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16.15: Solvent Activity Coefficients from Freezing-point Depression Measurements

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.15%3A_Solvent_Activity_Coefficients_from_Freezing-point_Depression_Measurements

The analysis of freezing-point depression that we present in Section 16.11 introduces a number of simplifying assumptions. We now undertake a more rigorous analysis of this phenomenon. This analysis is of practical importance. Measuring the freezing-point depression of a solution is one way that we can determine the activity and the activity coefficient of the solvent component. As we see in Section 16.7, if we have activity coefficients for the solvent over a range of solute concentrations, we can use the Gibbs-Duhem equation to find activity coefficients for the solute. Freezing-point depression measurements have been used extensively to determine the activity coefficients of aqueous solutes by measuring the activity of water in their solutions.As in our earlier discussion of freezing-point depression, the equilibrium system is a solution of solute \(A\) in solvent \(B\), which is in phase equilibrium with pure solid solvent \(B\). Our present objective is to determine the activity of the solvent in its solutions at the melting point of the pure solvent. Having obtained this information, we can use the Gibbs-Duhem relationship to find the activity of the solute, as a function of solute concentration, at the melting point of the pure solvent. Once we have the solute activity at the melting point of the pure solvent, we can use the methods developed in Section 14.14 to find the solute activity in a solution at any higher temperature.In Section 14.14, we find the temperature dependence of the natural logarithm of the chemical activity of a component of a solution. For a particular choice of activity standard states and enthalpy reference states, we develop a method to obtain the experimental data that we need to apply this equation. For brevity, let us refer to these choices as the infinite dilution standard states. In order to determine the activity of a solvent in its solutions at the melting point of the pure solvent, it is useful to define an additional standard state for the solvent. At temperatures below the normal melting point, which we again designate as \(T_F\), we let the activity standard state of the solvent be pure solid \(B\). Above the melting point, we use the infinite dilution standard state that we define in Section 14.14; that is, we let the activity standard state of the solvent be pure liquid solvent \(B\).At and below the melting point, \(T_F\), the activity standard state for the solvent, \(B\), is pure solid \(B\). At and above the melting point, the activity standard state for the solvent is pure liquid \(B\). At the melting point, pure solid solvent is in equilibrium with pure liquid solvent, which is also the solvent in an infinitely dilute solution. At \(T_F\), the activity standard state chemical potentials of the pure solid solvent, the pure liquid solvent, and the solvent in an infinitely dilute solution are all the same. It follows that the value that we obtain for the activity of the solvent at \(T_F\), for any particular solution, will be the same whether we determine it from measurements below \(T_F\) using the pure solid standard state or from measurements above \(T_F\) using the infinitely dilute solution standard state.Now let us consider the chemical potential of liquid solvent \(B\) in a solution whose composition is specified by the molality of solute \(A\), \({\underline{m}}_A\), when the activity standard state is pure solid \(B\). We want to find this chemical potential at temperatures in the range \(T_{fp}<t_f\)>, where \(T_{fp}\) is the freezing point of the solution whose composition is specified by \({\underline{m}}_A\). In this temperature range, we have\[{\mu }_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right) \nonumber \] \[={\widetilde{\mu }}^o_B\left(\mathrm{pure\ solid},\ T\right)+RT{ \ln {\tilde{a}}_B\ }\left(\mathrm{solution},\ {\underline{m}}_A,T\right) \nonumber \]Using the Gibbs-Helmholtz equation, we obtain\[{\left(\frac{\partial { \ln {\tilde{a}}_B\ }\left(\mathrm{solution},\ {\underline{m}}_A,T\right)}{\partial T}\right)}_{P,{\underline{m}}_A} \nonumber \]\[\ \ \ \ =\frac{-{\overline{H}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)}{RT^2}+\frac{{\tilde{H}}^o_B\left(T\right)}{RT^2} \nonumber \] \[=\frac{-\left[{\overline{H}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)-{\overline{H}}^{\textrm{⦁}}_B\left(\mathrm{solid},T\right)\right]}{RT^2} \nonumber \]where we have \({\tilde{H}}^o_B\left(T\right)={\overline{H}}^{\textrm{⦁}}_B\left(\mathrm{solid},T\right)\), because pure solid \(B\) is the activity standard state. Using the ideas developed in Section 14.14, we can use the thermochemical cycle shown in to evaluate\[{\overline{H}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)-{\overline{H}}^{\textrm{⦁}}_B\left(\mathrm{solid},T\right) \nonumber \]In this cycle, \({\Delta }_{\mathrm{fus}}{\overline{H}}_B\left(T_F\right)\) is the molar enthalpy of fusion of pure \(B\) at the melting point, \(T_F\). \({\overline{L}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)\) is the relative partial molar enthalpy of \(B\) at \(T_F\) in a solution whose composition is specified by \({\underline{m}}_A\). The only new quantity in this cycle is\[\int^T_{T_f}{{\left(\frac{\partial {\overline{H}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)}{\partial T}\right)}_{P,{\underline{m}}_A}dT} \nonumber \] We can use the relative partial molar enthalpy of the solution to find it. By definition,\[{\overline{L}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right) \nonumber \] \[={\left(\frac{\partial H\left(\mathrm{solution},\ {\underline{m}}_A,T\right)}{\partial n_B}\right)}_{P,T,n_A}-{\overline{H}}^{\mathrm{ref}}_B\left(T\right) \nonumber \]or, dropping the parenthetical information,\[{\overline{L}}_B={\overline{H}}_B-{\overline{H}}^{\mathrm{ref}}_B\left(T\right) \nonumber \]so that \[{\left(\frac{{\partial \overline{L}}_B}{\partial T}\right)}_P={\left(\frac{\partial {\overline{H}}_B}{\partial T}\right)}_P-{\left(\frac{\partial {\overline{H}}^{\mathrm{ref}}_B}{\partial T}\right)}_P \nonumber \]In Section 14.14, we introduce the relative partial molar heat capacity,\[{\overline{J}}_B\left(T\right)={\left(\frac{{\partial \overline{L}}_B}{\partial T}\right)}_P \nonumber \]Since the infinitely dilute solution is the enthalpy reference state for \(B\) in solution, we expect the molar enthalpy of pure liquid \(B\) to be a good approximation to the partial molar enthalpy of liquid \(B\) in the enthalpy reference state. Then, \({\left({\partial {\overline{H}}^{\mathrm{ref}}_B}/{\partial T}\right)}_P\) is just the molar heat capacity of pure liquid \(B\), \(C_P\left(B,\mathrm{liquid},T\right)\). (See problem 16-11.) We find\[{\left(\frac{\partial {\overline{H}}_B}{\partial T}\right)}_P={\overline{J}}_B\left(T\right)+C_P\left(B,\mathrm{liquid},T\right) \nonumber \]Using this result, the enthalpy changes around the cycle in yield\[{\overline{H}}_B\left(\mathrm{solution},\ {\underline{m}}_A,T\right)-{\overline{H}}^{\textrm{⦁}}_B\left(\mathrm{solid},T\right) \nonumber \]\[=\int^{T_F}_T{C_P\left(B,\mathrm{solid},T\right)}dT+{\Delta }_{\mathrm{fus}}{\overline{H}}_B\left(T_F\right)-L^o_B\left(T_F\right)+{\overline{L}}_B\left({\underline{m}}_A,T_F\right)+\int^T_{T_F}{\left[{\overline{J}}_B\left(T\right)+\mathrm{\ }C_P\left(B,\mathrm{liquid},T\right)\right]}dT \nonumber \]\[={\Delta }_{\mathrm{fus}}{\overline{H}}_B\left(T_F\right)+{\overline{L}}_B\left({\underline{m}}_A,T_F\right)-L^o_B\left(T_F\right)-\int^{T_F}_T{\left[C_P\left(B,\mathrm{liquid},T\right)-C_P\left(B,\mathrm{solid},T\right)+{\overline{J}}_B\right]}dT \nonumber \]Since we know how to determine \({\overline{J}}_B\) and the heat capacities as functions of temperature, we can evaluate this integral to obtain a function of temperature. For present purposes, let us assume that \({\overline{J}}_B\) and the heat capacities are essentially constant and introduce the abbreviation \(\Delta C_P=C_P\left(B,\mathrm{liquid},T\right)-C_P\left(B,\mathrm{solid},T\right)\), so that \[\int^{T_F}_T{\left[C_P\left(B,\mathrm{liquid},T\right)-C_P\left(B,\mathrm{solid},T\right)+{\overline{J}}_B\right]}dT=\left(\Delta C_P+{\overline{J}}_B\right)\left(T_F-T\right) \nonumber \]The temperature derivative of \({ \ln {\tilde{a}}_B\ }\left(\mathrm{solution},\ {\underline{m}}_A,T\right)\) becomes\[{\left(\frac{\partial { \ln {\tilde{a}}_B\ }\left(\mathrm{solution},\ {\underline{m}}_A,T\right)}{\partial T}\right)}_{P,{\underline{m}}_A} \nonumber \] \[=\frac{-{\Delta }_{\mathrm{fus}}{\overline{H}}_B\left(T_F\right)-{\overline{L}}_B\left({\underline{m}}_A,T_F\right)+L^o_B\left(T_F\right)}{RT^2}+\frac{\left(\Delta C_P+{\overline{J}}_B\right)\left(T_F-T\right)}{RT^2} \nonumber \]\(T_{fp}\) is the freezing point of the solution whose composition is specified by \({\underline{m}}_A\). At \(T_{fp}\) the solvent in this solution is at equilibrium with pure solid solvent. Hence, the chemical potential of the solution solvent is equal to that of the pure-solid solvent. Then, because the pure solid is the activity standard state for both solution solvent and pure-solid solvent at \(T_{fp}\), the activity of the solution solvent is equal to that of the pure-solid solvent. Because the pure solid is the activity standard state, the solvent activity is unity at \(T_{fp}\). This means that we can integrate the temperature derivative from \(T_{fp}\) to \(T_F\) to obtain\[\begin{aligned} \int^{T_F}_{T_{fp}} d ~ \ln \tilde{a}_B \left(\mathrm{solution},\ {\underline{m}}_A,T\right) & ~ \\ ~ & = \ln \tilde{a}_B \left(\mathrm{solution},\ {\underline{m}}_A,T_F\right) \\ ~ & = \left( \frac{\Delta _{\text{fus}} \overline{H}_B \left(T_F \right)+ \overline{L}_B \left( \underline{m}_A, T_F \right) +L^o_B\left(T_F\right)}{R} \right) \left(\frac{1}{T_F}-\frac{1}{T_{fp}}\right) -\left(\frac{\Delta C_P+ \overline{J}_B}{R}\right) \left(1-\frac{T_F}{T_{fp}}+ \ln \left(\frac{T_F}{T_{fp}} \right) \right) \end{aligned} \nonumber \]Thus, from the measured freezing point of a solution whose composition is specified by \({\underline{m}}_A\), we can calculate the activity of the solvent in that solution at \(T_F\).Several features of this result warrant mention. It is important to remember that we obtained it by assuming that \(\Delta C_P+{\overline{J}}_B\left(T\right)\) is a constant. This is usually a good assumption. It is customary to express experimental results as values of the freezing-point depression, \(\Delta T=T_F-T_{fp}\). The activity equation becomes\[\begin{aligned} \ln \tilde{a}_B \left(\mathrm{solution}, \underline{m}_A,T_F\right) & ~ \\ ~ & =-\left(\frac{\Delta _{\mathrm{fus}} \overline{H}_B\left(T_F\right)+\overline{L}_B\left( \underline{m}_A,T_F\right)+L^o_B\left(T_F\right)}{RT_FT_{fp}}\right)\Delta T \\ ~ & +\left(\frac{\Delta C_P+{\overline{J}}_B}{R}\right)\left(\frac{\Delta T}{T_{fp}}-{ \ln \left(1+\frac{\Delta T}{T_{fp}}\right)\ }\right) \end{aligned} \nonumber \]The terms involving \({\overline{L}}_B\), \(L^o_B\), \(\Delta C_P\), and \({\overline{J}}_B\) are often negligible, particularly when the solute concentration is low. When \({T_F}/{T_{fp}\approx 1}\), that is, when the freezing-point depression is small, the coefficient of \(\Delta C_P+{\overline{J}}_B\) is approximately zero. When these approximations apply, the activity equation is approximated by\[{ \ln {\tilde{a}}_B\ }\left(\mathrm{solution},\ {\underline{m}}_A,T_F\right)=-\left(\frac{{\Delta }_{\mathrm{fus}}{\overline{H}}_B\left(T_F\right)}{RT^2_F}\right)\Delta T \nonumber \]This page titled 16.15: Solvent Activity Coefficients from Freezing-point Depression Measurements is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

16.16: Electrolytic Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.16%3A_Electrolytic_Solutions

Thus far in our discussion of solute activities, we have assumed that the solute is a molecular species whose chemical structure is unchanged when the pure substance dissolves. This is not the case when salts dissolve in water and other polar solvents. A pure solid salt exists as a lattice of charged ions, rather than electrically neutral molecular moieties, and its solutions contain solvated ions. Since salt solutions conduct electricity, we often call them electrolytic solutions. Solutions of salts in water are extremely important from both practical and theoretical standpoints. Accordingly, we focus our discussion on aqueous solutions; however, the ideas that we develop apply to salt solutions in any solvent that supports the formation of solvated ions.We can apply the concepts that we develop in this chapter to measure the activities of aqueous salt solutions. When we do so, we find new features. These features arise from the formation of aquated ionic species and from electrical interactions among these species. In this chapter, we consider only the most basic issues that arise when we investigate the activities of dissolved salts. We consider only strong electrolytes; that is, salts that are completely dissociated in solution. In this section, we briefly review the qualitative features of such solutions.Departure from Henry’s law behavior begins at markedly lower concentrations when the solute is a salt than when it is a neutral molecular species. This general observation is easily explained: Departures from Henry’s law are caused by interactions among solution species. For neutral molecules separated by a distance \(r\), the variation of the interaction energy with distance is approximately proportional to \(r^{-6}\). This means that only the very closest molecules interact strongly with one another. For ions, Coulomb’s law forces give rise to interaction energies that vary as \(r^{-1}\). Compared to neutral molecules, ions interact with one another at much greater distances, so that departures from Henry’s law occur at much lower concentrations.Our qualitative picture of an aqueous salt solution is that the cations and anions that comprise the solid salt are separated from one another in the solution. Both the cations and the anions are surrounded by layers of loosely bound water molecules. The binding results from the electrical interaction between the ions and the water-molecule dipole. The negative (oxygen) end of the water dipole is preferentially oriented toward cations and the positive (hydrogen) end is preferentially oriented toward anions.In aqueous solution, simple metallic cations are coordinated to a first layer of water molecules that occupy well-defined positions around the cation. In this layer, the bonding can have a covalent component. Such combinations of metal and coordinated water molecules are called aquo complexes. For most purposes, we can consider that the aquo complex is the cationic species in solution. Beyond the layer of coordinated water molecules, a second layer of water molecules is less tightly bound. The positions occupied by these molecules are more variable. At still greater distances, water molecules interact progressively more weakly with the central cation. In general, when we consider the water molecules that surround a given anion, we find that even the closest solvent molecules do not occupy well-defined positions.In any macroscopic quantity of solution, each ion has a specific average concentration. On a microscopic level, the Coulomb’s law forces between dissolved ions operate to make the relative locations of cations and anions less random. It is useful to think about a spherical volume that surrounds a given ion. We suppose that the diameter of this sphere is several tens of nanometers. Within such a sphere centered on a particular cation, the concentration of anions will be greater than the average concentration of anions; the concentration of cations will be less than the average concentration of cations. Likewise, within a microscopic sphere centered on a given anion, the concentration of cations will be greater than the average concentration of cations; the concentration of anions will be below average.As the concentration of a dissolved salt increases, distinguishable species can be formed in which a cation and an anion are nearest neighbors. We call such species ion pairs. At sufficiently high salt concentrations, a significant fraction of the ions can be found in such ion-pair complexes. Compared to other kinds of chemical bonds, ion-pair bonds are weak. The ion-pair bond is labile; the lifetime of a given ion pair is short. At still higher salt concentrations, the formation of significant concentrations of higher aggregates becomes possible. Characterizing the species present in an electrolytic solution becomes progressively more difficult as the salt concentration increases.This page titled 16.16: Electrolytic Solutions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

16.17: Activities of Electrolytes - The Mean Activity Coefficient

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.17%3A_Activities_of_Electrolytes_-_The_Mean_Activity_Coefficient

We can find the activity of a salt in its aqueous salt solutions. For example, we can measure the freezing point depression for aqueous solutions of sodium chloride, find the activity of water in these solutions as a function of the sodium chloride concentration, and use the Gibbs-Duhem equation to find the activity of the dissolved sodium chloride as a function of its concentration. When we do so, we find some marked differences from our observations on molecular solutes.For molecular solutes, the activity approaches the solute concentration as the concentration approaches zero; that is, the activity coefficient for a molecular solute approaches unity as the concentration approaches zero. For sodium chloride, and other 1:1 electrolytes, we find that the activity we measure in this way approaches the square of the solute concentration as the concentration approaches zero. For other salts, the measured activity approaches other powers of the solute concentration as the concentration approaches zero.The dissociation of the solid salt into solvated ions explains these observations. Let us consider a solution made by dissolving \(n\) moles of a salt, \(A_pB_q\), in \(n_{\mathrm{solvent}}\) moles of solvent. (Let \(A\) be the cation and \(B\) the anion.) For present purposes, the cation and anion charges are not important. We use \(p\) and \(q\) to designate the composition of the salt. Typically, we are interested in dilute solutions, and it is convenient to use the hypothetical one-molal solution as the standard state for the activity of a solute species. We can represent the Gibbs free energy of this solution as\[G=n_{\mathrm{solvent}}{\mu }_{\mathrm{solvent}}+n{\mu }_{A_pB_q} \nonumber \]where \({\mu }_{\mathrm{solvent}}\) and \({\mu }_{A_pB_q}\) are the partial molar Gibbs free energies of the solvent and the solute in the solution. We can also write\[\mu_{A_pB_q}={\widetilde{\mu }}^o_{A_pB_q}+RT{ \ln {\tilde{a}}_{A_pB_q}\ } \nonumber \]where \({\widetilde{\mu }}^o_{A_pB_q}\) is the partial molar Gibbs free energy when \({\tilde{a}}_{A_pB_q}={\underline{m}}_{A_pB_q}{\gamma }_{A_pB_q}=1\) in the activity standard state of the salt.We assume that \(A_pB_q\) is a strong electrolyte; its solution contains \(np\) moles of the cation, \(A\), and \(nq\) moles of the anion, \(B\). In principle, we can also represent the Gibbs free energy of the solution as\[G=n_{\mathrm{solvent}}{\mu }_{\mathrm{solvent}}+np{\mu }_A+nq{\mu }_B \nonumber \]and the individual-ion chemical potentials as \({\mu }_A={\widetilde{\mu }}^o_A+RT{ \ln {\tilde{a}}_A\ }\) and \({\mu }_B={\widetilde{\mu }}^o_B+RT{ \ln {\tilde{a}}_B\ }\), where \({\widetilde{\mu }}^o_A\) and \({\widetilde{\mu }}^o_B\) are the partial molar Gibbs free energies of the ions \(A\) and \(B\) in their hypothetical one-molal activity standard states. Equating the two equations for the Gibbs free energy of the solution, we have\[{\mu }_{A_pB_q}=p{\mu }_A+q{\mu }_B \nonumber \]and\[\widetilde{\mu}^o_{A_pB_q}+RT{ \ln {\tilde{a}}_{A_pB_q}\ } ={p\widetilde{\mu }}^o_A+RT{ \ln {\tilde{a}}^p_A\ }+{q\widetilde{\mu }}^o_B+RT{ \ln {\tilde{a}}^q_B\ } \nonumber \]While it is often experimentally challenging to do so, we can measure \({\widetilde{\mu }}^o_{A_pB_q}\) and \({\tilde{a}}_{A_pB_q}\). In principle, the meanings of the individual-ion activities, \({\tilde{a}}_A\) and \({\tilde{a}}_B\), and their standard-state chemical potentials, \({\widetilde{\mu }}^o_A\) and \({\widetilde{\mu }}^o_B\), are unambiguous; however, since we cannot prepare a solution that contains cation \(A\) and no anion, we cannot make measurements of \({\tilde{a}}_A\) or \({\widetilde{\mu }}^o_A\) that are independent of the properties of \(B\), or some other anion. Consequently, we must adopt some conventions to relate these properties of the ions, which we cannot measure, to those of the salt solution, which we can.The universally adopted convention for the standard chemical potentials is to equate the sum of the standard chemical potentials of the constituent ions to that of the salt. We can think of this as assigning an equal share of the standard-state chemical potential of the salt to each of its ions; that is, we let\[{\widetilde{\mu }}^o_A={\widetilde{\mu }}^o_B=\frac{\widetilde{\mu }^o_{A_pB_q}}{p+q} \nonumber \]Then,\[{\widetilde{\mu }}^o_{A_pB_q}=p{\widetilde{\mu }}^o_A+q{\widetilde{\mu }}^o_B \nonumber \]and the activities of the individual ions are related to that of the salt by\[{\tilde{a}}_{A_pB_q}={\tilde{a}}^p_A{\ \tilde{a}}^q_B \nonumber \]We can develop the convention for the activities of the individual ions by representing each activity as the product of a concentration and an activity coefficient. That is, we represent the activity of each individual ion in the same way that we represent the activity of a molecular solute. In effect, this turns the problem of developing a convention for the activities of the individual ions into the problem of developing a convention for their activity coefficients. Using the hypothetical one-molal standard state for each ion, we write \({\tilde{a}}_A={\underline{m}}_A{\gamma }_A\) and \({\tilde{a}}_B={\underline{m}}_B{\gamma }_B\), where \({\underline{m}}_A\), \({\gamma }_A\), \({\underline{m}}_B\), and \({\gamma }_B\) are the molalities and activity coefficients for ions \(A\) and \(B\), respectively. Let the molality of the salt, \(A_pB_q\), be \(\underline{m}\). Then \({\underline{m}}_A=p\underline{m}\) and \({\underline{m}}_B=q\underline{m}\), and\[{\tilde{a}}_{A_pB_q}={\tilde{a}}^p_A{\ \tilde{a}}^q_B={\left(p\underline{m}{\gamma }_A\right)}^p{\left(q\underline{m}{\gamma }_B\right)}^q=\left(p^pq^q\right){\underline{m}}^{p+q}{\gamma }^p_A{\gamma }^q_B \nonumber \]Now we introduce the geometric mean of the activity coefficients \({\gamma }_A\) and \({\gamma }_B\); that is, we define the geometric mean activity coefficient, \({\gamma }_{\pm }\), by\[\gamma_{\pm}=\left(\gamma^p_A \gamma^q_B\right)^{1/{\left(p+q\right)}} \nonumber \]The activity of the dissolved salt is then given by\[\tilde{a}_{A_pB_q}=\left(p^pq^q\right) \underline{m}^{p+q}{\gamma_{\pm }}^{p+q} \nonumber \]The mean activity coefficient, \({\gamma }_{\pm }\), can be determined experimentally as a function of \({\underline{m}}_{A_pB_q}\), but the individual activity coefficients, \({\gamma }_A\) and \({\gamma }_B\), cannot. It is common to present the results of activity measurements on electrolytic solutions as a table or a graph that shows the mean activity coefficient as a function of the salt molality.While we cannot determine the activity or activity coefficient for an individual ion experimentally, no principle prohibits a theoretical model that estimates individual ion activities. Debye and Hückel developed such a theory. The Debye-Hückel theory gives reasonably accurate predictions for the activity coefficients of ions for solutions in which the total ion concentration is about 0.01 molal or less. We summarize the results of the Debye-Hückel theory in Section 16.18.This page titled 16.17: Activities of Electrolytes - The Mean Activity Coefficient is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

