodynamic (molar) property \(\mathrm{P}\) of an ideal binary liquid mixture (e.g. volume) can be expressed as a function of the mole fraction composition using equation (a). \[\mathrm{P}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{P}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{P}_{2}^{*}(\ell)\]Here \(\mathrm{P}_{1}^{*}(\ell)\) and \(\mathrm{P}_{2}^{*}(\ell)\) are the properties of the two pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). In many cases equation (a) is taken as a pattern on which to base a description of other properties of liquid mixtures; e.g. relative permitivities, surface tensions and viscosities. There is often no thermodynamic basis for this description although it has to be admitted that such an equation has an intuitively attractive form. In the next stage the difference between measured property \(\mathrm{P}(\mathrm{mix})\) and \(\mathrm{P}(\text { mix } ; \mathrm{id})\) leads to a defined pseudo-excess property. \(\mathrm{P}^{\mathrm{E}}\).For the sake of completeness, the use of molar changes on mixing is recommended in the present context. Thus, \[X_{m}(n o-\operatorname{mix})=x_{1} \, X_{1}^{*}(\ell)+x_{2} \, X_{2}^{*}(\ell)\]Then by definition at common temperature and pressure, \[\Delta_{\text {mix }} X_{m}(\operatorname{mix})=X_{m}(\operatorname{mix})-X_{m}(n o-m i x)\]This page titled 1.18.5: Liquid Mixtures: Binary: Pseudo-Excess Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.