16.18: Activities of Electrolytes - The Debye-Hückel Theory

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/16%3A_The_Chemical_Activity_of_the_Components_of_a_Solution/16.18%3A_Activities_of_Electrolytes_-_The_Debye-Huckel_Theory

In earlier sections, we introduce some basic methods for the experimental measurement of activities and activity coefficients. The Debye-Hückel theory leads to an equation for the activity coefficient of an ion in solution. The theory gives accurate values for the activity of an ion in very dilute solutions. As salt concentrations become greater, the accuracy of the Debye-Hückel model decreases. As a rough rule of thumb, the theory gives useful values for the activity coefficients of dissolved ions in solutions whose total salt concentrations are less than about 0.01 molal.\({}^{2}\) The theory is based on an electrostatic model. We describe this model and present the final result. We do not, however, present the argument by which the result is obtained.We begin by reviewing some necessary ideas from electrostatics. When point charges \(q_1\) and \(q_2\) are embedded in a continuous medium, the Coulomb’s law force exerted on \(q_1\) by \(q_2\) is\[{\mathop{F}\limits^{\rightharpoonup}}_{21}=\frac{q_1q_2{\hat{r}}_{21}}{4\pi \varepsilon_0Dr^2_{12}} \nonumber \]where \(\varepsilon_0\) is a constant called the permittivity of free space, and \(D\) is a constant called the dielectric coefficient of the continuous medium. \({\hat{r}}_{21}\) is a unit vector in the direction from the location of \(q_2\) to the location of \(q_1\). When \(q_1\) and \(q_2\) have the same sign, the force is positive and acts to increase the separation between the charges. The force exerted on \(q_2\) by \(q_1\) is \({\mathop{F}\limits^{\rightharpoonup}}_{12}=-{\mathop{F}\limits^{\rightharpoonup}}_{21};\) the net force on the system of charges is\[{\mathop{F}\limits^{\rightharpoonup}}_{net}={\mathop{F}\limits^{\rightharpoonup}}_{12}+{\mathop{F}\limits^{\rightharpoonup}}_{21}=0. \nonumber \]When the force is expressed in newtons, the point charges are expressed in coulombs, and distance is expressed in meters, \(\varepsilon_0=8.854\times {10}^{-12}\ {\mathrm{C}^2}{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}\). The dielectric coefficient is a dimensionless quantity whose value in a vacuum is unity. In liquid water at 25 ºC, \(D=78.4\) We are interested in the interactions between ions whose charges are multiples of the fundamental unit of charge, \(e\). We designate the charge on a proton and an electron as \(e\) and \(-e\), respectively, where \(e=1.602\times {10}^{-19\ }\mathrm{C}\). We express the charge on a cation, say \(A^{m+}\), as \(z_Ae\), and that on an anion, say \(B^{n-}\), as \(z_Be\), where \(z_A=+m>0\) and \(z_B=-n<0\).The Debye-Hückel theory models the environment around a particular central ion—the ion whose activity coefficient we calculate. We assume that the interactions between the central ion and all other ions result exclusively from Coulomb’s law forces. We assume that the central ion is a hard sphere whose charge, \(q_C\), is located at the center of the sphere. We let the radius of this sphere be \(a_C\). Focusing on the central ion makes it possible to simplify the mathematics by fixing the origin of the coordinate system at the center of the central ion; as the central ion moves through the solution, the coordinate system moves with it. The theory develops a relationship between the activity coefficient of the central ion and the electrical work that is done when the central ion is brought into the solution from an infinite distance—where its potential energy is taken to be zero.The theory models the interactions of the central ion with the other ions in the solution by supposing that, for every type of ion, \(k\), in the solution, there is a spherically symmetric function, \({\rho }_k\left(r\right)\), which specifies the concentration of \(k\)-type ions at the location specified by \(r\), for \(r\ge a_C\). That is, we replace our model of mobile point-charge ions with a model in which charge is distributed continuously. The physical picture corresponding to this assumption is that the central ion remains discrete while all of the other ions are “ground up” into tiny charged bits that are spread smoothly—but not uniformly—throughout the solution that surrounds the central ion. The introduction of \({\rho }_k\left(r\right)\) changes our model from one involving point-charge neighbor ions—whose effects would have to be obtained by summing an impracticably large number of terms and whose locations are not well defined anyway—to one involving a mathematically continuous function. From this perspective, we adopt, for the sake of a quantitative mathematical treatment, a physical model that violates the atomic description of everything except the central ion.It is useful to have a name for the collection of charged species around the central ion; we call it the ionic atmosphere. The ionic atmosphere occupies a microscopic region around the central ion in which ionic concentrations depart from their macroscopic-solution values. The magnitudes of these departures depend on the sign and magnitude of the charge on the central ion.The essence of the Debye-Hückel model is that the charge of the central ion gives rise to the ionic atmosphere. To appreciate why this is so, we can imagine introducing an uncharged moiety, otherwise identical to the central ion, into the solution. In such a process, no ionic atmosphere would form. As far as long-range Coulombic forces are concerned, no work would be done.When we imagine introducing the charged central ion into the solution in this way, Coulombic forces lead to the creation of the ionic atmosphere. Since formation of the ionic atmosphere entails the separation of charge, albeit on a microscopic scale, this process involves electrical work. Alternatively, we can say that electrical work is done when a charged ion is introduced into a salt solution and that this work is expended on the creation of the ionic atmosphere.In the Debye-Hückel model, this electrical work is the energy change associated with the process of solvating the ion. Since the reversible, non-pressure–volume work done in a constant-temperature, constant-pressure process is also the Gibbs free energy change for that process, the work of forming the ionic atmosphere is the same thing as the Gibbs free energy change for introducing the ion into the solution.The Debye-Hückel theory makes these ideas quantitative by finding the work done in creating the ionic atmosphere. To do this, it proves to be useful to define a quantity that we call the ionic strength of the solution. By definition, the ionic strength is\[I=\sum^n_{k=1} {z^2_k{\underline{m}}_k/{2}} \nonumber \]where the sum is over all of the ions present in the solution. The factor of \(1/2\) is essentially arbitrary. We introduce it in order to make the ionic strength of a 1:1 electrolyte equal to its molality. (\(z_k\) is dimensionless.)For the hypothetical one-molal standard state that we consider in §6, the activity coefficient for solute \(C\), \(\gamma_C\), is related to the chemical potential of the real substance, \(\mu_C\left(P,{\underline{m}}_C\right)\), and that of a hypothetical ideal solute \(C\) at the same concentration, \(\mu_C\left(\mathrm{Hyp\ solute},P,{\underline{m}}_C\right)\), by\[{ \ln \gamma_C\ }=\frac{\mu_C\left(P,{\underline{m}}_C\right)-\mu_C\left(\mathrm{Hyp\ solute},P,{\underline{m}}_C\right)}{RT} \nonumber \]The Debye-Hückel model equates this chemical-potential difference to the electrical work that accompanies the introduction of the central ion into a solution whose ionic strength is I. The final result is\[{ \ln \gamma_C\ }=\frac{-z^2_Ce^2\kappa \overline{N}}{8\pi \varepsilon_0D\left(1+\kappa a_C\right)} \nonumber \](While it is not obvious from our discussion, the parameter,\[\kappa ={\left(\frac{2e^2\overline{N}d_wI}{\varepsilon_0DkT}\right)}^{1/2} \nonumber \]characterizes the ionic atmosphere around the central ion. The quantity \(d_w\) is the density of the pure solvent, which is usually water.)For sufficiently dilute solutions, \(1+\kappa a_C\approx 1\). (See problem 14.) Introducing this approximation, substituting for \(\kappa\), and dividing by 2.303 to convert to base-ten logarithms, we obtain the Debye-Hückel limiting law in the form in which it is usually presented:\[\log_{10} \gamma_C =-A_\gamma z^2_CI^{1/2} \nonumber \]where\[A_\gamma=\frac{\left(2d_w\right)^{1/2}\overline{N}^2}{2.303\left(8\pi \right)}{\left(\frac{e^2}{\varepsilon_0DRT}\right)}^{3/2} \nonumber \]For aqueous solutions at 25 C, \(A_\gamma=0.510\).The Debye-Hückel model finds the activity of an individual ion. In §18, we note that the activity of an individual ion cannot be determined experimentally. We introduce the mean activity coefficient, \(\gamma_{\pm }\), for a strong electrolyte to as a way to express the departure of a salt solution from ideal-solution behavior. Adopting the hypothetical one-molal ideal-solution state as the standard state for the salt, \(A_pB_q\), we develop conventions that express the Gibbs free energy of a real salt solution and find\[\gamma_{\pm}=\left(\gamma^p_A \gamma^q_B \right)^{1/{\left(p+q\right)}}. \nonumber \]Using the Debye-Hückel limiting law values for the individual-ion activity coefficients, we find\[\log_{10} \gamma_{\pm } =\frac{p \log_{10} \gamma_A+q \log_{10} \gamma_B }{p+q}=\frac{-pA_\gamma z^2_AI^{1/2}-qA_\gamma z^2_BI^{1/2}}{p+q}=-\left(\frac{pz^2_A+qz^2_B}{p+q}\right)A_\gamma I^{1/2}=-A_\gamma z_Az_BI^{1/2} \nonumber \]where we use the identity\[\frac{pz^2_A+qz^2_B}{p+q}=-z_Az_B \nonumber \](See problem 16.12.)This page titled 16.18: Activities of Electrolytes - The Debye-Hückel Theory is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

16.19: Finding Solute Activity Using the Hypothetical One-molal Standard State

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In this chapter, we introduce several ways to measure the activities and chemical potentials of solutes. In Sections 16.1–16.6 we consider the determination of the activities and chemical potentials of solutes with measurable vapor pressures. To do so, we use the ideal behavior expressed by Raoult’s Law and Henry’s Law. In Section 16.15 we discuss the determination of solvent activity coefficients from measurements of the decrease in the freezing point of the solvent. In Section 16.7 we discuss the mathematical analysis by which we can obtain solute activity coefficients from measured solvent activity coefficients. Electrical potential measurements on electrochemical cells are an important source of thermodynamic data. In Chapter 17, we consider the use of electrochemical cells to measure the Gibbs free energy difference between two systems that contain the same substances but at different concentrations.We define the activity of substance \(A\) in a particular system such that\[{\overline{G}}_A=\mu_A={\widetilde\mu}^o_A+RT{\ln \tilde{a}_A\ }. \nonumber \]In the activity standard state the chemical potential is \({\widetilde\mu}^o_A\) and the activity is unity, \(\tilde{a}_A=1\). It is often convenient to choose the standard state of the solute to be the hypothetical one-molal solution, particularly for relatively dilute solutions. In the hypothetical one-molal standard state, the solute molality is unity and the environment of a solute molecule is the same as its environment at infinite dilution. The solute activity is a function of its molality, \(\tilde{a}_A\left(\underline{m}_A\right)\). We let the molality of the actual solution of unit activity be \(\underline{m}^o_A\). That is, we let \(\tilde{a}_A\left(\underline{m}^o_A\right)=1\); consequently, we have \(\mu_A\left(\underline{m}^o_A\right)={\widetilde\mu}^o_A\) even though the actual solution whose molality is \(\underline{m}^o_A\) is not the standard state. To relate the solute activity and chemical potential in the actual solution to the solute molality, we must find the activity coefficient, \(\gamma_A\), as a function of the solute molality,\[\gamma_A=\gamma_A\left(\underline{m}_A\right) \nonumber \]Then\[\tilde{a}_A\left(\underline{m}_A\right)=\underline{m}_A\gamma_A\left(\underline{m}_A\right) \nonumber \] and \[\tilde{a}_A\left(\underline{m}^o_A\right)=\underline{m}^o_A\gamma_A\left(\underline{m}^o_A\right)=1 \nonumber \]To introduce some basic approaches to the determination of activity coefficients, let us assume for the moment that we can measure the actual chemical potential, \(\mu_A\), in a series of solutions where \(\underline{m}_A\) varies. We have\[\mu_A={\widetilde\mu}^o_A+RT{\ln \tilde{a}_A\ }={\widetilde\mu}^o_A+RT{\ln \underline{m}_A\ }+RT{\ln \gamma_A\ } \nonumber \]We know \(\underline{m}_A\) from the preparation of the system—or by analysis. If we also \({\widetilde\mu}^o_A\), we can calculate \(\gamma_A\left(\underline{m}_A\right)\) from our experimental values of \(\mu_A\). If we don’t know \({\widetilde\mu}^o_A\), we need to find it before we can proceed. To find it, we recall that\[{\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} RT\ln \gamma_A }=0 \nonumber \] Then\[{\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \left(\mu_A-RT{\ln \underline{m}_A\ }\right)\ }={\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \left({\widetilde\mu}^o_A+RT{\ln \gamma_A\ }\right)\ }={\widetilde\mu}^o_A \nonumber \]and a plot of \(\left(\mu_A-RT{\ln \underline{m}_A\ }\right)\) versus \(\underline{m}_A\) will intersect the line \(\underline{m}_A=0\) at \({\widetilde\mu}^o_A\).Now, in fact, we can measure only Gibbs free energy differences. In the best of circumstances what we can measure is the difference between the chemical potential of \(A\) at two different concentrations. If we choose a reference molality, \(\underline{m}^{\mathrm{ref}}_A\), the chemical potential difference \(\Delta \mu_A\left(\underline{m}_A\right)=\mu_A\left(\underline{m}_A\right)-\mu_A\left(\underline{m}^{\mathrm{ref}}_A\right)\) is a measurable quantity. A series of such results can be displayed as a plot of \(\Delta \mu_A\left(\underline{m}_A\right)\) versus \(\underline{m}_A\)—or any other function of \(\underline{m}_A\) that proves to suit our purposes. The reverence molality, \(\underline{m}^{\mathrm{ref}}_A\), can be chosen for experimental convenience.If our theoretical structure is valid, the results are represented by the equations\[\Delta \mu_A\left(\underline{m}_A\right)=\mu_A\left(\underline{m}_A\right)-\mu_A\left(\underline{m}^{\mathrm{ref}}_A\right)=RT{\ln \frac{\tilde{a}_A\left(\underline{m}_A\right)}{\tilde{a}_A\left(\underline{m}^{\mathrm{ref}}_A\right)}\ }=RT{\ln \underline{m}_A+RT{\ln \gamma_A\left(\underline{m}_A\right)\ }\ }-RT{\ln \tilde{a}_A\left(\underline{m}^{\mathrm{ref}}_A\right)\ } \nonumber \]When \(\underline{m}_A=\underline{m}^o_A\), we have\[\Delta \mu_A\left(\underline{m}^o_A\right)=\mu_A\left(\underline{m}^o_A\right)-\mu_A\left(\underline{m}^{\mathrm{ref}}_A\right)={\widetilde\mu}^0_A-\mu_A\left(\underline{m}^{\mathrm{ref}}_A\right)=RT{\ln \frac{\tilde{a}_A\left(\underline{m}^o_A\right)}{\tilde{a}_A\left(\underline{m}^{\mathrm{ref}}_A\right)}\ }=-RT{\ln \tilde{a}_A\left(\underline{m}^{\mathrm{ref}}_A\right)\ } \nonumber \]and\[\Delta \mu_A\left(\underline{m}_A\right)-\Delta \mu_A\left(\underline{m}^o_A\right)=RT{\ln \underline{m}_A\ }+RT{\ln \gamma_A\left(\underline{m}_A\right)\ } \nonumber \] so that\[RT{\ln \gamma_A\left(\underline{m}_A\right)\ }=-\Delta \mu_A\left(\underline{m}^o_A\right)+\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ } \nonumber \]Since \({\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \gamma_A\left(\underline{m}_A\right)=1\ }\), we have\[0={\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} RT{\ln \gamma_A\left(\underline{m}_A\right)\ }\ }=-\Delta \mu_A\left(\underline{m}^o_A\right)+{\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \left[\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ }\right]\ } \nonumber \]Letting \[\beta \left(\underline{m}_A\right)=\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ } \nonumber \]and \[{\beta }^o={\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \left[\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ }\right]\ } \nonumber \]we have \[\Delta \mu_A\left(\underline{m}^o_A\right)={\beta }^o \nonumber \]Then\[RT{\ln \gamma_A\left(\underline{m}_A\right)\ }=-\Delta \mu_A\left(\underline{m}^o_A\right)+\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ }=-{\beta }^o+\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ } \nonumber \]so that we know both the activity coefficient, \(\gamma_A=\gamma_A\left(\underline{m}_A\right)\), and the activity, \(\tilde{a}_A\left(\underline{m}_A\right)=\underline{m}_A\gamma_A\left(\underline{m}_A\right)\), of \(A\) as a function of its molality. Consequently, we know the value of \(\Delta \mu_A\left(\underline{m}_A\right)-\Delta \mu_A\left(\underline{m}^o_A\right)\) as a function of molality. Since this difference vanishes when \(\underline{m}_A=\underline{m}^o_A\), we can find \(\underline{m}^0_A\) from our experimental data. Finally, the activity equation becomes\[RT{\ln \tilde{a}_A\left(\underline{m}_A\right)\ }=\Delta \mu_A\left(\underline{m}_A\right)-{\beta }^o \nonumber \]This procedure yields the activity of \(A\) as a function of the solute molality. We obtain this function from measurements of \(\Delta \mu_A\left(\underline{m}_A\right)=\mu_A\left(\underline{m}_A\right)-\mu_A\left(\underline{m}^{\mathrm{ref}}_A\right)\). These measurements do not yield a value for \(\mu_A\left(\underline{m}_A\right)\); what we obtain from our analysis is an alternative expression,\[RT{\ln \frac{\tilde{a}_A\left(\underline{m}_A\right)}{\tilde{a}_A\left(\underline{m}^{\mathrm{ref}}_A\right)}\ } \nonumber \]for the chemical potential difference, \(\mu_A\left(\underline{m}_A\right)-\mu_A\left(\underline{m}^{ref}_A\right)\) between two states of the same substance. \(\mu_A\left(\underline{m}_A\right)\) is the difference between the chemical potential of solute A at \(\underline{m}_A\) and the chemical potential of its constituent elements in their standard states at the same temperature. To find this difference is a separate experimental undertaking. If, however, we can find \(\mu_A\left(\underline{m}^*_A\right)\) for some \(\underline{m}^*_A\), our activity equation yields \({\widetilde\mu}^o_A\) as\[{\widetilde\mu}^0_A=\mu_A\left(\underline{m}^*_A\right)-RT{\ln \tilde{a}_A\left(\underline{m}^*_A\right)\ } \nonumber \]This analysis of the \(\Delta \mu_A\left(\underline{m}_A\right)\) data assumes that we can find \({\beta }^o={\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} \left[\Delta \mu_A\left(\underline{m}_A\right)-RT{\ln \underline{m}_A\ }\right]\ }\). To find an accurate value for \({\beta }^o\), it is important to collect data for \(\Delta \mu_A\left(\underline{m}_A\right)\) at the lowest possible values for \(\underline{m}_A${\mathrm{m}}_{\mathrm{A}}. Inevitably, however, the experimental error in \(\Delta \mu_A\left(\underline{m}_A\right)\) increases as \(\underline{m}_A\) decreases. Our theory requires that \(\beta \left(\underline{m}_A\right)={\beta }^o+f\left(\underline{m}_A\right)\), where \({\mathop{\mathrm{lim}}_{\underline{m}_A\to 0} f\left(\underline{m}_A\right)=0\ }\), so that the graph of \(\beta \left(\underline{m}_A\right)\) versus \(f\left(\underline{m}_A\right)\) has an intercept at \({\beta }^o\). Accurate extrapolation of the data to the intercept at \(\underline{m}_A=0\) is greatly facilitated if we can choose \(f\left(\underline{m}_A\right)\) so that the graph is linear. In practice, the increased experimental error in \(\beta \left(\underline{m}_A\right)\) at the lowest values of \(\underline{m}_A\) causes the uncertainty in the extrapolated value of \({\beta }^o\) for a given choice of \(f\left(\underline{m}_A\right)\) to be similar to the range of \({\beta }^o\) values estimated using different functions. For some \(p\) in the range \(0.5 < p < 2\), letting \(f \left( \underline{m}_A \right) = \underline{m}_A^P\) often provides a fit that is as satisfactorily linear as the experimental uncertainty can justify.This procedure yields the activity of \(A\) as a function of the solute molality. We obtain this function from measurements of \(\Delta \mu_A \left( \underline{m}_A \right) = \mu_A \left( \underline{m}_A \right) - \mu_A \left) \underline{m}_A^{ \text{ref}} \right)\). These measurements do not yield a value for \(\mu_A \left( \underline{m}_A \right)\); what we obtain from our analysis is an alternative expression,\[ RT \ln \frac{ \tilde{a}_A \left( \underline{m}_A \right)}{ \tilde{a}_A \left( \underline{m}_A^{ \text{ref}} \right)} \nonumber \]for the chemical potential difference, \(\mu_A \left( \underline{m}_A \right) - \mu_A \left) \underline{m}_A^{ \text{ref}} \right)\) between two states of the same substance. \(\mu_A \left( \underline{m}_A \right)\) is the difference between the chemical potential of solute A at \(\underline{m}_A\) and the chemical potential of its constituent elements in their standard states at the same temperature. To find this difference is a separate experimental undertaking. If, however, we can find \(\mu_A \left( \underline{m}_A^* \right)\) for some \(\underline{m}_A^*\), our activity equation yields \(\tilde{\mu}^o_A\) as\[ \tilde{ \mu}_A^o = \mu_A \left( \underline{m}_A^* \right) - RT \ln \tilde{a}_A \left( \underline{m}_A^* \right) \nonumber \]Finally, let us contrast this analysis to the analysis of chemical equilibrium that we discuss briefly in Chapter 15. In the present analysis, we use an extrapolation to infinite dilution to derive activity values from the difference between the chemical potentials of the same substance at different concentrations. In the chemical equilibrium analysis for \(aA+bB\rightleftharpoons cD+dD\), we have\[\Delta_r\mu =\Delta_r{\widetilde\mu}^o+RT{\ln \frac{\tilde{a}^c_C\tilde{a}^d_D}{\tilde{a}^a_A\tilde{a}^b_B}\ }=\Delta_r{\widetilde\mu}^o+RT{\ln \frac{\underline{m}^c_C\underline{m}^d_D}{\underline{m}^a_A\underline{m}^b_B}\ }+RT{\ln \frac{\gamma^c_C{\widetilde{\gamma}}^d_D}{\gamma ^a_A\gamma^b_B}\ } \nonumber \]When the system is at equilibrium, we have \(\Delta_r\mu =0\). Since \({\mathop{\mathrm{lim}}_{\underline{m}_i\to 0} \gamma_i=1\ }\), we have, in the limit that all of the concentrations go to zero in an equilibrium system,\[0={\mathop{\mathrm{lim}}_{\underline{m}_{i\to 0}} RT{\ln \frac{\gamma^c_C\gamma^d_D}{\gamma^a_A\gamma^b_B}\ }\ }=\Delta_r{\widetilde\mu}^o+{\mathop{\mathrm{lim}}_{\underline{m}_{i\to 0}} RT{\ln \frac{\underline{m}^c_C\underline{m}^d_D}{\underline{m}^a_A\underline{m}^b_B}\ }\ } \nonumber \]Letting\[K_c=\frac{\underline{m}^c_C\underline{m}^d_D}{\underline{m}^a_A\underline{m}^b_B} \nonumber \]We have\[\Delta_r{\widetilde\mu}^o=-RT{\mathop{\mathrm{lim}}_{\underline{m}_i\to 0} K_c\ } \nonumber \]Since \(\Delta_r\mu =0\) whenever the system is at equilibrium, measurement of \(K_c\) for any equilibrium state of the reaction yields the corresponding ratio of activity coefficients: \[RT{\ln \frac{\gamma^c_C\gamma^d_D}{\gamma^a_A\gamma^b_B}\ }=-\Delta_r{\widetilde\mu}^o-RT{\ln K_c\ } \nonumber \]This page titled 16.19: Finding Solute Activity Using the Hypothetical One-molal Standard State is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

16.20: Problems

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1. At 100 C, the enthalpy of vaporization of water is \(40.657\ \mathrm{kJ}\ {\mathrm{mol}}^{-1}\). Calculate the boiling-point elevation constant for water when the solute concentration is expressed in molality units.2. At 0 C, the enthalpy of fusion of water is \(6.009\ \mathrm{kJ}\ {\mathrm{mol}}^{-1}\). Calculate the freezing-point depression constant for water when the solute concentration is expressed in molality units.3. A solution is prepared by dissolving 20.0 g of ethylene glycol (1,2-ethanediol) in 1 kg of water. Estimate the boiling point and the freezing point of this solution.4. A biopolymer has a molecular weight of 250,000 dalton. At 300 K, estimate the osmotic pressure of a solution that contains 1 g of this substance in 10 mL of water.5. Cyclohexanol melts at 25.46 C; the enthalpy of fusion is \(1.76\ \mathrm{kJ}\ {\mathrm{mol}}^{-1}\). Estimate the freezing-point depression constant when the solute concentration is expressed as a mole fraction and when it is expressed in molality units. A solution is prepared by mixing 1 g of ethylene glycol with 50 g of liquid cyclohexanol. How much is the freezing point of this solution depressed relative to the freezing point of pure cyclohexanol?6. Freezing-point depression data for numerous solutes in aqueous solution\({}^{3}\) are reproduced below. Calculate the freezing-point depression, \(-{\Delta T_{fp}}/{\underline{m}}\), for each of these solutes. Compare these values to the freezing-point depression constant that you calculated in problem 2. Explain any differences.7. In a binary solution of solute \(A\) in solvent \(B\), the mole fractions in the pure solvent are \(y_A=0\) and \(y_B=1\). We let the pure solvent be the solvent standard state; when \(y_A=0\), \(y_B={\tilde{a}}_B=1\), and \({ \ln {\tilde{a}}_B\ }=0\). What happens to the value of \({ \ln {\tilde{a}}_B\ }\) as \(y_A\to 1\)? Sketch the graph of \({\left(1-y_A\right)}/{y_A}\) versus \({ \ln {\tilde{a}}_B\ }\). For \(0<y^*_a><1\)>, shade the area on this graph that represents the integral \[\int^{y^*_A}_{y_A}{\left(\frac{1-y_A}{y_A}\right)d{ \ln {\gamma }_A\ }} \nonumber \] Is this area greater or less than zero?8. In a binary solution of solute \(A\) in solvent \(B\), the activity coefficient of the solvent can be modeled by the equation \({ \ln {\gamma }_B\ }\left(y_A\right)=cy^p_A\), where the constants \(c\) and \(p\) are found by using least squares to fit experimental data to the equation. Find an equation for \({ \ln {\gamma }_A\ }\left(y_A\right)\). For \(c=8.4\) and \(p=2.12\), plot \({ \ln {\gamma }_B\ }\left(y_A\right)\) and \({ \ln {\gamma }_A\ }\left(y_A\right)\) versus \(y_A\).9. A series of solutions contains a non-volatile solute, \(A\), dissolved in a solvent, \(B\). At a fixed temperature, the vapor pressure of solvent \(B\) is measured for these solutions and for pure \(B\) (\(y_A=0\)). At low solute concentrations, the vapor pressure varies with the solute mole fraction according to \(P=P^{\textrm{⦁}}\left(1-y_A\right)\mathrm{exp}\left(-\alpha y^{\beta }_A\right)\).(a) If the pure solvent at one bar is taken as the standard state for liquid \(A\), and gaseous \(B\) behaves as an ideal gas, now does the activity of solvent \(B\) vary with \(y_A\)?(b) How does \({ \ln {\gamma }_B\ }\) vary with \(y_A\)?(c) Find \({ \ln {\gamma }_A\ }\left(y_A\right)\).10. In Section 14.14, we find for liquid solvent \(B\),\[{\overline{L}}^o_B=H^o_B-{\overline{H}}^{ref}_B={\mathop{\mathrm{lim}}_{T\to 0} {\left(-\frac{\partial {\Delta }_{\mathrm{mix}}\overline{H}}{\partial n_B}\right)}_{P,T,n_A}\ } \nonumber \]Since \({\overline{H}}^{\textrm{⦁}}_B\) is the molar enthalpy of pure liquid \(B\), we have \[{\left(\frac{\partial {\overline{H}}^{\textrm{⦁}}_B}{\partial T}\right)}_P=C_P\left(B,\mathrm{liquid},T\right) \nonumber \] In Section 16.15, we set \[{\left(\frac{\partial {\overline{H}}^{ref}_B}{\partial T}\right)}_P=C_P\left(B,\mathrm{liquid},T\right) \nonumber \] Show that this is equivalent to the condition \[{\left(\frac{\partial {\overline{L}}^o_B}{\partial T}\right)}_P\ll C_P\left(B,\mathrm{liquid},T\right) \nonumber \]11. If \(pz_A=-qz_B\), prove that \[\frac{pz^2_A+qz^2_B}{p+q}=-z_Az_B \nonumber \]12. At temperatures of 5 C, 25 C, and 45 C, evaluate Debye-Hückel parameter \(\kappa\) for aqueous sodium chloride solutions at concentrations of \({10}^{-3}\ \underline{m}\), \({10}^{-2}\ \underline{m}\), and \({10}^{-1}\ \underline{m}\).13. Introducing the approximation \(1+\kappa a_c\approx 1\) produces the Debye-Hückel limiting law, which is strictly applicable only in the limiting case of an infinitely dilute solution. Introducing the approximation avoids the problem of choosing an appropriate value for \(a_c\). If \(a_c=0.2\ \mathrm{nm}\), calculate \(1+\kappa a_c\) for aqueous solutions in which the ionic strength, \(I\), is \({10}^{-3}\ \underline{m}\), \({10}^{-2}\ \underline{m}\), and \({10}^{-1}\ \underline{m}\). What does the result suggest about the ionic-strength range over which the limiting law is a good approximation?14. The solubility product for barium sulfate, \(K_{sp}= \tilde{a}_{Ba^{2+}} \tilde{a}_{SO^{2-}_4}\), is \(1.08\times {10}^{-10}\). Estimate the solubility of barium sulfate in pure water and in \({10}^{-2}\ \underline{m}\) potassium perchlorate.15. The enthalpy of vaporization\({}^{3}\) of n-butane at its normal boiling point, 272.65 K, is \(22.44\ \mathrm{kJ}\ {\mathrm{mol}}^{-1}\).\({}^{\ }\) In the temperature range \(273.15, the solubility\({}^{3}\) of n-butane in water is given by\[{ \ln y_A\ }=A+\frac{100\ B}{T}+C\ { \ln \left(\frac{T}{100}\right)\ } \nonumber \]where \(A=-102.029\), \(B=146.040\ {\mathrm{K}}^{-1}\), and \(C=38.7599\). From the result we develop in Section 16.14, calculate \({\Delta }_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\) for n-butane at is normal boiling point. (Note that the normal boiling temperature is slightly below the temperature range to which the equation for \({ \ln y_A\ }\) is valid.) Comment.16. The enthalpy of vaporization\({}^{3}\) of molecular oxygen at its normal boiling point, 90.02 K, is \(6.82\ \mathrm{kJ}\ {\mathrm{mol}}^{-1}\).\({}^{\ }\) In the temperature range \(273.15<\)>348.15, the solubility\({}^{3}\) of oxygen in water is given by\[{ \ln y_A\ }=A+\frac{100\ B}{T}+C\ { \ln \left(\frac{T}{100}\right)\ } \nonumber \]where \(A=-66.7354\), \(B=87.4755\ {\mathrm{K}}^{-1}\), and \(C=24.4526\). From the result we develop in Section 16.14, calculate \({\Delta }_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\) for oxygen at 273.25 K and at its normal boiling point, 90.02 K. Comment.Notes\({}^{1}\) Raoult’s law and ideal solutions can be defined using fugacities in place of partial pressures. The result is more general but—for those whose intuition has not yet embraced fugacity—less transparent.\({}^{2}\) For a discussion of the concentration range in which the Debeye-Huckel model is valid and of various supplemental models that allow for the effects of forces that are specific to the chemical characteristics of the interacting ions, see Lewis and Randall, Pitzer and Brewer, Thermodynamics, 2\({}^{nd}\) Edition, McGraw Hill Book Company, New York, 1961, Chapter 23.\({}^{3}\) Data from CRC Handbook of Chemistry and Physics, 79\({}^{th}\) Edition, David R. Lide, Ed., CRC Press, 1998-1999.This page titled 16.20: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.1: Oxidation-reduction Reactions

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We find it useful to classify reactions according to the type of change that the reagents undergo. Many classification schemes exist, often overlapping one another. The three most commonly used classifications are acid–base reactions, substitution reactions, and oxidation–reduction reactions.Acids and bases can be defined in several ways, the most common being the Brønsted-Lowry definition, in which acids are proton donors and bases are proton acceptors. The Brønsted-Lowry definition is particularly useful for reactions that occur in aqueous solutions. A prototypical example is the reaction of acetic acid with hydroxide ion to produce the acetate ion and water.\[CH_3CO_2H+OH^-\to CH_3CO^-_2+H_2O \nonumber \]Here acetic acid is the proton donor, hydroxide ion is the proton acceptor. The products are also an acid and a base, since water is a proton donor and acetate ion is a proton acceptor.When we talk about substitution reactions, we focus on a particular substituent in a chemical compound. The original compound is often called the substrate. In a substitution reaction, the original substituent is replaced by a different chemical moiety. A prototypical example is the displacement of one substituent on a tetrahedral carbon atom by a more nucleophilic group, as in the reaction of methoxide ion with methyl iodide to give dimethyl ether and iodide ion.\[CH_3I+CH_3O^-\to CH_3OCH_3+I^- \nonumber \]We could view a Brønsted-Lowry acid-base reaction as a substitution reaction in which one group (the acetate ion in the example above) originally bonded to a proton is replaced by another (hydroxide ion). Whether we use one classification scheme or another to describe a particular reaction depends on which is better suited to our immediate purpose.In acid-base reactions and substitution reactions, we focus on the transfer of a chemical moiety from one chemical environment to another. In a large and important class of reactions we find it useful to focus on the transfer of one or more electrons from one chemical moiety to another. For example, copper metal readily reduces aqueous silver ion. If we place a piece of clean copper wire in an aqueous silver nitrate solution, reaction occurs according to the equation\[2Ag^++Cu^0\to 2Ag^0+Cu^{2+} \nonumber \]We have no trouble viewing this reaction as the transfer of two electrons from the copper atom to the silver ions. In consequence, a cupric ion, formed at the copper surface, is released into the solution. Two atoms of metallic silver are deposited at the copper surface. Reactions in which electrons are transferred from one chemical moiety to another are called oxidation–reduction reactions, or redox reactions, for short.We define oxidation as the loss of electrons by a chemical moiety. Reduction is the gain of electrons by a chemical moiety. Since a moiety can give up electrons only if they have some place to go, oxidation and reduction are companion processes. Whenever one moiety is oxidized, another is reduced. In the reduction of silver ion by copper metal, it is easy to see that silver ion is gaining electrons and copper metal is losing them. In other reactions, it is not always so easy to see which moieties are gaining and losing electrons, or even that electron transfer is actually involved. As an adjunct to our ideas about oxidation and reduction, we develop a scheme for formally assigning electrons to the atoms in a molecule or ion. This is called the oxidation state formalism and comprises a series of rules for assigning a number, which we call the oxidation state (or oxidation number), to every atom in the molecule. When we adopt this scheme, the redox character of a reaction is determined by which atoms increase their oxidation state and which decrease their oxidation state as a consequence of the reaction. Those whose oxidation state increases lose electrons and are oxidized, while those whose oxidation state decreases gain electrons and are reduced.A process of electron loss is called an oxidation because reactions with elemental oxygen are viewed as prototypical examples of such processes. Since many observations are correlated by supposing that oxygen atoms in compounds are characteristically more electron-rich than the atoms in elemental oxygen, it is useful to regard a reaction of a substance with oxygen as a reaction in which the atoms of the substance surrender electrons to oxygen atoms. It is then a straightforward generalization to say that a substance is oxidized whenever it loses electrons, whether oxygen atoms or some other chemical moiety takes up those electrons. So, for example, the reaction of sodium metal with oxygen in a dry environment produces sodium oxide, \({Na}_2O\), in which the sodium is usefully viewed as carrying a positive charge. (The oxidation state of sodium is 1+; the oxidation state of oxygen is 2–.)The conversion of a metal oxide to the corresponding metal is described as reducing the oxide. Since converting a metal oxide to the metal reverses the change that occurs when we oxidize it, generalization of this idea leads us to apply the term reduction to any process in which a chemical moiety gains electrons. It is a fortunate coincidence that a reduction process is one in which the oxidation number of an atom becomes smaller (more negative) and is therefore reduced, in the sense of being decreased.Another feature of oxidation–reduction reactions, and one that relates to the utility of viewing these reactions in terms of electron gain and loss, emerges when we observe the reaction of aqueous silver ions with copper metal closely. As the reaction proceeds, the aqueous solution becomes blue as cupric ions accumulate. Long needle-like crystals of silver metal grow out from the copper surface. The simplest mechanism that we can imagine for the growth of well-formed silver crystals is that silver ions from the solution plate out on the surface of the growing silver crystal, accepting an electron from the metallic crystal as they do so. The silver metal acquires this electron from the copper metal, with which it is in contact, but at a large (on an atomic scale) distance from the site at which the new atom of silver is deposited. Evidently the processes of electron loss and gain that characterize an overall reaction can occur at different locations, if there is a suitable process for moving the electron from one location to the other.This page titled 17.1: Oxidation-reduction Reactions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.2: Electrochemical Cells

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We can extend the idea of carrying out the electron loss and electron gain steps in different physical locations. Suppose that the only aqueous species in contact with the silver metal are silver ions and nitrate ions; the silver metal is also in contact with a length of copper wire, whose other end dips into a separate reservoir containing an aqueous solution of sodium nitrate. This arrangement is sketched in When we create this arrangement, nothing happens. We do not see any visible change in the silver metal, and the water contacting the copper wire never turns blue. On the one hand, this result does not surprise us. We are accustomed to the idea that reactants must be able to contact one another in order for reaction to occur.On the other hand, the original experiment really does show that silver ions can accept electrons in one location while copper atoms give them up in another, so long as we provide a metal bridge on which the electrons can move between the two locations. Why should this not continue to happen in the new experimental arrangement? In fact, it does. It is just that the reaction occurs to only a very small extent before stopping altogether. The reason is easy to appreciate. After a very small number of silver ions are reduced, the silver nitrate solution contains more nitrate ions than silver ions; the solution as a whole has a negative charge. In the other reservoir, a small number of cupric ions dissolve, but there is no increase in the number of counter ions, so this solution acquires a positive charge. These net charges polarize the metal that connects them; the metal has an excess of positive charge at the copper-solution end and an excess of negative charge at the silver-solution end. This polarization opposes the motion of a negatively charged electron from the copper-solution end toward the silver-solution end. When the polarization becomes sufficiently great, electron flow ceases and no further reaction can occur.By this analysis, the anions that the cupric solution needs in order to achieve electroneutrality are present in the silver-ion solution. The reaction stops because the anions have no way to get from one solution to the other. Evidently, the way to make the reaction proceed is to modify the two-reservoir experiment so that nitrate ions can move from the silver-solution reservoir to the copper-solution reservoir. Alternatively, we could introduce a modification that allows copper ions to move in the opposite direction or one that allows both kinds of movement. We can achieve the latter by connecting the two solutions with a tube containing sodium nitrate solution, as diagrammed in Now, nitrate ions can move between the reservoirs and maintain electroneutrality in both of them. However, silver ions can also move between the reservoirs. When we do this experiment, we observe that electrons do flow through the wire, indicating that silver-ion reduction and copper-atom oxidation are occurring at the separated sites. However, after a short time, the solutions mix; silver ions migrate through the aqueous medium and react directly with the copper metal. Because the mixing is poorly controllable, the reproducibility of this experiment is poor.Evidently, we need a way to permit the exchange of ions between the two reservoirs that does not permit the wholesale transfer of reactive species. One device that accomplishes this is called a salt bridge. The requirement we face is that ions should be able to migrate from reservoir to reservoir so as to maintain electroneutrality. However, we do not want ions that participate in electrode reactions to migrate. A salt bridge is simply a salt solution that we use to connect the two reservoirs. To avoid introducing unwanted ions into the reservoir solutions, we prepare the salt-bridge solution using a salt whose ions are not readily oxidized or reduced. Alkali metal salts with nitrate, perchlorate, or halide anions are often used. To avoid mixing the reservoir solutions with the salt bridge solution, we plug each end of the salt bridge with a porous material that permits diffusion of ions but inhibits bulk movement of solution in or out of the bridge. The inhibition of bulk movement can be made much more effective by filling the bridge with a gel, so that the solution is unable to undergo bulk motion in any part of the bridge.With a salt bridge in place, inert ions can move from one reservoir to the other to maintain electroneutrality. Under these conditions, we see an electrical current through the external circuit and a compensating diffusion of ions through the salt bridge. The salt bridge completes the circuit. Transport of electrons from one electrode to the other carries charge in one direction; motion of ionic species through the salt bridge carries negative charge through the solution in the opposite direction. This compensating ionic motion has anions moving opposite to the electron motion and cations moving in the same direction as the electrons.We have just described one kind of electrochemical cell. As diagrammed in We then vary the potential of the reference device until current flow in the circuit stops. When this occurs the potential drop being supplied by the reference device must be precisely equal to the potential drop across the electrochemical cell, which is the datum we want.In practice, the reference device is another “standard” electrochemical cell, whose potential drop is defined to have a particular value at specified conditions. Modern electronics make it possible to do the actual measurements with great sophistication. The necessary measurements can also be done with very basic equipment. The principles remain the same. In the basic experiment, a variable resistor is used to adjust the potential drop across the standard cell until it exactly matches that of the cell being studied. When this potential is reached, current flow ceases. Current flow is monitored using a sensitive galvanometer. It is not necessary to actually measure the current. Since we are interested in locating the potential drop at which the current flow is zero, it is sufficient to find the potential drop at which the galvanometer detects no current. The accuracy of the potential measurement depends on the stability of the standard cell potential, the accuracy of the variable resistor, and the sensitivity of the galvanometer.This page titled 17.2: Electrochemical Cells is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.3: Defining Oxidation States

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We introduce oxidation states to organize our thinking about oxidation–reduction reactions and electrochemical cells. When we define oxidation states, we create a set of rules for allocating the electrons in a molecule or ion to the individual atoms that make it up. The definition of oxidation states is therefore an accounting exercise. The definition of oxidation states predates our ability to estimate electron densities through quantum mechanical calculations. As it turns out, however, the ideas that led to the oxidation state formalism are directionally correct; atoms that have high positive oxidation states according to the formalism also have relatively high positive charges by quantum mechanical calculation. In general, the absolute values of oxidation states are substantially larger than the absolute values of the partial charges found by quantum-mechanical calculation; however, there is no simple quantitative relationship between oxidation states and the actual distribution of electrons in real chemical moieties. It is a serious mistake to think that our accounting system provides a quantitative description of actual electron densities.It is a serious mistake to think that the Oxidation State system provides a quantitative description of actual electron densities.The rules for assigning oxidation states grow out of the primitive (and quantitatively incorrect) idea that oxygen atoms usually acquire two electrons and hydrogen atoms usually lose one electron in forming chemical compounds and ionic moieties. The rest of the rules derive from a need to recognize some exceptional cases and from applying the basic ideas to additional elements. The rules of the oxidation state formalism are these:This page titled 17.3: Defining Oxidation States is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.4: Balancing Oxidation-reduction Reactions

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Having defined oxidation states, we can now redefine an oxidation–reduction reaction as one in which at least one element undergoes a change of oxidation state. For example, in the reaction between permanganate ion and oxalate ion, the oxidation states of manganese and carbon atoms change. In the reactants, the oxidation state of manganese is 7+; in the products, it is 2+. In the reactants, the oxidation state of carbon is 3+; in the products, it is 4+.\[\begin{array}{c c c c c c c} 7+ & ~ & 3+ & ~ & 2+ & ~ & 4+ \\ MnO^-_4 & + & C_2O^{2-}_4 & \to & {Mn}^{2+} & + & CO_2 \end{array} \nonumber \]These oxidation state changes determine the stoichiometry of the reaction. In terms of the oxidation state formalism, each manganese atom gains five electrons and each carbon atom loses one electron. Thus the reaction must involve five times as many carbon atoms as manganese atoms. Allowing for the presence of two carbon atoms in the oxalate ion, conservation of electrons requires that the stoichiometric coefficients be\[2\ MnO^-_4+5\ C_2O^-_4\to 2\ {Mn}^{2+}+10\ CO_2 \nonumber \]Written this way, two \(MnO^-_4\) moieties gain ten electrons, and five \(C_2O^{2-}_4\) moieties lose ten electrons. When we fix the coefficients of the redox reactants, we also fix the coefficients of the redox products. However, inspection shows that both charge and the number of oxygen atoms are out of balance in this equation.The reaction occurs in acidic aqueous solution. This means that enough water molecules must participate in the reaction to achieve oxygen-atom balance. Adding eight water molecules to the product brings oxygen into balance. Now, however, charge and hydrogen atoms\[2\ MnO^-_4+5\ C_2O^-_4\to 2\ {Mn}^{2+}+10\ CO_2+8\ H_2O \nonumber \]do not balance. Since the solution is acidic, we can bring hydrogen into balance by adding sixteen protons to the reactants. When we do so, we find that charge balances also.\[2\ MnO^-_4+5\ C_2O^-_4+16\ H^+\to 2\ {Mn}^{2+}+10\ CO_2+8\ H_2O \nonumber \]Evidently, this procedure achieves charge balance because the oxidation state formalism enables us to find the correct stoichiometric ratio between oxidant and reductant.We can formalize this thought process in a series of rules for balancing oxidation–reduction reactions. In doing this, we can derive some advantage from splitting the overall chemical change into two parts, which we call half-reactions. It is certainly not necessary to introduce half-reactions just to balance equations; the real advantage is that a half-reaction describes the chemical change in an individual half-cell. The rules for balancing oxidation–reduction reactions using half-cell reactions are these:When we apply this method to the permanganate–oxalate reaction, we have\[2\ MnO^-_4+16\ H^++10\ e^-\to 2\ {Mn}^{2+}+8\ H_2O \nonumber \]reduction half-reaction\[5\ C_2O^-_4\to 10\ CO_2+10\ e^- \nonumber \]oxidation half-reaction\[2\ MnO^-_4+5\ C_2O^-_4+16\ H^+\to 2\ {Mn}^{2+}+10\ CO_2+8\ H_2O \nonumber \]balanced reactionThe half-reactions sum to the previously obtained result; the electrons cancel. For an example of a reaction in basic solution, consider the disproportionation of chloride dioxide to chlorite and chlorate ions:\[ \begin{array}{c c c c c} 4+ & ~ & 3+ & ~ & 5+ \\ ClO_2 & \to & ClO^-_2 & + & ClO^-_3 \end{array} \nonumber \]skeletal reaction\[ClO_2+e^-\to ClO^-_2 \nonumber \]reduction half-reaction\[ClO_2+2OH^-\to ClO^-_3+H_2O+e^- \nonumber \]oxidation half-reaction\[2\ ClO_2+2OH^-\to ClO^-_2+ClO^-_3+H_2O \nonumber \]balanced equationThis page titled 17.4: Balancing Oxidation-reduction Reactions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.5: Electrical Potential

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Electrical potential is measured in volts. If a system comprising one coulomb of charge passes through a potential difference of one volt, one joule of work is done on the system. The work done on the system is equal to the change in the energy of the system. For \(Q\) coulombs passing through a potential difference of \(\mathcal{E}\) volts, we have \(\Delta E=w_{elec}=Q\mathcal{E}\). Whether this represents an increase or a decrease in the energy of the system depends on the sign of the charge and on the sign of the potential difference.Electrical potential and gravitational potential are analogous. The energy change associated with moving a mass from one elevation to another in the earth’s gravitational field is\[\Delta E=w_{grav}=mgh_{final}-mgh_{initial}=m{\mathit{\Phi}}_{grav} \nonumber \]where \({\mathit{\Phi}}_{grav}=g\left(h_{final}-h_{initial}\right)\), which is the gravitational potential difference.The role played by charge in the electrical case is played by mass in the gravitational case. The energies of these systems change because charge or mass moves in response to the application of a force. In the electrical case, the force is the electrical force that arises from the interaction between charges. In the gravitational case, the force is the gravitational force that arises from the interaction between masses. A notable difference is that mass is always a positive quantity, whereas charge can be positive or negative.The distinguishing feature of an electrochemical cell is that there is an electrical potential difference between the two terminals. For any given cell, the magnitude of the potential difference depends on the magnitude of the current that is flowing. (Making the general problem even more challenging, we find that it depends also on the detailed history of the conditions under which electrical current has been drawn from the cell.) Fortunately, if we keep the cell’s temperature constant and measure the potential at zero current, the electrical potential is constant. Under these conditions, the cell’s characteristics are fixed, and potential measurements give reproducible results. We want to understand the origin and magnitude of this potential difference. Experimentally, we find:We can summarize these experimental observations by saying that the central issue in electrochemistry is the interrelation of three characteristics of an electrochemical cell: the electrical-potential difference between the terminals of the cell, the flow of electrons in the external circuit, and the chemical changes inside the cell that accompany this electron flow.This page titled 17.5: Electrical Potential is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.6: Electrochemical Cells as Circuit Elements

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Suppose we use a wire to connect the terminals of the cell built from the silver–silver ion half-cell and the copper–cupric ion half-cell. This wire then constitutes the external circuit, the path that the electrons follow as chemical change occurs within the cell. When the external circuit is simply a low-resistance wire, the cell is short-circuited. The external circuit can be more complex. For example, when we want to know the direction of electron flow, we incorporate a galvanometer.If the reaction between silver ions and copper metal is to occur, electrons must pass through the external circuit from the copper terminal to the silver terminal. An electron that is free to move in the presence of an electrical potential must move away from a region of more negative electrical potential and toward a region of more positive electrical potential. Since the electron-flow is away from the copper terminal and toward the silver terminal, the copper terminal must be electrically negative and the silver terminal must be electrically positive. Evidently, if we know the chemical reaction that occurs in an electrical cell, we can immediately deduce the direction of electron flow in the external circuit. Knowing the direction of electron flow in the external circuit immediately tells us which is the negative and which the positive terminal of the cell.The converse is also true. If we know which cell terminal is positive, we know that electrons in the external circuit flow toward this terminal. Even if we know nothing about the composition of the cell, the fact that electrons are flowing toward a particular terminal tells us that the reaction occurring in that half-cell is one in which a solution species, or the electrode material, takes up electrons. That is to say, some chemical entity is reduced in a half-cell whose potential is positive. It can happen that we know the half-reaction that occurs in a given half-cell, but that we do not know which direction the reaction goes. For example, if we replace the silver–silver ion half cell with a similar cell containing an aqueous zinc nitrate solution and a zinc electrode, we are confident that the half-cell reaction is either\[\ce{Zn^{0} \to Zn^{2+} + 2e^{-}} \nonumber \]or\[\ce{Zn^{2+} + 2 e^{-} \to Zn^{0}} \nonumber \]When we determine experimentally that the copper electrode is electrically positive with respect to the zinc electrode, we know that electrons are leaving the zinc electrode and flowing to the copper electrode. Therefore, the cell reaction must be\[\ce{Zn^{0} + Cu^{2+} \to Zn^{2+} + Cu^{0}} \nonumber \]It is convenient to have names for the terminals of an electrochemical cell. One naming convention is to call one terminal the anode and the other terminal the cathode. The definition is that the cathode is the electrode at which a reacting species is reduced. In the silver–silver ion containing cell, the silver electrode is the cathode. In the zinc–zinc ion containing cell, the copper electrode is the cathode. In these cells, the cathode is the electrically positive electrode. An important feature of these experiments is that the direction of the electrical potential in the external circuit is established by the reactions that occur spontaneously in the cells. The cells are sources of electrical current. Cells that operate to produce current are called galvanic cells.This page titled 17.6: Electrochemical Cells as Circuit Elements is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.7: The Direction of Electron Flow and its Implications

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We can incorporate another potential source into the external circuit of an electrochemical cell. If we do so in such a way that the two electrical potentials augment one another, as diagrammed in , the cell reaction is a spontaneous process. Since, as the cell reaction proceeds, electrons move through a potential difference in the external circuit, the reaction releases energy in the cell’s surroundings. If the external circuit is simply a resistor, as when the terminals are short-circuited, the energy is released as heat. Let \(q\) be the heat released and let \(Q\) be the amount of charge passed through the external circuit in a time interval \(\Delta t\). The heat-release rate is given by\[\frac{q}{\Delta t}=\frac{\Delta E}{\Delta t}=\frac{Q\mathcal{E}}{\Delta t} \nonumber \]The electrical current is \(I={Q}/{\Delta t}\). If the resistor follows Ohm’s law, \(\mathcal{E}=IR\), where \(R\) is the magnitude of the resistance, the heat release rate becomes\[\frac{q}{\Delta t}=I^2R \nonumber \]As the reaction proceeds and energy is dissipated in the external circuit, the ability of the cell to supply further energy is continuously diminished. The energy delivered to the surroundings through the external circuit comes at the expense of the cell’s internal energy and corresponds to the depletion of the cell reactants.When the chemical reaction occurring within a cell is driven by the application of an externally supplied potential difference, the opposite occurs. In the driven (electrolytic) cell, the direction of the cell reaction is opposite the direction of the spontaneous reaction that occurs when the cell operates galvanically. The electrolytic process produces the chemical reagents that are consumed in the spontaneous cell reaction. The external circuit delivers energy to the electrolytic cell, increasing its content of spontaneous-direction reactants and thereby increasing its ability to do work.In summary, the essential difference between electrolytic and galvanic cells lies in the factor that determines the direction of current flow and, correspondingly, the direction in which the cell reaction occurs. In a galvanic cell, a spontaneous chemical reaction occurs and this reaction determines the direction of current flow and the signs of the electrode potentials. In an electrolytic cell, the sign of the electrode potentials is determined by an applied potential source, which determines the direction of current flow; the cell reaction proceeds in the non-spontaneous direction.This page titled 17.7: The Direction of Electron Flow and its Implications is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.8: Electrolysis and the Faraday

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Electrolytic cells are very important in the manufacture of products essential to our technology-intensive civilization. Only electrolytic processes can produce many materials, notably metals that are strong reducing agents. Aluminum and the alkali metals are conspicuous examples. Many manufacturing processes that are not themselves electrolytic utilize materials that are produced in electrolytic cells. These processes would not be possible if the electrolytic products were not available. For example, elemental silicon, the essential precursor of most contemporary computer chips, is produced from silicon tetrachloride by reduction with sodium.\[SiCl_4+4\ Na^0\to Si^0+4\ NaCl \nonumber \](The silicon so produced is intensively refined, formed into large single crystals, and sliced into wafers before the chip-manufacturing process begins.) Elemental sodium is produced by the electrolysis of molten sodium chloride.Successful electrolytic processes involve artful selection of the current-collector material and the reaction conditions. The design of the cell is often crucial. Since sodium metal reacts violently with water, we recognize immediately that electrolysis of aqueous sodium chloride solutions cannot produce sodium metal. What products are obtained depends on numerous factors, notably the composition of the electrodes, the concentration of the salt solution, and the potential that is applied to the cell.Electrolysis of concentrated, aqueous, sodium chloride solutions is used on a vast scale in the chlor-alkali process for the co-production of chlorine and sodium hydroxide, both of which are essential for the manufacture of many common products.\[2\ NaCl\left(aq\right)+2\ H_2O\left(\ell \right)\to 2\ NaOH\left(aq\right)+Cl_2\left(g\right)+H_2\left(g\right) \nonumber \]Hydrogen is a by-product. The overall process does not involve sodium ion; rather, the overall reaction is an oxidation of chloride ion and a reduction of water.\[2\ Cl^-\left(aq\right)\to Cl_2\left(g\right)+2\ e^- \nonumber \] oxidation half-reaction \[2\ H_2O\left(\ell \right)+2\ e^-\to 2\ OH^-\left(aq\right)+H_2\left(g\right) \nonumber \]reduction half-reactionThe engineering difficulties associated with the chlor-alkali process are substantial. They occur because hydroxide ion reacts with chlorine gas; a practical cell must be designed to keep these two products separate. Commercially, two different designs have been successful. The diaphragm-cell process uses a porous barrier to separate the anodic and cathodic cell compartments. The mercury-cell process uses elemental mercury as the cathodic current collector; in this case, sodium ion is reduced, but the product is sodium amalgam (sodium–mercury alloy) not elemental sodium. Like metallic sodium, sodium amalgam reduces water, but the amalgam reaction is much slower. The amalgam is removed from the cell and reacted with water to produce sodium hydroxide and regenerate mercury for recycle to the electrolytic cell.Elemental sodium is manufactured by the electrolysis of molten sodium chloride. This is effected commercially using an iron cathode and a carbon anode. The reaction is\[NaCl\left(\ell \right)\to {Na}^0+Cl_2\left(g\right) \nonumber \]Such a cell is diagrammed in A mechanical barrier suffices to keep the products separate and prevent their spontaneous reaction back to the salt. A more significant problem in the design of the cell was to find an anode material that did not react with the chlorine produced. From the cell reaction, we see that one electron passes through the external circuit for every sodium atom that is produced. The charge that passes through the external circuit during the production of one mole of sodium metal is, therefore, the charge on one mole of electrons.In honor of Michael Faraday, the magnitude of the charge carried by a mole of electrons is called the faraday. The faraday constant is denoted by the symbol “\(\mathcal{F}\).” That is,\[1\mathcal{F}=\frac{6.02214\times {10}^{23}\mathrm{\ electrons}}{\mathrm{mol}}\times \frac{1.602187\times {10}^{-19}\ C}{\mathrm{electron}}=96,485\ C\ {\mathrm{mol}}^{-1} \nonumber \]The faraday is a useful unit in electrochemical calculations. The unit of electrical current, the ampere, is defined as the passage of one coulomb per second. Knowing the current in a circuit and the time for which it is passed, we can calculate the number of coulombs that are passed. Remembering the value of one faraday enables us to do stoichiometric calculations without bringing in Avogadro’s number and the electron charge every time.Tabulated information about the thermodynamic characteristics of half-reactions enable us to make useful predictions about what can and cannot occur in various cells that we might think of building. This information can be used to predict the potential difference that will be observed in a galvanic cell made by connecting two arbitrarily selected half-cells. In any electrolytic cell, more than one electron-transfer reaction can usually occur. In the chlor-alkali process, for example, water rather than chloride ion might be oxidized at the anode. In such cases, tabulated half-cell data enable us to predict which species can react at a particular applied potential.This page titled 17.8: Electrolysis and the Faraday is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.9: Electrochemistry and Conductivity

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From the considerations we have discussed, it is evident that any electrolytic cell involves a flow of electrons in an external circuit and a flow of ions within the materials comprising the cell. The function of the current collectors is to transfer electrons back and forth between the external circuit and the cell reagents.The measurement of solution conductivity is a useful technique for determining the concentrations and mobilities of ions in solution. Since conductivity measurements involve the passage of electrical current through a liquid medium, the process must involve electrode reactions as well as motion of ions through the liquid. Normally, the electrode reactions are of little concern in conductivity measurements. The applied potential is made large enough to ensure that some electrode reaction occurs. When the liquid medium is water, the electrode reactions are usually the reduction of water at the cathode and its oxidation at the anode. The conductivity attributable to a given ionic species is approximately proportional to its concentration. In the absence of dissolved ions, little current is passed. For aqueous solutions, this just restates the familiar observation that pure water is a poor electrical conductor. When few ions are present, it is not possible to move charge through the cell quickly enough to support a significant current in the external circuit.This page titled 17.9: Electrochemistry and Conductivity is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.10: The Standard Hydrogen Electrode (S.H.E)

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In Section 17.4, we introduce the idea of a half-reaction and a half-cell in the context of balancing equations for oxidation–reduction reactions. The real utility of these ideas is that they correspond to distinguishable parts of actual electrochemical cells. Information about the direction of a spontaneous reaction enables us to predict the relative electrical potentials of the half-cells that make up the corresponding electrochemical cell. Conversely, given information about the characteristic electrical potentials of half-cells, we can predict what chemical reactions can occur spontaneously. In short, there is a relationship between the electrical potential of an electrochemical cell at a particular temperature and pressure and the Gibbs free energy change for the corresponding oxidation–reduction reaction.Since cell potentials vary with the concentrations of the reactive components, we can simplify our record-keeping requirements by defining standard reference conditions that apply to a standard electrode of any type. We adopt the convention that a standard electrochemical cell contains all reactive components at unit activity. The vast majority of electrochemical cells that have been studied contain aqueous solutions. In data tables, the activity standard state for solute species is nearly always the hypothetical one-molal solution. For many purposes, it is an adequate approximation to say that all solutes are present at a concentration of one mole per liter, and all reactive gases at a pressure of one bar. (In Section 17.15, we see that the dependence of cell potential on reagent concentration is logarithmic.) In Sections 17.2 and 17.7, we discuss the silver–silver ion electrode; in this approximation, a standard silver–silver ion electrode is one in which the silver ion is present in the solution at a concentration of one mole per liter. Likewise, a standard copper–cupric ion electrode is one in which cupric ion is present in the solution at one mole per liter.We also need to choose an arbitrary reference half-cell. The choice that has been adopted is the Standard Hydrogen Electrode, often abbreviated the S.H.E. The S.H.E. is defined as a piece of platinum metal, immersed in a unit-activity aqueous solution of a protonic acid, and over whose surface hydrogen gas, at unit fugacity, is passed continuously. These concentration choices make the electrode a standard electrode. Frequently, it is adequate to approximate the S.H.E. composition by assuming that the hydrogen ion concentration is one molar and the hydrogen gas pressure is one bar. The half-reaction associated with the S.H.E. is\[\ce{ H^{+} + e^{-} \to 1/2 H2} \nonumber \]We define the electrical potential of this half-cell to be zero volts.This page titled 17.10: The Standard Hydrogen Electrode (S.H.E) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.11: Half-reactions and Half-cells

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Let us consider some standard electrochemical cells we could construct using the S.H.E. Two possibilities are electrochemical cells in which the second electrode is the standard silver–silver ion electrode or the standard copper–cupric ion electrode. The diagrams in summarize the half-reactions and the electrical potentials that we find when we construct these cells.We can also connect these cells so that the two S.H.E. are joined by one wire, while a second wire joints the silver and copper electrodes. This configuration is sketched in Whatever happens at one S.H.E. happens in the exact reverse at the other S.H.E. The net effect is essentially the same as connecting the silver– silver ion half-cell to the copper–cupric ion half-cell by a single salt bridge. If we did not already know what reaction occurs, we could figure it out from the information we have about how each of these two cells performs when it operates against the S.H.E.This page titled 17.11: Half-reactions and Half-cells is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.12: Standard Electrode Potentials

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We adopt a very useful convention to tabulate the potential drops across standard electrochemical cells, in which one half-cell is the S.H.E. Since the potential of the S.H.E. is zero, we define the standard electrode potential, \({\mathcal{E}}^o\), of any other standard half-cell (and its associated half-reaction) to be the potential difference when the half-cell operates spontaneously versus the S.H.E. The electrical potential of the standard half-cell determines both the magnitude and sign of the standard half-cell potential.If the process that occurs in the half-cell reduces a solution species or the electrode material, electrons traverse the external circuit toward the half-cell. Hence, the electrical sign of the half-cell terminal is positive. By the convention, the algebraic sign of the cell potential is positive \(\left({\mathcal{E}}^o>0\right)\). If the process that occurs in the half-cell oxidizes a solution species or the electrode, electrons traverse the external circuit away from the half-cell and toward the S.H.E. The electrical sign of the half-cell is negative, and the algebraic sign of the cell potential is negative \(\left({\mathcal{E}}^o<0\right)\).If we know the standard half-cell potential, we know the essential electrical properties of the standard half-cell operating spontaneously versus the S.H.E. at zero current. In particular, the algebraic sign of the standard half-cell potential tells us the direction of current flow and hence the direction of the reaction that occurs spontaneously.An older convention associates the sign of the standard electrode potential with the direction in which an associated half-reaction is written. This convention is compatible with the definition we have chosen; however, it creates two ways of expressing the same information. The difference is whether we write the direction of the half-reaction with the electrons appearing on the right or on the left side of the equation.When the half-reaction is written as a reduction process, with the electrons appearing on the left, the associated half-cell potential is called the reduction potential of the half-cell. Thus we would convey the information we have developed about the silver–silver ion and the copper–copper ion half cells by presenting the reactions and their associated potentials as\[{Ag}^++e^-\to {Ag}^0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ +0.7992\ \mathrm{volts} \nonumber \]\[{Cu}^{2+}+2\ e^-\to {Cu}^0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ +0.3394\ \mathrm{volts} \nonumber \]When the half-reaction is written as a reduction process, the sign of the electrode potential is the same as the sign of the electrical potential of the half-cell when the half-cell operates spontaneously versus the S.H.E. Thus, the reduction potential has the same algebraic sign as the electrode potential of our definition.We can convey the same information by writing the half-reaction in the reverse direction; that is, as an oxidation process in the left-to-right direction so that the electrons appear on the right. The agreed-upon convention is that we reverse the sign of the half-cell potential when we reverse the direction in which we write the equation. When the half-reaction is written as an oxidation process, the associated half-cell potential is called the oxidation potential of the half-cell. Older tabulations of electrochemical data often present half-reactions written as oxidation processes, with the electrons on the right, and present the potential information using the oxidation potential convention.\[{Ag}^++e^-\to {Ag}^0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ +0.7992\ \mathrm{volts} \nonumber \]reduction potential\[{Ag}^0\to {Ag}^++e^-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ -0.7992\ \mathrm{volts} \nonumber \]oxidation potentialNote that, in the convention that we have adopted, the term half-cell potential always denotes the potential of the half-cell when it operates spontaneously versus the S.H.E. In this convention, we do not need to write the half-reaction in order to specify the standard potential. It is sufficient to specify the chemical constituents of the half-cell. This is achieved using another representational convention.This cell-describing convention lists the active components of a half-cell, using a vertical line to indicate the presence of a phase boundary like that separating silver metal from an aqueous solution containing silver ion. The silver–silver ion cell is denoted \({Ag}^+\mid {Ag}^0\). (Using the superscript zero on the symbol for elemental silver is redundant; however, it does promote clarity.) The copper–cupric ion cell is denoted\[{Cu}^{2+}\mid {Cu}^0. \nonumber \]The S.H.E. is denoted \(H^+\mid H_2\mid {Pt}^0\), reflecting the presence of three distinct phases in the operating electrode. A complete electrochemical cell can be described using this convention. When the complete cell contains a salt bridge, this is indicated with a pair of vertical lines, \(\mathrm{\textrm{⃦}}\). A cell composed of a silver–silver ion half-cell and a S.H.E. is denoted\[{Pt}^0\mid H_2\ \mid H^+\ \ \ \textrm{⃦}\ \ {Ag}^+\mid \ {Ag}^0. \nonumber \]A further convention stipulates that the half-cell with the more positive electrode potential is written on the right. Under this convention, spontaneous operation of the standard full cell transfers electrons through the external circuit from the terminal shown on the left to the terminal shown on the right.We can now present our information about the behavior of the silver–silver ion half-cell versus the S.H.E. by writing that the standard potential of the \({Ag}^+\mid {Ag}^0\) half-cell is +0.7792 volts. The standard potential of the \({Cu}^{2+}\mid {Cu}^0\) half-cell is +0.3394 volts. The standard potential of the \(H^+\mid H_2\mid {Pt}^0\) (the S.H.E.) half-cell is 0.0000 volts. Again, our definition of the standard electrode potential makes the sign of the standard electrode potential independent of the direction in which the equation of the corresponding half-reaction is written.This page titled 17.12: Standard Electrode Potentials is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.13: Predicting the Direction of Spontaneous Change

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While our convention does not use the equation that we write for the half-reaction to establish the algebraic sign of the standard electrode potential, it is useful to associate the standard electrode potential with the half-reaction written as a reduction, that is, with the electrons written on the left side of the equation. We also establish the convention that reversing the direction of the half-reaction reverses the algebraic sign of its potential. When these conventions are followed, the overall reaction and the full-cell potential can be obtained by adding the corresponding half-cell information. If the resulting full-cell potential is greater than zero, the spontaneous overall reaction proceeds in the direction it is written, from left to right. If the full-cell potential is negative, the direction of spontaneous reaction is opposite to that written; that is, a negative full cell potential corresponds to the spontaneous reaction occurring from right to left. For example,\[2\ {Ag}^++2\ e^-\to 2\ {Ag}^0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ +0.7992\ \mathrm{volts} \nonumber \]\[{Cu}^0\to {Cu}^{2+}+2\ e^-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ -0.3394\ \mathrm{volts} \nonumber \]\[2\ {Ag}^++{Cu}^0\to 2\ {Ag}^0+{Cu}^{2+} \nonumber \] \[{\mathcal{E}}^0=\ +0.4598\ \mathrm{volts} \nonumber \]yields the equation corresponding to the spontaneous reaction and a positive full-cell potential. Writing\[2\ {Ag}^0\to 2\ {Ag}^+\ +2\ e^-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=\ -0.7992\ \mathrm{volts} \nonumber \]\[{Cu}^{2+}+2\ e^-\to {Cu}^0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{E}}^0=+0.3394\ \mathrm{volts} \nonumber \]\[2\ {Ag}^0+{Cu}^{2+}\to 2\ {Ag}^++{Cu}^0 \nonumber \] \[{\mathcal{E}}^0=\ -0.4598\ \mathrm{volts} \nonumber \]yields the equation for the non-spontaneous reaction and, correspondingly, the full-cell potential is less than zero.Note that when we multiply a chemical equation by some factor, we do not apply the same factor to the corresponding potential. The electrical potential of the corresponding electrochemical cell is independent of the number of moles of reactants and products that we choose to write. The cell potential is an intensive property. It has the same value for a small cell as for a large one, so long as the other intensive properties (temperature, pressure, and concentrations) are the same.This page titled 17.13: Predicting the Direction of Spontaneous Change is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.14: Cell Potentials and the Gibbs Free Energy

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In Section 17.11, we see that the electrical potential drop across the standard cell \({Pt}^0\mid H_2\ \mid H^+\ \ \ \textrm{⃦}\ \ {Ag}^+\mid \ {Ag}^0\) is 0.7992 volts. We measure this potential under conditions in which no current is flowing. That is, we find the counter-potential at which no current flows through the cell in either direction. An arbitrarily small change in the counter-potential away from this value, in either direction, is sufficient to initiate current flow. This means that the standard potential is measured when the cell is operating reversibly. By the definition of a standard cell, all of the reactants are at the standard condition of unit activity. If any finite current is drawn from a cell of finite size, the concentrations of the reagents will no longer be exactly the correct values for a standard cell. Nevertheless, we can calculate the energy that would be dissipated in the surroundings if the cell were to pass one mole of electrons (corresponding to consuming one mole of silver ions and one-half mole of hydrogen gas) through the external circuit while the cell conditions remain exactly those of the standard cell. This energy is\[96,485\ \mathrm{C}\ {\mathrm{mol}}^{-1}\times 0.7992\ \mathrm{V}=77,110\ \mathrm{J}\ {\mathrm{mol}}^{-1} \nonumber \]The form in which this energy appears in the surroundings depends on the details of the external circuit. However, we know that this energy represents the reversible work done on electrons in the external circuit as they traverse the path from the anode to the cathode. We call this the electrical work. Above we describe this as the energy change for a hypothetical reversible process in which the composition of the cell does not change. We can also view it as the energy change per electron for one electron-worth of real process, multiplied by the number of electrons in a mole. Finally, we can also describe it as the reversible work done on electrons during the reaction of one mole of silver ions in an infinitely large standard cell.The Gibbs free energy change for an incremental reversible process is \(dG = VdP + SdT + dw_{NPV}\), where \(dw_{NPV}\) is the increment of non-pressure–volume work. In the case of an electrochemical cell, the electrical work is non-pressure–volume work. In the particular case of an electrochemical cell operated at constant temperature and pressure, \(dP = dT\mathrm{=0}\), and \(dG = dw_{NPV} = dw_{\mathrm{elect}}\).The electrical work is just the charge times the potential drop. Letting \(n\) be the number of moles of electrons that pass through the external circuit for one unit of reaction, the total charge is \(Q=-n\mathcal{F}\), where \(\mathcal{F}\) is one faraday. For a standard cell, the potential drop is \({\mathcal{E}}^0\), so the work done on the electrons is \(Q{\mathcal{E}}^0=-n{\mathcal{F}\mathcal{E}}^0\). Since the standard conditions for Gibbs free energies are the same as those for electrical cell potentials, we have\[{w^{rev}_{elect}={\Delta }_rG}^o={-n\mathcal{F}\mathcal{E}}^o \nonumber \]If the reaction occurs spontaneously when all of the reagents are in their standard states, we have \({\mathcal{E}}^o>0\). For a spontaneous process, the work done on the system is less than zero, \(w^{rev}_{elect}<0\); the work done on the surroundings is \({\hat{w}}^{rev}_{elect}=-w^{rev}_{elect}>0\); and the energy of the surroundings increases as the cell reaction proceeds. The standard potential is an intensive property; it is independent of the size of the cell and of the way we write the equation for the chemical reaction. However, the work and the Gibbs free energy change depend on the number of electrons that pass through the external circuit. We usually specify the number of electrons by specifying the chemical equation to which the Gibbs free energy change applies. That is, if the associated reaction is written as\[Ag^+ + \frac{1}{2} H_2\to Ag^0+H^+ \nonumber \]we understand that one mole of silver ions are reduced and one mole of electrons are transferred; \(n=1\) and\(\Delta G^o=-\mathcal{F}{\mathcal{E}}^o\). If the reaction is written\[2\ {Ag}^++H_2\to 2\ {Ag}^0+2\ H^+ \nonumber \]we understand that two moles of silver ions are reduced and two moles of electrons are transferred, so that \(n=2\) and \(\Delta G^o=-2\mathcal{F}{\mathcal{E}}^o\).The same considerations apply to measurement of the potential of electrochemical cells whose component are not at the standard condition of unit activity. If the cell is not a standard cell, we can still measure its potential. We use the same symbol to denote the potential, but we omit the superscript zero that denotes standard conditions. These are, of course, just the conventions we have been using to distinguish the changes in other thermodynamic functions that occur at standard conditions from those that do not. We have therefore, for the Gibbs free energy change for the reaction occurring in an electrochemical cell that is not at standard conditions,\[w^{rev}_{elect}={\Delta }_rG=-n\mathcal{F}\mathcal{E} \nonumber \]This page titled 17.14: Cell Potentials and the Gibbs Free Energy is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.15: The Nernst Equation

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In Chapter 14, we find that the Gibbs free energy change is a function of the activities of the reactants and products. For the general reaction \(aA+bB\to cC+dD\)we have \[{\Delta }_rG={\Delta }_rG^o+RT{ \ln \frac{{\tilde{a}}^c_C{\tilde{a}}^d_D}{{\tilde{a}}^a_A{\tilde{a}}^b_B}\ } \nonumber \]Using the relationship between cell potentials and the Gibbs free energy, we find\[-n\mathcal{F}\mathcal{E}=-n\mathcal{F} \mathcal{E}^o+RT \ln \frac{\tilde{a}^c_C \tilde{a}^d_D}{\tilde{a}^a_A \tilde{a}^b_B} \nonumber \]or\[\mathcal{E}= \mathcal{E}^o-\frac{RT}{n\mathcal{F}} \ln \frac{\tilde{a}^c_C \tilde{a}^d_D}{ \tilde{a}^a_A \tilde{a}^b_B} \nonumber \]This is the Nernst equation. We derive it from our previous results for the activity dependence of the Gibbs free energy, which makes no explicit reference to electrochemical measurements at all. When we make the appropriate experimental measurements, we find that the Nernst equation accurately represents the temperature and concentration dependence of electrochemical-cell potentials.Reagent activities are often approximated adequately by molalities or molarities, for solute species, and by partial pressures—expressed in bars—for gases. The activities of pure solid and liquid phases can be taken as unity. For example, if we consider the reaction \[Ag^+ + \frac{1}{2} H_2\to Ag^0+H^+ \nonumber \]it is often sufficiently accurate to approximate the Nernst equation as\[\mathcal{E}= \mathcal{E}^o-\frac{RT}{n\mathcal{F}} \ln \frac{\left[H^+\right]}{\left[{Ag}^+\right] P^{1/2}_{H_2}} \nonumber \]This page titled 17.15: The Nernst Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.16: The Nernst Equation for Half-cells

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If the S.H.E. is one of the half-cells, the corresponding Nernst equation can be viewed as a description of the other half-cell. Using the cell in which the silver–silver ion electrode opposes the S.H.E., as in the preceding example, the cell potential is the algebraic sum of the potential of the silver terminal and the potential of the platinum terminal. We can represent the potential of the silver–silver ion electrode as \({\mathcal{E}}_{Ag\mid {Ag}^+}\). Since the S.H.E. is always at standard conditions, its potential, which we can represent as \({\mathcal{E}}^o_{Pt\mid H_2\mid H^+}\), is zero by definition. The cell potential is\[\mathcal{E}={\mathcal{E}}_{Ag\mid {Ag}^+}+{\mathcal{E}}^o_{Pt\mid H_2\mid H^+} \nonumber \]The potential of the cell with both half-cells at standard conditions is\[{\mathcal{E}^o={\mathcal{E}}^o_{Ag\mid {Ag}^+}+\mathcal{E}}^o_{Pt\mid H_2\mid H^+} \nonumber \]and, again since the S.H.E. is at standard conditions, \({\tilde{a}}_{H^+}=1\) and \(P_{H_2}=1\). Substituting into the Nernst equation for the full cell, we have\[\mathcal{E}_{Ag\mid {Ag}^+}+ \mathcal{E}^o_{Pt\mid H_2\mid H^+}= \mathcal{E}^o_{Ag\mid Ag^+}+\mathcal{E}^o_{Pt\mid H_2\mid H^+}-\frac{RT}{\mathcal{F}} \ln \frac{1}{\tilde{a}_{Ag}^+} \nonumber \]or\[\mathcal{E}_{Ag\mid {Ag}^+}= \mathcal{E}^o_{Ag\mid {Ag}^+}-\frac{RT}{\mathcal{F}} \ln \frac{1}{\tilde{a}_{Ag^+}} \nonumber \]where the algebraic signs of \(\mathcal{E}_{Ag\mid {Ag}^+}\) and \(\mathcal{E}^o_{Ag\mid {Ag}^+}\) correspond to writing the half-reaction in the direction \(Ag^++e^-\to Ag^0\). Note that this is precisely the equation that we would obtain by writing out the Nernst equation corresponding to the chemical equation \(Ag^++e^-\to Ag^0\).To see how these various conventions work together, let us consider the oxidation of hydroquinone \(\left(H_2Q\right)\) to quinone \(\left(Q\right)\) by ferric ion in acidic aqueous solutions:\[2\ {Fe}^{3+}+H_2Q\rightleftharpoons \ 2\ {Fe}^{2+}+Q+2H^+ \nonumber \]The quinone–hydroquinone couple isand the ferric ion–ferrous ion couple is\[Fe^{3+}+e^-\rightleftharpoons Fe^{2+} \nonumber \]The standard electrode potentials are \(\mathcal{E}_{Pt\mid Q,H_2Q,H^+}=+0.699\ \mathrm{v}\) and \(\mathcal{E}_{Pt\mid Fe^{3+},Fe^{2+}}=+0.783\ \mathrm{v}\). In each case, the numerical value is the potential of a full cell in which the other electrode is the S.H.E. The algebraic sign of the half-cell potential is equal to the sign of the half-cell’s electrical potential when it operates versus the S.H.E.To carry out this reaction in an electrochemical cell, we can use a salt bridge to join a \(Pt\mid Fe^{3+},Fe^{2+}\) cell to a \(Pt\mid Q,H_2Q,H^+\) cell. To construct a standard \(Pt\mid Fe^{3+},Fe^{2+}\) cell, we need only insert a platinum wire into a solution containing ferric and ferrous ions, both at unit activity. To construct a standard \(Pt\mid Q,H_2Q,H^+\) cell, we insert a platinum wire into a solution containing quinone, hydroquinone, and hydronium ion, all at unit activity. For standard half-cells, the cathode and anode reactions are\[Fe^{3+}+e^-\rightleftharpoons Fe^{2+} \nonumber \]and\[H_2Q\rightleftharpoons Q+2H^++2e^- \nonumber \]We can immediately write the Nernst equation for each of these half-reactions as\[\mathcal{E}_{Pt\mid Fe^{3+},Fe^{2+}}=\mathcal{E}^o_{Pt\mid Fe^{3+},Fe^{2+}}-\frac{RT}{\mathcal{F}} \ln \frac{\tilde{a}_{Fe^{2+}}}{\tilde{a}_{Fe^{3+}}} \nonumber \]and\[\left(-\mathcal{E}_{Pt\mid Q,H_2Q,H^+}\right)=\left(- \mathcal{E}^o_{Pt\mid Q,H_2Q,H^+}\right)-\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{\tilde{a}_Q \tilde{a}^2_{H^+}}{\tilde{a}_{H_2Q}} \nonumber \]If we add the equations for these half-reactions, the result does not correspond to the original full-cell reaction, because the number of electrons does not cancel. This can be overcome by multiplying the ferric ion–ferrous ion half-reaction by two. What do we then do about the corresponding half-cell Nernst equation? Clearly, the values of \({\mathcal{E}}_{Pt\mid {Fe}^{3+},{Fe}^{2+}}\) and \({\mathcal{E}}^o_{Pt\mid {Fe}^{3+},{Fe}^{2+}}\) do not depend on the stoichiometric coefficients in the half-reaction equation. However, the activity terms in the logarithm’s argument do, as does the number of electrons taking part in the half-reaction. We have\[2Fe^{3+}+2e^-\rightleftharpoons 2Fe^{2+} \nonumber \]with\[\begin{aligned} \mathcal{E}_{Pt\mid Fe^{3+},Fe^{2+}} & = \mathcal{E}^o_{Pt\mid Fe^{3+},Fe^{2+}}-\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{\tilde{a}^2_{Fe^{2+}}}{\tilde{a}^2_{Fe^{3+}}} \\ ~ & =\mathcal{E}^o_{Pt\mid Fe^{3+},Fe^{2+}}-\frac{RT}{\mathcal{F}} \ln \frac{\tilde{a}_{Fe^{2+}}}{\tilde{a}_{Fe^{3+}}} \end{aligned} \nonumber \]We see that we can apply any factor we please to the half-reaction. The Nernst equation gives the same dependence of the half-cell potential on reagent concentrations no matter what factor we choose. This is true also of the Nernst equation for any full-cell reaction. In the present example, adding the appropriate half-cell equations and their corresponding Nernst equations gives\[2\ Fe^{3+}+H_2Q\rightleftharpoons \ 2\ Fe^{2+}+Q+2H^+ \nonumber \]and\[ \begin{aligned} \mathcal{E} & = \mathcal{E}_{Pt\mid Fe^{3+},Fe^{2+}}- \mathcal{E}_{Pt\mid Q,H_2Q,H^+} \\ ~ & = \mathcal{E}^o_{Pt\mid Fe^{3+},Fe^{2+}}- \mathcal{E}^o_{Pt\mid Q,H_2Q,H^+}-\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{\tilde{a}^2_{Fe^{2+}}}{\tilde{a}^2_{Fe^{3+}}} -\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{\tilde{a}_Q \tilde{a}^2_{H^+}}{\tilde{a}_{H_2Q}} \\ ~ & =\mathcal{E}^0-\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{\tilde{a}_Q \tilde{a}^2_{H^+} \tilde{a}^2_{Fe^{2+}}}{\tilde{a}_{H_2Q} \tilde{a}^2_{Fe^{3+}}} \end{aligned} \nonumber \]This page titled 17.16: The Nernst Equation for Half-cells is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.17: Combining two Half-cell Equations to Obtain a new Half-cell Equation

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The same chemical species can be a reactant or product in many different half-cells. Frequently, data on two different half-cells can be combined to give information about a third half-cell. Let us consider two half-cells that involve the ferrous ion, \({Fe}^{2+}\). Ferrous ion and elemental iron form a redox couple. The half-cell consists of a piece of pure ion in contact with aqueous ferrous ion at unit activity. Our notation for this half-cell and its potential are \(Fe\mid {Fe}^{2+}\) and \({\mathcal{E}}_{Fe\mid {Fe}^{2+}}\). The corresponding half-reaction and its potential are\[Fe^{2+}+2e^-\rightleftharpoons Fe^0 \nonumber \]and\[\mathcal{E}_{Fe\mid Fe^{2+}}= \mathcal{E}^o_{Fe\mid Fe^{2+}}-\frac{RT}{\mathrm{2}\mathcal{F}} \ln \frac{1}{\tilde{a}_{Fe^{2+}}} \nonumber \]Ferrous ion can also give up an electron at an inert electrode, forming ferric ion, \(Fe^{3+}\). This process is reversible. Depending on the potential of the half-cell with which it is paired, the inert electrode can either accept an electron from the external circuit and deliver it to a ferric ion, or take an electron from a ferrous ion and deliver it to the external circuit. Thus, ferrous and ferric ions form a redox couple. Platinum metal functions as an inert electrode in this reaction. The half-cell consists of a piece of pure platinum in contact with aqueous ferrous and ferric ions, both present at unit activity. Our notation for this half-cell and potential are \(Pt\mid Fe^{2+},Fe^{3+}\) and \(\mathcal{E}_{Pt\mid Fe^{2+},Fe^{3+}}\). The corresponding half-reaction and its potential are\[Fe^{3+}+e^-\rightleftharpoons Fe^{2+} \nonumber \]and\[\mathcal{E}_{Pt\mid Fe^{2+},Fe^{3+}}= \mathcal{E}^o_{Pt\mid Fe^{2+},Fe^{3+}}-\frac{RT}{\mathcal{F}} \ln \frac{\tilde{a}_{Fe^{2+}}}{\tilde{a}_{Fe^{3+}}} \nonumber \]We can add these two half-reactions, to obtain\[Fe^{3+}+3e^-\rightleftharpoons Fe^0 \nonumber \]The Nernst equation for this half-reaction is\[\mathcal{E}_{Fe\mid Fe^{3+}}=\mathcal{E}^o_{Fe\mid Fe^{3+}}-\frac{RT}{\mathrm{3}\mathcal{F}} \ln \frac{1}{\tilde{a}_{Fe^{3+}}} \nonumber \]From our past considerations, both of these equations are clearly correct. However, in this case, the Nernst equation of the sum is not the sum of the Nernst equations. Nor should we expect it to be. The half-cell Nernst equations are really shorthand notation for the behavior of the half-cell when it is operated against a S.H.E. Adding half-cell Nernst equations corresponds to creating a new system by connecting the two S.H.E. electrodes of two separate full cells, as we illustrate in In the present instance, we are manipulating two half-reactions to obtain a new half-reaction; this manipulation does not correspond to any possible way of interconnecting the corresponding half-cells.Nevertheless, if we know the standard potentials for the first two reactions (\(\mathcal{E}^o_{Fe\mid Fe^{2+}}\) and \(\mathcal{E}^o_{Pt\mid Fe^{2+},Fe^{3+}}\)), we can obtain the standard potential for their sum (\(\mathcal{E}^o_{Fe\mid Fe^{3+}}\)). To do so, we exploit the relationship we found between electrical potential and Gibbs free energy. The first two reactions represent sequential steps that jointly achieve the same net change as the third reaction. Therefore, the sum of the Gibbs free energy changes for the first two reactions must be the same as the Gibbs free energy change for the third reaction. The standard potentials are not additive, but the Gibbs free energy changes are. We have\[\begin{array}{l l} Fe^{3+}+e^-\rightleftharpoons Fe^{2+} & \Delta G^o_{Fe^{3+}\to Fe^{2+}}=\ \mathcal{F} \mathcal{E}^o_{Pt\mid Fe^{2+},Fe^{3+}} \\ Fe^{2+}+2e^-\rightleftharpoons Fe^0 & \Delta G^o_{Fe^{2+}\to Fe^0} =2 \mathcal{F} \mathcal{E}^o_{Fe\mid Fe^{2+}} \\ \hline Fe^{3+}+3e^-\rightleftharpoons Fe^0 & \Delta G^o_{Fe^{3+}\to Fe^0} =3 \mathcal{F} \mathcal{E}^o_{Fe\mid Fe^{3+}} \end{array} \nonumber \]Since also\[\Delta G^o_{Fe^{3+}\to Fe^{2+}}+ \Delta G^o_{Fe^{2+} \to Fe^0}=\Delta G^o_{Fe^{3+} \to Fe^0} \nonumber \]we have\[\mathcal{F} \mathcal{E}^o_{Pt\mid Fe^{2+},Fe^{3+}}+2\mathcal{F} \mathcal{E}^o_{Fe\mid Fe^{2+}}=3\mathcal{F} \mathcal{E}^o_{Fe\mid Fe^{3+}} \nonumber \]and\[\mathcal{E}^o_{Fe\mid Fe^{3+}}=\frac{\mathcal{E}^o_{Pt\mid Fe^{2+},Fe^{3+}}+2 \mathcal{E}^o_{Fe\mid Fe^{2+}}}{3} \nonumber \]This page titled 17.17: Combining two Half-cell Equations to Obtain a new Half-cell Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.18: The Nernst Equation and the Criterion for Equilibrium

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In Section 17.15 we find for the general reaction \(aA+bB\to cD+dD\) that the Nernst equation is\[\mathcal{E}= \mathcal{E}^o-\frac{RT}{n\mathcal{F}} \ln \frac{\tilde{a}^c_C \tilde{a}^d_D}{\tilde{a}^a_A \tilde{a}^b_B} \nonumber \]We now want to consider the relationship between the potential of an electrochemical cell and the equilibrium position of the cell reaction. If the potential of the cell is not zero, short-circuiting the terminals of the cell will cause electrons to flow in the external circuit and reaction to proceed spontaneously in the cell. Since a spontaneous reaction occurs, the cell is not at equilibrium with respect to the cell reaction.As we draw current from any electrochemical cell, cell reactants are consumed and cell products are produced. Experimentally, we see that the cell voltage decreases continuously, and inspection of the Nernst equation shows that it predicts a potential decrease. Eventually, the voltage of a short-circuited cell decreases to zero. No further current is passed. The cell reaction stops; it has reached chemical equilibrium. If the cell potential is zero, the cell reaction must be at equilibrium, and vice versa.We also know that, at equilibrium, the activity ratio that appears as the argument of the logarithmic term is a constant—the equilibrium constant. So when \(\mathcal{E}=0\), we have also that\[K_a=\frac{{\tilde{a}}^c_C{\tilde{a}}^d_D}{{\tilde{a}}^a_A{\tilde{a}}^b_B} \nonumber \]Substituting these conditions into the Nernst equation, we obtain \[0={\mathcal{E}}^o-\frac{RT}{n\mathcal{F}}{ \ln K_a\ } \nonumber \] or \[K_a=\mathrm{exp}\frac{\left(n\mathcal{F}{\mathcal{E}}^o\right)}{RT} \nonumber \]We can obtain this same result if we recall that \(\Delta G^o=-RT{ \ln K_a\ }\) and that \(\Delta G^o=-n\mathcal{F}{\mathcal{E}}^o\). We can determine equilibrium constants by measuring the potentials of standard cells. Alternatively, we can measure an equilibrium constant and determine the potential of the corresponding cell without actually constructing it. Standard potentials and equilibrium constants are both measures of the Gibbs free energy change when the reaction occurs under standard conditions.This page titled 17.18: The Nernst Equation and the Criterion for Equilibrium is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

17.19: Problems

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1. Balance the following chemical equations assuming that they occur in aqueous solution.(a) \({Cu}^0+{Ag}^+\to {Cu}^{2+}+{Ag}^0\)(b) \({Fe}^{2+}+{Cr}_2O^{2-}_7\to {Fe}^{3+}+{Cr}^{3+}\)(c) \({Cr}^{2+}+{Cr}_2O^{2-}_7\to {Cr}^{3+}\)(d) \({Cl}_2+{Br}^-\to {Cl}^-+{Br}_2\)(e) \(ClO_3\to {Cl}^-+ClO^-_4\)(f) \(I^-+{IO}^-_3\to I_2\)(g) \(I^-+O_2\to I_2+{OH}^-\) (basic solution)(h) \(H_2C_2O^{2-}_4+MnO^-_4\to {CO}_2+{Mn}^{2+}\)(i) \({Fe}^{2+}+MnO^-_4\to {Fe}^{3+}+{Mn}^{2+}\)(j) \(H_2O_2+MnO^-_4\to {Mn}^{2+}+O_2\)(k) \(PbO_2+{Pb}^0+H_2SO_4\to {PbSO}_4\)(l) \({Fe}^{2+}\to {Fe}^0+{Fe}^{3+}\)(m) \({Cu}^{2+}+{Fe}^0\to {Cu}^0+{Fe}^{3+}\)(n) \({Al}^0+{OH}^-\to H_2+Al{\left(OH\right)}^-_4\) (basic solution)(o) \({Au}^0+{CN}^-+O_2\to Au{\left(CN\right)}^-_4\) (basic solution)(p) \({Cu}^0+{HNO}_3\to {Cu}^{2+}+NO_2\)(q) \({Al}^0+{Fe}_2O_3\to {Al}_2O_3+{Fe}^0\)(r) \(I^-+H_2O_2\to I_2+{OH}^-\) (basic solution)(s) \(HFeO^-_4+{Mn}^{2+}\to {Fe}^{3+}+{MnO}_2\)(t) \({Fe}^{2+}+S_2O^{2-}_8\to {Fe}^{3+}+SO^{2-}_4\)(u) \({Cu}^++O_2\to {Cu}^{2+}+{OH}^-\) (basic solution)2. Calculate the equilibrium constant,\(K_a\), for the reaction \({Cu}^0+2{Ag}^+\to {Cu}^{2+}+2{Ag}^0\). An excess of clean copper wire is placed in a \({10}^{-1}\ \underline{\mathrm{M}}\) silver nitrate solution. Assuming that molarities adequately approximate the activities of the ions, find the equilibrium concentrations of \({Ag}^+\) and \({Cu}^{2+}\).3. The standard potentials for reduction of \({Fe}^{2+}\) and \({Fe}^{3+}\) to \({Fe}^0\) are\[{Fe}^{2+}+2e^-\to {Fe}^0 {\mathcal{E}}^o=-0.447\ \mathrm{v} \nonumber \] \[{Fe}^{3+}+3e^-\to {Fe}^0 {\mathcal{E}}^o=-0.037\ \mathrm{v} \nonumber \](a) Find the standard potential for the disproportionation of \({Fe}^{2+}\) to \({Fe}^{3+}\) and \({Fe}^0\): \(\ \ \ {Fe}^{2+}\to {{Fe}^0+Fe}^{3+}\).(b) Find the standard half-cell potential for the reduction of \({Fe}^{3+}\) to \({Fe}^{2+}\): \({Fe}^{3+}+e^-\to {Fe}^{2+}\).4. The standard potential for reduction of tris-ethylenediamineruthenium (III) to tris-ethylenediamineruthenium (II) is\[{\left[Ru{\left(en\right)}_3\right]}^{3+}+e^-\to {\left[Ru{\left(en\right)}_3\right]}^{2+} {\mathcal{E}}^o=+0.210\ \mathrm{v} \nonumber \]Half-cell potential data are given below for several oxidants. Which of them can oxidize \({\left[Ru{\left(en\right)}_3\right]}^{2+}\) to \({\left[Ru{\left(en\right)}_3\right]}^{3+}\) in acidic (\(\left[H^+\right]\approx {\tilde{a}}_{H^+}={10}^{-1}\)) aqueous solution?(a) \(UO^{2+}_2+e^-\to UO^+_2\) \({\mathcal{E}}^o=+0.062\ \mathrm{v}\)(b) \({\left[Ru{\left(NH_3\right)}_6\right]}^{3+}+e^-\to {\left[Ru{\left(NH_3\right)}_6\right]}^{2+}\) \[{\mathcal{E}}^o=+0.10\ \mathrm{v} \nonumber \] (c) \({Cu}^{2+}+e^-\to {Cu}^+\) \({\mathcal{E}}^o=+0.153\ \mathrm{v}\)(d) \(AgCl+e^-\to {Ag}^0+{Cl}^-\) \({\mathcal{E}}^o=+0.222\ \mathrm{v}\)(e) \({Hg}_2{Cl}_2+2e^-\to 2{Hg}^0+2{Cl}^-\) \[{\mathcal{E}}^o=+0.268\ \mathrm{v} \nonumber \] (f) \(AgCN+e^-\to {Ag}^0+{CN}^-\) \({\mathcal{E}}^o=-0.017\ \mathrm{v}\)(g) \(SnO_2+4H^++4e^-\to {Sn}^0+2H_2O\) \[{\mathcal{E}}^o=-0.117\ \mathrm{v} \nonumber \]5. An electrochemical cell is constructed in which one half cell is a standard hydrogen electrode and the other is a hydrogen electrode immersed in a solution of \(pH=7\) \(\left(\left[H^+\right]\approx {\tilde{a}}_{H^+}={10}^{-7}\right)\). What is the potential difference between the terminals of the cell? What chemical change occurs in this cell?6. The standard half-cell potential for the reduction of oxygen gas at an inert electrode (like platinum metal) is\[O_2+4H^++4e^-\to 2H_2O {\mathcal{E}}^o=+1.229\ \mathrm{v} \nonumber \]An electrochemical cell is constructed in which one half cell is a standard hydrogen electrode and the other cell is a piece of platinum metal, immersed in a \(1\ \underline{M}\) solution of \(HClO_4\), which is continuously in contact with bubbles of oxygen gas at a pressure of \(1\) bar.(a) What is the potential difference between the terminals of the cell? What chemical change occurs in this cell?(b) The \(1\ \underline{M}\) \(HClO_4\) solution in part (a) is replaced with pure water \(\left(\left[H^+\right]\approx {\tilde{a}}_{H^+}={10}^{-7}\right)\). What is the potential difference between the terminals of this cell?(c) The \(1\ \underline{M}\) \(HClO_4\) solution in part (a) is replaced with \(1\ \underline{M}\) \(NaOH\) \(\left(\left[H^+\right]\approx {\tilde{a}}_{H^+}={10}^{-14}\right)\). What is the potential difference between the terminals of this cell?7. A variable electrical potential source is introduced into the external circuit of the cell in part (a) of problem6. The negative terminal of the potential source is connected to the oxygen electrode and the positive terminal of the potential source is connected to the standard hydrogen electrode. If the applied electrical potential is 1.3 v, what chemical change occurs? What is the minimum electrical potential that must be applied to electrolyze water if the oxygen electrode contains a \(1\ \underline{M}\) \(HClO_4\) solution? A neutral (\(pH=7\)) solution? A \(1\ \underline{M}\) \(NaOH\) solution?8. Two platinum electrodes are immersed in \(1\ \underline{M}\) \(HClO_4\). What potential difference must be applied between these electrodes in order to electrolyze water? (Assume that \(P_{O_2}=1\ \mathrm{bar}\) and \(P_{H_2}=1\ \mathrm{bar}\) at their respective electrodes, as will be the case as soon as a few bubbles of gas have accumulated at each electrode.) What potential difference is required if the electrodes are immersed in pure water? In \(1\ \underline{M}\) \(NaOH\)?This page titled 17.19: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.1: Energy Distributions and Energy Levels

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Beginning in Chapter 20, we turn our attention to the distribution of energy among the molecules in a closed system that is immersed in a constant-temperature bath, that is at equilibrium, and that contains a large number of molecules. We want to find the probability that the energy of a molecule in such a system is in a particular interval of energy values. This probability is also the fraction of the molecules whose energies are in the specified interval, since we assume that these statements mean the same thing for a system at equilibrium.The probability that the energy of a particular molecule is in a particular interval is intimately related to the energies that it is possible for a molecule to have. Before we can make further progress in describing molecular energy distributions, we must discuss atomic and molecular energies. For our development of the Boltzmann equation, we need to introduce the idea of quantized energy states. This requires a short digression on the basic ideas of quantum mechanics and the quantized energy levels of atoms and molecules.We have derived two expressions that relate the energy of a molecule to the probability that the molecule will have that energy. One follows from the barometric formula\[ \begin{align*} \eta \left(h\right) &=\eta \left(0\right)\mathrm{exp}\left(\frac{-mgh}{kT}\right) \\[4pt] &=\eta \left(0\right)\mathrm{exp}\left(\frac{-{\epsilon }_{potential}}{kT}\right) \end{align*} \]in which the number density of molecules depends exponentially on their gravitational potential energy, \(mgh\), and the reciprocal of the temperature. From the barometric formula, we can find the probability density function\[\frac{df}{dh}=\frac{mg}{kT}\mathrm{exp}\left(\frac{-mgh}{kT}\right) \nonumber \](See problem 3.22) The other is the Maxwell-Boltzmann distribution function\[\frac{df}{dv}=4\pi {\left(\frac{m}{2\pi kT}\right)}^{3/2}v^2\mathrm{exp}\left(\frac{-mv^2}{2kT}\right)=4\pi {\left(\frac{m}{2\pi kT}\right)}^{3/2}v^2\mathrm{exp}\left(\frac{-{\epsilon }_{kinetic}}{2kT}\right) \nonumber \]in which the probability density of molecular velocities depends exponentially on their kinetic energies, \({{mv}^2}/{2}\), and the reciprocal of the temperature. We will see that this dependence is very general. Any time the molecules in a system can have a range of energies, the probability that a molecule has energy \(\epsilon\) is proportional to \(\mathrm{exp}\left({-\epsilon }/{kT}\right)\). The exponential term, \(\mathrm{exp}\left({-\epsilon }/{kT}\right)\), is often called the Boltzmann factor.We might try to develop a more general version of the Maxwell-Boltzmann distribution function by an argument that somehow parallels our derivation of the Maxwell-Boltzmann equation. It turns out that any such attempt is doomed to failure, because it is based on a fundamentally incorrect view of nature. In developing the barometric formula and the Maxwell-Boltzmann distribution, we assume that the possible energies are continuous; a molecule can be at any height above the surface of the earth, and its translational velocity can have any value. When we turn to the distribution of other ways in which molecules can have energy, we find that this assumption produces erroneous predictions about the behavior of macroscopic collections of molecules.The failure of such attempts led Max Planck to the first formulation of the idea that energy is quantized. The spectrum of light emitted from glowing-hot objects (so-called “black bodies”) depends on the temperature of the emitting object. Much of the experimentally observable behavior of light can be explained by the hypothesis that light behaves like a wave. Mechanical (matter-displacement) waves carry energy; the greater the amplitude of the wave, the more energy it carries. Now, light is a form of energy, and a spectrum is an energy distribution. It was a challenge to late nineteenth century physics to use the wave model for the behavior of light to predict experimentally observed emission spectra. This challenge went unmet until Planck introduced the postulate that such “black-body radiators” absorb or emit electromagnetic radiation only in discrete quantities, called quanta. Planck proposed that the energy of one such quantum is related to the frequency, \(\nu\), of the radiation by the equation \(E=h\nu\), where the proportionality constant, \(h\), is now called Planck’s constant. In Planck’s model, the energy of an electromagnetic wave depends on its frequency, not its amplitude.In the years following Planck’s hypothesis, it became clear that many properties of atoms and molecules are incompatible with the idea that an atom or molecule can have any arbitrary energy. We obtain agreement between experimental observations and theoretical models only if we assume that atoms and molecules can have only very particular energies. This is observed most conspicuously in the interactions of atoms and molecules with electromagnetic radiation. One such interaction gives rise to a series of experimental observations known as the photoelectric effect. In order to explain the photoelectric effect, Albert Einstein showed that it is necessary to extend Planck’s concept to assume that light itself is a stream of discrete energy quanta, called photons. In our present understanding, it is necessary to describe some of the properties of light as wave-like and some as particle-like.In many absorption and emission spectra, we find that a given atom or molecule can emit or absorb electromagnetic radiation only at very particular frequencies. For example, the light emitted by atoms excited by an electrical discharge contains a series of discrete emission lines. When it is exposed to a continuous spectrum of frequencies, an atom is observed to absorb light at precisely the discrete frequencies that are observed in emission. Niels Bohr explained these observations by postulating that the electrons in atoms can have only particular energies. The absorption of visible light by atoms and molecules occurs when an electron takes up electromagnetic energy and moves from one discrete energy level to a second, higher, one. (Absorption of a continuous range of frequencies begins to occur only when the light absorbed provides sufficient energy to separate an electron from the original chemical species, producing a free electron and a positively charged ion. At the onset frequency, neither of the product species has any kinetic energy. Above the onset frequency, spectra are no longer discrete, and the species produced have increasingly greater kinetic energies.) Similar discrete absorption lines are observed for the absorption of infrared light and microwave radiation by diatomic or polyatomic gas molecules. Infrared absorptions are associated with vibrational motions, and microwave absorptions are associated with rotational motions of the molecule about its center of mass. These phenomena are explained by the quantum theory.The quantized energy levels of atoms and molecules can be found by solving the Schrödinger equation for the system at hand. To see the basic ideas that are involved, we discuss the Schrödinger equation and some of the most basic approximations that are made in applying it to the description of atomic and molecular systems. But first, we should consider one more preliminary question: If the quantum hypothesis is so important to obtaining valid equations for the distribution of energies, why are the derivations of the Maxwell-Boltzmann equation and the barometric formula successful? Maxwell’s derivation is successful because the quantum mechanical description of a molecule’s translational kinetic energy is very well approximated by the assumption that the molecule’s kinetic energy can have any value. In the language of quantum mechanics, the number of translational energy levels available to a molecule is very large, and successive energy levels are very close together—so close together that it is a good approximation to say that they are continuous. Similarly, the gravitational potential energies available to a molecule in the earth’s atmosphere are well approximated by the assumption that they belong to a continuous distribution.This page titled 18.1: Energy Distributions and Energy Levels is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.2: Quantized Energy - De Broglie's Hypothesis and the Schroedinger Equation

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Subsequent to Planck’s proposal that energy is quantized, the introduction of two further concepts led to the theory of quantum mechanics. The first was Einstein’s relativity theory, and his deduction from it of the equivalence of matter and energy. The relativistic energy of a particle is given by\[E^2=p^2c^2+m^2_0c^4 \nonumber \]where \(p\) is the momentum and \(m_0\) is the mass of the particle when it is at rest. The second was de Broglie’s hypothesis that any particle of mass \(m\) moving at velocity \(v\), behaves like a wave. De Broglie’s hypothesis is an independent postulate about the structure of nature. In this respect, its status is the same as that of Newton’s laws or the laws of thermodynamics. Nonetheless, we can construct a line of thought that is probably similar to de Broglie’s, recognizing that these are heuristic arguments and not logical deductions.We can suppose that de Broglie’s thinking went something as follows: Planck and Einstein have proposed that electromagnetic radiation—a wave-like phenomenon—has the particle-like property that it comes in discrete lumps (photons). This means that things we think of as waves can behave like particles. Conversely, the lump-like photons behave like waves. Is it possible that other lump-like things can behave like waves? In particular is it possible that material particles might have wave-like properties? If a material particle behaves like a wave, what wave-like properties should it exhibit?Well, if we are going to call something a wave, it must have a wavelength, \(\lambda\), a frequency, \(\nu\), and a propagation velocity, \(v\), and these must be related by the equation \(v=\lambda \nu\). The velocity of propagation of light is conventionally given the symbol \(c\), so \(c=\lambda \nu\). The Planck-Einstein hypothesis says that the energy of a particle (photon) is \(E=h\nu ={hc}/{\lambda }\). Einstein proposes that the energy of a particle is given by \(E^2=p^2c^2+m^2_0c^4\). A photon travels at the speed of light. This is compatible with other relativistic equations only if the rest mass of a photon is zero. Therefore, for a photon, we must have \(E=pc\). Equating these energy equations, we find that the momentum of a photon is\[p={h}/{\lambda } \nonumber \]Now in a further exercise of imagination, we can suppose that this equation applies also to any mass moving with any velocity. Then we can replace \(p\) with \(mv\), and write \[mv={h}/{\lambda } \nonumber \]We interpret this to mean that any mass, \(m\), moving with velocity, \(v\), has a wavelength, \(\lambda\), given by\[\lambda ={h}/{mv} \nonumber \]This is de Broglie’s hypothesis. We have imagined that de Broglie found it by a series of imaginative—and not entirely logical—guesses and suppositions. The illogical parts are the reason we call the result a hypothesis rather than a derivation, and the originality of the guesses and suppositions is the reason de Broglie’s hypothesis was new. It is important physics, because it turns out to be experimentally valid. Very small particles do exhibit wave-like properties, and de Broglie’s hypothesis correctly predicts their wavelengths.In a similar vein, we can imagine that Schrödinger followed a line of thought something like this: de Broglie proposes that any moving particle behaves like a wave whose wavelength depends on its mass and velocity. If a particle behaves as a wave, it should have another wave property; it should have an amplitude. In general, the amplitude of a wave depends on location and time, but we are thinking about a rather particular kind of wave, a wave that—so to speak—stays where we put it. That is, our wave is supposed to describe a particle, and particles do not dissipate themselves in all directions like the waves we get when we throw a rock in a pond. We call a wave that stays put a standing wave; it is distinguished by the fact that its amplitude depends on location but not on time.Mathematically, the amplitude of any wave can be described as a sum of (possibly many) sine and cosine terms. A single sine term describes a simple wave. If it is a standing wave, its amplitude depends only on distance, and its amplitude is the same for any two points separated by one wavelength. Letting the amplitude be \(\psi\), this standing wave is described by \(\psi \left(x\right)=A{\mathrm{sin} \left(ax\right)\ }\), where \(x\) is the location, expressed as a distance from the origin at \(x=0\). In this wave equation, \(A\) and \(a\) are parameters that fix the maximum amplitude and the wavelength, respectively. Requiring the wavelength to be \(\lambda\) means that \(a\lambda =2\pi\). (Since \(\psi\) is a sine function, it repeats every time its argument increases by \(2\pi\) radians. We require that \(\psi\) repeat every time its argument increases by \(a\lambda\) radians, which requires that \(a\lambda =2\pi\).) Therefore, we have \[a={2\pi }/{\lambda } \nonumber \]and the wave equation must be\[\psi \left(x\right)=A{\mathrm{sin} \left({2\pi x}/{\lambda }\right)\ } \nonumber \]Equations, \(\psi\), that describe standing waves satisfy the differential equation\[\frac{d^2\psi }{dx^2}=-C\psi \nonumber \]where \(C\) is a constant. In the present instance, we see that\[\frac{d^2\psi }{dx^2}=-{\left(\frac{2\pi }{\lambda }\right)}^2A{\mathrm{sin} \left(\frac{2\pi x}{\lambda }\right)\ }=-{\left(\frac{2\pi }{\lambda }\right)}^2\psi \nonumber \]From de Broglie’s hypothesis, we have \(\lambda ={h}/{mv}\), so that the constant \(C\) can be written as\[C={\left(\frac{2\pi }{\lambda }\right)}^2={\left(\frac{2\pi mv}{h}\right)}^2={\left(\frac{2\pi }{h}\right)}^2\left(2m\right)\left(\frac{{mv}^2}{2}\right)=\left(\frac{8{\pi }^2m}{h^2}\right)\left(\frac{{mv}^2}{2}\right) \nonumber \]Let \(T\) be the kinetic energy, \({{mv}^2}/{2}\), and let \(V\) be the potential energy of our wave-like particle. Then its energy is \(E=T+V\), and we have \({{mv}^2}/{2}=T=E-V\).The constant \(C\) becomes\[C=\left(\frac{8{\pi }^2m}{h^2}\right)T=\left(\frac{8{\pi }^2m}{h^2}\right)\left(E-V\right) \nonumber \]Making this substitution for \(C\), we find a differential equation that describes a standing wave, whose wavelength satisfies the de Broglie equation. This is the time-independent Schrödinger equation in one dimension:\[\frac{d^2\psi }{dx^2}=-\left(\frac{8{\pi }^2m}{h^2}\right)\left(E-V\right)\psi \nonumber \] or \[-\left(\frac{h^2}{8{\pi }^2m}\right)\frac{d^2\psi }{dx^2}+V\psi =E\psi \nonumber \]Often the latter equation is written as\[\left[-\left(\frac{h^2}{8{\pi }^2m}\right)\frac{d^2}{dx^2}+V\right]\psi =E\psi \nonumber \]where the expression in square brackets is called the Hamiltonian operator and abbreviated to \(H\), so that the Schrödinger equation becomes simply, if cryptically,\[H\psi =E\psi \nonumber \]If we know how the potential energy of a particle, \(V\), depends on its location, we can write down the Hamiltonian operator and the Schrödinger equation that describe the wave properties of the particle. Then we need to find the wave equations that satisfy this differential equation. This can be difficult even when the Schrödinger equation involves only one particle. When we write the Schrödinger equation for a system containing multiple particles that interact with one another, as for example an atom containing two or more electrons, analytical solutions become unattainable; only approximate solutions are possible. Fortunately, a great deal can be done with approximate solutions.The Schrödinger equation identifies the value of the wavefunction, \(\psi \left(x\right)\), with the amplitude of the particle wave at the location x. Unfortunately, there is no physical interpretation for \(\psi \left(x\right)\); that is, no measurable quantity corresponds to the value of \(\psi \left(x\right)\). There is, however, a physical interpretation for the product \(\psi \left(x\right)\psi \left(x\right)\) or \({\psi }^2\left(x\right)\). [More accurately, the product \(\psi \left(x\right){\psi }^*\left(x\right)\), where \({\psi }^*\left(x\right)\) is the complex conjugate of \(\psi \left(x\right)\). In general, \(x\) is a complex variable.] \({\psi }^2\left(x\right)\) is the probability density function for the particle whose wavefunction is \(\psi \left(x\right)\). That is, the product \({\psi }^2\left(x\right)dx\) is the probability of finding the particle within a small distance, \(dx\), of the location \(x\). Since the particle must be somewhere, we also have\[1=\int^{+\infty }_{-\infty }{{\psi }^2\left(x\right)}dx \nonumber \]This page titled 18.2: Quantized Energy - De Broglie's Hypothesis and the Schroedinger Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.3: The Schrödinger Equation for A Particle in A Box

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A problem usually called the particle in a box provides a convenient illustration of the principles involved in setting up and solving the Schrödinger equation. Besides being a good illustration, the problem also proves to be a useful approximation to many physical systems. The statement of the problem is simple. We have a particle of mass \(m\) that is constrained to move only in one dimension. For locations in the interval \(0\le x\le \ell\), the particle has zero potential energy. For locations outside this range, the particle has infinite potential energy. Since the particle cannot have infinite energy, this means that it can never find its way into locations outside of the interval \(0\le x\le \ell\). We can think of this particle as a bead moving on a wire, with stops located on the wire at \(x=0\) and at \(x=\ell\). We can also think of it as being confined to a one-dimensional box of length \(\ell\), which is the viewpoint represented by the name. The particle in a box model is diagrammed in =0\), when the value of \(x\) lies in the interval\[ - \infty x < 0 \nonumber \]or\[ \ell < x < + \infty \nonumber \]We assume that the probability of finding the particle cannot change abruptly when its location changes by an arbitrarily small amount. This means that the wavefunction must be continuous, and it follows that \(\psi \left(0\right)=0\) and \(\psi \left(\ell \right)=0\). Inside the box, the particle’s Schrödinger equation is\[-\left(\frac{h^2}{8{\pi }^2m}\right)\frac{d^2\psi }{dx^2}=E\psi \nonumber \]and we seek those functions \(\psi \left(x\right)\) that satisfy both this differential equation and the constraint equations \(\psi \left(0\right)=0\) and \(\psi \left(\ell \right)=0\). It turns out that there are infinitely many such solutions, \({\psi }_n\), each of which corresponds to a unique energy level, \(E_n\).To find these solutions, we first guess—guided by our considerations in §2—that solutions will be of the form \[\psi \left(x\right)=A{\mathrm{sin} \left(ax\right)\ }+B{\mathrm{cos} \left(bx\right)\ } \nonumber \]A solution must satisfy\[\psi \left(0\right)=A{\mathrm{sin} \left(0\right)\ }+B{\mathrm{cos} \left(0\right)\ }=B{\mathrm{cos} \left(0\right)=0\ } \nonumber \]so that \(B=0\). At the other end of the box, we must have\[\psi \left(\ell \right)=A{\mathrm{sin} \left(a\ell \right)\ }=0 \nonumber \]which means that \(a\ell =n\pi\), where \(n\) is any integer: \(n\ =1,\ 2,\ \dots .\) Hence, we have\[a={n\pi }/{\ell } \nonumber \]and the only equations of the proposed form that satisfy the conditions at the ends of the box are\[{\psi }_n\left(x\right)=A{\mathrm{sin} \left({n\pi x}/{\ell }\right)\ } \nonumber \]To test whether these equations satisfy the Schrödinger equation, we check\[-\left(\frac{h^2}{8{\pi }^2m}\right)\frac{d^2}{dx^2}\left[A{\mathrm{sin} \left(\frac{n\pi x}{\ell }\right)\ }\right]=E_n{\psi }_n \nonumber \]and find\[\left(\frac{h^2}{8{\pi }^2m}\right){\left(\frac{n\pi }{\ell }\right)}^2\left[A{\mathrm{sin} \left(\frac{n\pi x}{\ell }\right)\ }\right]=\left(\frac{n^2h^2}{8{m\ell }^2}\right){\psi }_n=E_n{\psi }_n \nonumber \]so that the wavefunctions \({\psi }_n\left(x\right)=A{\mathrm{sin} \left({n\pi x}/{\ell }\right)\ }\) are indeed solutions and the energy, \(E_n\), associated with the wavefunction \({\psi }_n\left(x\right)\) is\[E_n=\frac{n^2h^2}{8{m\ell }^2} \nonumber \]We see that the energy values are quantized; although there are infinitely many energy levels, \(E_n\), only very particular real numbers—those given by the equation above—correspond to energies that the particle can have. If we sketch the first few wavefunctions, \({\psi }_n\left(x\right)\), we see that there are always \(n-1\) locations inside the box at which \({\psi }_n\left(x\right)\) is zero. These locations are called nodes. Once we know \(n\), we know the number of nodes, and we can sketch the general shape of the corresponding wavefunction. The first three wavefunctions and their squares are sketched in . To determine \(A\), we interpret \({\psi }^2\left(x\right)\) as a probability density function, and we require that the probability of finding the particle in the box be equal to unity. This means that\[1=\int^{\ell }_0{A^2{{\mathrm{sin}}^2 \left(\frac{n\pi x}{\ell }\right)\ }dx}=A^2\int^{\ell }_0{\left[\frac{1}{2}-\frac{1}{2}{\mathrm{cos} \left(\frac{2n\pi x}{\ell }\right)\ }\right]dx}=A^2\left(\frac{\ell }{2}\right) \nonumber \]so that \(A=\sqrt{2/\ell }\), and the final wavefunctions are\[{\psi }_n\left(x\right)=\sqrt{\frac{2}{\ell }}{\mathrm{sin} \left(\frac{n\pi x}{\ell }\right)\ } \nonumber \]This page titled 18.3: The Schrödinger Equation for A Particle in A Box is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.4: The Schrödinger Equation for a Molecule

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Molecules are composed of atoms, and atoms are composed of nuclei and electrons. When we consider the internal motions of molecules, we have to consider the motions of a large number of charged particles with respect to one another. In principle, we can write down the potential function (the \(V\) in the Schrödinger equation) that describes the Coulomb’s law based potential energy of the system of charged particles. In principle, we can then solve the Schrödinger equation and obtain a series of wavefunctions, \({\psi }_n\left(x\right)\), and their corresponding energies, \(E_n\), that completely characterize the motions of the molecule’s constituent particles. Each of the \(E_n\) is an energy value that the molecule can have. Often we say that it is an energy level that the molecule can occupy.Since every distance between two charged particles is a variable in the Schrödinger equation, the number of variables increases dramatically as the size of the molecule increases. The two-particle hydrogen-atom problem has been solved analytically. For any chemical species larger than the hydrogen atom, only approximate solutions are possible. Nevertheless, approximate results can be obtained to very high accuracy. Greater accuracy comes at the expense of more extensive calculations.Let us look briefly at the more fundamental approximations that are made. One is called the Born-Oppenheimer approximation; it states that the motions of the nuclei in a molecule are too slow to affect the motions of the electrons. This occurs because nuclei are much more massive than electrons. The Born-Oppenheimer approximation assumes that the electronic motions can be calculated as if the nuclei are fixed at their equilibrium positions without introducing significant error into the result. That is, there is an approximate wavefunction describing the motions of the electrons that is independent of a second wavefunction that describes the motions of the nuclei.The mathematical description of the nuclear motions can be further simplified using additional approximations; we can separate the nuclear motions into translational, rotational, and vibrational modes. Translational motion is the three-dimensional displacement of an entire molecule. It can be described by specifying the motion of the molecule’s center of mass. The motions of the constituent nuclei with respect to one another can be further subdivided: rotational motions change the orientation of the whole molecule in space; vibrational motions change distances between constituent nuclei.The result is that the wavefunction for the molecule as a whole can be approximated as a product of a wavefunction (\({\psi }_{electronic}\) or \({\psi }_e\)) for the electronic motions, a wavefunction (\({\psi }_{vibration}\) or \({\psi }_v\)) for the vibrational motions, a wavefunction (\({\psi }_{rotation}\) or \({\psi }_r\)) for the rotational motions, and a wavefunction (\({\psi }_{translation}\) or \({\psi }_t\)) for the translational motion of the center of mass. We can write\[{\psi }_{molecule}={\psi }_e{\psi }_v{\psi }_r{\psi }_t \nonumber \](None of this is supposed to be obvious. We are merely describing the essential results of a considerably more extensive development.)When we write the Hamiltonian for a molecule under the approximation that the electronic, vibrational, rotational, and translational motions are independent of each other, we find that the Hamiltonian is a sum of terms. In some of these terms, the only independent variables are those that specify the locations of the electrons. We call these variables electronic coordinates. Some of the remaining terms involve only vibrational coordinates, some involve only rotational coordinates, and some involve only translational coordinates. That is, we find that the Hamiltonian for the molecule can be expressed as a sum of terms, each of which is the Hamiltonian for one of the kinds of motion:\[H_{molecule}=H_e+H_v+H_r{+H}_t \nonumber \]where we have again abbreviated the subscripts denoting the various categories of motion.Consequently, when we write the Schoedinger equation for the molecule in this approximation, we have\[\begin{align*} H_{molecule}{\psi }_{molecule} &=\left(H_e+H_v+H_r{+H}_t\right){\psi }_e{\psi }_v{\psi }_r{\psi }_t \\[4pt] &={\psi }_v{\psi }_r{\psi }_tH_e{\psi }_e+{\psi }_e{\psi }_r{\psi }_tH_v{\psi }_v+{\psi }_e{\psi }_v{\psi }_tH_r{\psi }_r+{\psi }_e{\psi }_v{\psi }_rH_t{\psi }_t \\[4pt] &={\psi }_v{\psi }_r{\psi }_tE_e{\psi }_e+{\psi }_e{\psi }_r{\psi }_tE_v{\psi }_v+{\psi }_e{\psi }_v{\psi }_tE_r{\psi }_r+{\psi }_e{\psi }_v{\psi }_rE_t{\psi }_t \\[4pt] &=\left(E_e+E_v+E_r{+E}_t\right){\psi }_e{\psi }_v{\psi }_r{\psi }_t \end{align*} \]We find that the energy of the molecule as a whole is simply the sum of the energies associated with the several kinds of motion\[E_{molecule}=E_e+E_v+E_r{+E}_t \nonumber \]\({\psi }_t\), \({\psi }_v\), \({\psi }_r\), and \({\psi }_e\) can be further approximated as products of wavefunctions involving still smaller numbers of coordinates. We can have a component wavefunction for every distinguishable coordinate that describes a possible motion of a portion of the molecule. The three translational modes are independent of one another. It is a good approximation to assume that they are also independent of the rotational and vibrational modes. Frequently, it is a good approximation to assume that the vibrational and rotational modes are independent of one another. We can deduce the number of one-dimensional wavefunctions that are required to give an approximate wavefunction that describes all of the molecular motions, because this will be the same as the number of coordinates required to describe the nuclear motions. If we have a collection of \(N\) atoms that are not bonded to one another, each atom is free to move in three dimensions. The number of coordinates required to describe their motion is \(3N\). When the same atoms are bonded to one another in a molecule, the total number of motions remains the same, but it becomes convenient to reorganize the way we describe them.First, we recognize that the atomic nuclei in a molecule occupy positions that are approximately fixed relative to one another. Therefore, to a good approximation, the motion of the center of mass is independent of the way that the atoms move relative to one another or relative to the center of mass. It takes three coordinates to describe the motion of the center of mass, so there are \(3N-3\) coordinates left over after this is done.The number of rotational motions available to a molecule depends upon the number of independent axes about which it can rotate. We can imagine a rotation of a molecule about any axis we choose. In general, in three dimensions, we can choose any three non-parallel axes and imagine that the molecule rotates about each of them independently of its rotation about the others. If we consider a set of more than three non-parallel axes, we find that any of the axes can be expressed as a combination of any three of the others. This means that the maximum number of independent rotational motions for the molecule as a whole is three.If the molecule is linear, we can take the molecular axis as one of the axes of rotation. Most conveniently, we can then choose the other two axes to be perpendicular to the molecular axis and perpendicular to each other. However, rotation about the molecular axis does not change anything about the molecule’s orientation in space. If the molecule is linear, rotation about the molecular axis is not a rotation at all! So, if the molecule is linear, only two coordinates are required to describe all of the rotational motions, and there are \(3N-5\) coordinates left over after we allocate those needed to describe the translational and rotational motions.The coordinates left over after we describe the translational and rotational motions must be used to describe the motion of the atoms with respect to one another. These motions are called vibrations, and hence the number of coordinates needed to describe the vibrations of a non-linear molecule is \(3N-6\). For a linear molecule, \(3N-5\) coordinates are needed to describe the vibrations.This page titled 18.4: The Schrödinger Equation for a Molecule is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.5: Solutions to Schroedinger Equations for Harmonic Oscillators and Rigid Rotors

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We can approximate the wavefunction for a molecule by partitioning it into wavefunctions for individual translational, rotational, vibrational, and electronic modes. The wavefunctions for each of these modes can be approximated by solutions to a Schrödinger equation that approximates that mode. Our objective in this chapter is to introduce the quantized energy levels that are found.Translational modes are approximated by the particle in a box model that we discuss above.Vibrational modes are approximated by the solutions of the Schrödinger equation for coupled harmonic oscillators. The vibrational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the vibration of two masses linked by a spring. Let the distance between the masses be \(r\) and the equilibrium distance be \(r_0\). Let the reduced mass of the molecule be \(\mu\), and let the force constant for the spring be \(\lambda\). From classical mechanics, the potential energy of the system is\[V\left(r\right)=\frac{\lambda {\left(r-r_0\right)}^2}{2} \nonumber \]and the vibrational frequency of the classical oscillator is \[\nu =\frac{1}{2\pi }\sqrt{\frac{\lambda }{\mu }} \nonumber \]The Schrödinger equation is\[-\left(\frac{h^2}{8{\pi }^2\mu }\right)\frac{d^2\psi }{dr^2}+\frac{\lambda {\left(r-r_0\right)}^2}{2}\psi =E\psi \nonumber \]The solutions to this equation are wavefunctions and energy levels that constitute the quantum mechanical description of the classical harmonic oscillator. The energy levels are given by\[E_n=h\nu \left(n+\frac{1}{2}\right) \nonumber \]where the quantum numbers, \(n\), can have any of the values \(n=0,\ 1,\ 2,\ 3,\ \dots .\) The lowest energy level, that for which \(n=0\), has a non-zero energy; that is,\[E_0={h\nu }/{2} \nonumber \]The quantum mechanical oscillator can have infinitely many energies, each of which is a half-integral multiple of the classical frequency, \(\nu\). Each quantum mechanical energy corresponds to a quantum mechanical frequency:\[{\nu }_n=\nu \left(n+\frac{1}{2}\right) \nonumber \]A classical rigid rotor consists of two masses that are connected by a weightless rigid rod. The rigid rotor is a dumbbell. The masses rotate about their center of mass. Each two-dimensional rotational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the motion of a rigid rotor in a plane. The simplest model assumes that the potential term is zero for all angles of rotation. Letting \(I\) be the molecule’s moment of inertia and \(\varphi\) be the rotation angle, the Schrödinger equation is\[-\left( \frac{h^2}{8\pi ^2I}\right) \frac{d^2\psi }{d \varphi ^2}=E\psi \nonumber \]The energy levels are given by\[E_m=\frac{m^2h^2}{8\pi ^2I} \nonumber \]where the quantum numbers, \(m\), can have any of the values \(m=1,\ 2,\ 3,\ \dots .,\)(but not zero). Each of these energy levels is two-fold degenerate. That is, two quantum mechanical states of the molecule have the energy \(E_m\).The three-dimensional rotational motion of a diatomic molecule is approximated by the solutions of the Schrödinger equation for the motion of a rigid rotor in three dimensions. Again, the simplest model assumes that the potential term is zero for all angles of rotation. Letting \(\theta\) and \(\varphi\) be the two rotation angles required to describe the orientation in three dimensions, the Schrödinger equation is\[-\frac{h^2}{8{\pi }^2I}\left(\frac{1}{\mathrm{sin} \theta} \frac{\partial }{\partial \theta } \left(\mathrm{sin} \theta \frac{\partial \psi }{\partial \theta }\right)+\frac{1}{\mathrm{sin}^2 \theta }\frac{d^2\psi }{d{\varphi }^2}\right)=E\psi \nonumber \]The energy levels are given by\[E_J=\frac{h^2}{8{\pi }^2I}J\left(J+1\right) \nonumber \]where the quantum numbers, \(J\), can have any of the values \(J=0,\ 1,\ 2\ ,3,\ \dots .\) \(E_J\) is \(\left(2J+1\right)\)-fold degenerate. That is, there are \(2J+1\) quantum mechanical states of the molecule all of which have the same energy, \(E_J\).Equations for the rotational energy levels of larger molecules are more complex.This page titled 18.5: Solutions to Schroedinger Equations for Harmonic Oscillators and Rigid Rotors is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.6: Wave Functions, Quantum States, Energy Levels, and Degeneracies

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We approximate the wavefunction for a molecule by using a product of approximate wavefunctions, each of which models some subset of the motions that the molecule undergoes. In general, the wavefunctions that satisfy the molecule’s Schrödinger equation are degenerate; that is, two or more of these wavefunctions have the same energy. (The one-dimensional particle in a box and the one-dimensional harmonic oscillator have non-degenerate solutions. The rigid-rotor in a plane has doubly degenerate solutions; two wavefunctions have the same energy. The \(J\)-th energy level of the three-dimensional rigid rotor is \(\left(2J+1\right)\)-fold degenerate; there are \(\left(2J+1\right)\) wavefunctions whose energy is \(E_J\).) We use doubly subscripted symbols to represent the wavefunctions that satisfy the molecule’s Schrödinger equation. We write \({\psi }_{i,j}\) to represent all of the molecular wavefunctions whose energy is \({\epsilon }_i\). We let \(g_i\) be the number of wavefunctions whose energy is \({\epsilon }_i\). We say that the energy level \({\epsilon }_i\) is \(g_i\)-fold degenerate. The wavefunctions\[{\psi }_{i,1},\ {\psi }_{i,2},\ \dots ,\ {\psi }_{i,j},\dots ,\ {\psi }_{i,g_i} \nonumber \]are all solution to the molecule’s Schrödinger equation; we have\[H_{molecule}{\psi }_{i,j}={\epsilon }_i{\psi }_{i,j} \nonumber \]for \(j=1,\ 2,\ \dots ,\ g_i\). Every energy level \({\epsilon }_i\) is associated with \(g_i\) quantum states. For simplicity, we can think of each of the \(g_i\) wavefunctions, \({\psi }_{i,j}\), as a quantum state; however, the molecule’s Schrödinger equation is also satisfied by any set of \(g_i\) independent linear combinations of the \({\psi }_{i,j}\). For present purposes, all that matters is that there are \(g_i\) quantum-mechanical descriptions—quantum states—all of which have energy \({\epsilon }_i\).This page titled 18.6: Wave Functions, Quantum States, Energy Levels, and Degeneracies is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

18.7: Particle Spins and Statistics- Bose-Einstein and Fermi-Dirac Statistics

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Our goal is to develop the theory of statistical thermodynamics from Boltzmann statistics. In this chapter, we explore the rudiments of quantum mechanics in order to become familiar with the idea that we can describe a series of discrete energy levels for any given molecule. For our purposes, that is all we need. We should note, however, that we are not developing the full story about the relationship between quantum mechanics and statistical thermodynamics. The spin of a particle is an important quantum mechanical property. It turns out that quantum mechanical solutions depend on the spin of the particle being described. Particles with integral spins behave differently from particles with half-integral spins. When we treat the statistical distribution of these particles, we need to treat particles with integral spins differently from particles with half-integral spins. Particles with integral spins are said to obey Bose-Einstein statistics; particles with half-integral spins obey Fermi-Dirac statistics.Fortunately, both of these treatments converge to the Boltzmann distribution if the number of quantum states available to the particles is much larger than the number of particles. For macroscopic systems at ordinary temperatures, this is the case. In Chapters 19 and 20, we introduce the ideas underlying the theory of statistical mechanics. In Chapter 21, we derive the Boltzmann distribution from a set of assumptions that does not correspond to either the Bose-Einstein or the Fermi-Dirac requirement. In Chapter 25, we derive the Bose-Einstein and Fermi-Dirac distributions and show how they become equivalent to the Boltzmann distribution for most systems of interest in chemistry.This page titled 18.7: Particle Spins and Statistics- Bose-Einstein and Fermi-Dirac Statistics is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

19.1: Distribution of Results for Multiple Trials with Two Possible Outcomes

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Suppose that we have two coins, one minted in 2001 and one minted in 2002. Let the probabilities of getting a head and a tail in a toss of the 2001 coin be \(P_{H,1}\) and \(P_{T,1}\), respectively. We assume that these outcomes exhaust the possibilities. From the laws of probability, we have: \(1=\left(P_{H,1}+P_{T,1}\right)\). For the 2002 coin, we have \(1=\left(P_{H,2}+P_{T,2}\right)\). The product of these two probabilities must also be unity. Expanding this product gives\[ \begin{align*} 1 &=\left(P_{H,1}+P_{T,1}\right)\left(P_{H,2}+P_{T,2}\right) \\[4pt] &=P_{H,1}P_{H,2}+P_{H,1}P_{T,2}+P_{T,1}P_{H,2}+P_{T,1}P_{T,2} \end{align*} \]This equation represents the probability of a trial in which we toss the 2001 coin first and the 2002 coin second. The individual terms are the probabilities of the possible outcomes of such a trial. It is convenient to give a name to this latter representation of the product; we will call it the expanded representation of the total probability sum.Our procedure for multiplying two binomials generates a sum of four terms. Each term contains two factors. The first factor comes from the first binomial; the second term comes from the second binomial. Each of the four terms corresponds to a combination of an outcome from tossing the 2001 coin and an outcome from tossing the 2002 coin. Conversely, every possible combination of outcomes from tossing the two coins is represented in the sum. \(P_{H,1}P_{H,2}\) represents the probability of getting a head from tossing the 2001 coin and a head from tossing the 2002 coin. \(P_{H,1}P_{T,2}\) represents the probability of getting a head from tossing the 2001 coin and a tail from tossing the 2002 coin, etc. In short, there is a one-to-one correspondence between the terms in this sum and the possible combinations of the outcomes of tossing these two coins.This analysis depends on our ability to tell the two coins apart. For this, the mint date is sufficient. If we toss the two coins simultaneously, the four possible outcomes remain the same. Moreover, if we distinguish the result of a first toss from the result of a second toss, etc., we can generate the same outcomes by using a single coin. If we use a single coin, we can represent the possible outcomes from two tosses by the ordered sequences \(HH\), \(HT\), \(TH\), and \(TT\), where the first symbol in each sequence is the result of the first toss and the second symbol is the result of the second toss. The ordered sequences \(HT\) and \(TH\) differ only in the order in which the symbols appear. We call such ordered sequences permutations.Now let us consider a new problem. Suppose that we have two coin-like slugs that we can tell apart because we have scratched a “\(1\)” onto the surface of one and a “\(2\)” onto the surface of the other. Suppose that we also have two cups, one marked “\(H\)” and the other marked “\(T\).” We want to figure out how many different ways we can put the two slugs into the two cups. We can also describe this as the problem of finding the number of ways we can assign two distinguishable slugs (objects) to two different cups (categories). There are four such ways: Cup \(H\) contains slugs \(1\) and \(2\); Cup \(H\) contains slug \(1\) and Cup \(T\) contains slug \(2\); Cup \(H\) contains slug \(2\) and Cup \(T\) contains slug \(1\); Cup \(T\) contains slugs \(1\) and\(\ 2\).We note that, given all of the ordered sequences for tossing two coins, we can immediately generate all of the ways that two distinguishable objects (numbered slugs) can be assigned to two categories (Cups \(H\) and \(T\)). For each ordered sequence, we assign the first object to the category corresponding to the first symbol in the sequence, and we assign the second object to the category corresponding to the second symbol in the sequence.In short, there are one-to-one correspondences between the sequences of probability factors in the total probability sum, the possible outcomes from tossing two distinguishable coins, the possible sequences of outcomes from two tosses of a single coin, and the number of ways we can assign two distinguishable objects to two categories. (See Table 1.)If the probability of tossing a head is constant, we have \(P_{H,1}=P_{H,2}=P_H\) and \(P_{T,1}=P_{T,2}=P_T\). Note that we are not assuming \(P_H=P_T\). If we do not care about the order in which the heads and tails appear, we can simplify our equation for the product of probabilities to\[1=P^2_H+2P_HP_T+P^2_T \nonumber \]\(P^2_H\) is the probability of tossing two heads, \(P_HP_T\) is the probability of tossing one head and one tail, and \(P^2_T\) is the probability of tossing two tails. We must multiply the \(P_HP_T\)-term by two, because there are two two-coin outcomes and correspondingly two combinations, \(P_{H,1}P_{T,2}\) and \(P_{T,1}P_{H,2}\), that have the same probability, \(P_HP_T\). Completely equivalently, we can say that the reason for multiplying the \(P_HP_T\)-term by two is that there are two permutations, \(HT\) and \(TH\), which correspond to one head and one tail in successive tosses of a single coin.We have lavished considerable attention on four related but very simple problems. Now, we want to extend this analysis—first to tosses of multiple coins and then to situations in which multiple outcomes are possible for each of many independent events. Eventually we will find that understanding these problems enables us to build a model for the behavior of molecules that explains the observations of classical thermodynamics.If we extend our analysis to tossing \(n\) coins, which we label coins \(1\), \(2\), etc., we find:\[ \begin{align*} 1 &=\left(P_{H,1}+P_{T,1}\right)\left(P_{H,2}+P_{T,2}\right)\dots \left(P_{H,n}+P_{T,n}\right) \\[4pt] &=\left(P_{H,1}P_{H,2}\dots P_{H,n}\right)+\left(P_{H,1}P_{H,2}{\dots P}_{H,i}\dots P_{T,n}\right)+\dots +\left(P_{T,1}P_{T,2}\dots P_{T,i}\dots P_{T,n}\right) \end{align*} \]We write each of the product terms in this expanded representation of the total-probability sum with the second index, \(r\), increasing from \(1\) to \(n\) as we read through the factors, \(P_{X,r}\), from left to right. Just as for tossing only two coins:In Section 3.9, we introduce the term population set to denote a set of numbers that represents a possible combination of outcomes. Here the possible combinations of outcomes are the numbers of heads and tails. If in five tosses we obtain \(3\) heads and \(2\) tails, we say that this group of outcomes belongs to the population set \(\{3,2\}\). If in \(n\) tosses, we obtain \(n_H\) heads and \(n_T\) tails, this group of outcomes belongs to the population set \(\{n_H,n_T\}\). For five tosses, the possible population sets are \(\left\{5,0\right\}\), \(\left\{4,1\right\}\), \(\left\{3,2\right\}\), \(\left\{2,3\right\}\), \(\left\{1,4\right\}\), and \(\left\{5,0\right\}\). Beginning in the next chapter, we focus on the energy levels that are available to a set of particles and on the number of particles that has each of the available energies. Then the number of particles, \(N_i\), that have energy \({\epsilon }_i\) is the population of the \({\epsilon }_i\)-energy level. The set of all such numbers is the energy-level population set for the set of particles.If we cannot distinguish one coin from another, the sequence \(P_{H,1}P_{T,2}P_{H,3}P_{H,4}\) becomes \(P_HP_TP_HP_H\). We say that \(P_HP_TP_HP_H\) is distinguishable from \(P_HP_HP_TP_H\) because the tails-outcome appears in the second position in \(P_HP_TP_HP_H\) and in the third position in \(P_HP_HP_TP_H\). We say that \(P_{H,1}P_{T,2}P_{H,3}P_{H,4}\) and \(P_{H,3}P_{T,2}P_{H,1}P_{H,4}\)are indistinguishable, because both become \(P_HP_TP_HP_H\). In general, many terms in the expanded form of the total probability sum belong to the population set corresponding to \(n_H\) heads and \(n_T\) tails. Each such term corresponds to a distinguishable permutation of \(n_H\) heads and \(n_T\) tails and the corresponding distinguishable permutation of \(P_H\) and \(P_T\) terms.We use the notation \(C\left(n_H,n_T\right)\) to denote the number of terms in the expanded form of the total probability sum in which there are \(n_H\) heads and \(n_T\) tails. \(C\left(n_H,n_T\right)\) is also the number of distinguishable permutations of \(n_H\) heads and \(n_T\) tails or of \(n_H\) P\({}_{H}\)-terms and \(n_T\) P\({}_{T}\)-terms. The principal goal of our analysis is to find a general formula for \(C\left(n_H,n_T\right)\). To do so, we make use of the fact that \(C\left(n_H,n_T\right)\) is also the number of ways that we can assign \(n\) objects (coins) to two categories (heads or tails) in such a way that \(n_H\) objects are in one category (heads) and \(n_T\) objects are in the other category (tails). We also call \(C\left(n_H,n_T\right)\) the number of combinations possible for distinguishable coins in the population set \(\{n_H,n_T\}\).The importance of \(C\left(n_H,n_T\right)\) is evident when we recognize that, if we do not care about the sequence (permutation) in which a particular number of heads and tails occurs, we can represent the total-probability sum in a much compressed form:\[1=P^n_H+nP^{n-1}_HP_T+\dots +C\left(n_H,n_T\right)P^{n_H}_HP^{n_T}_T+nP_HP^{n-1}_T+P^n_T \nonumber \]In this representation, there are \(n\) terms in the total-probability sum that have \(n_H=n-1\) and \(n_T=1\). These are the terms\[P_{H,1}P_{H,2}P_{H,3}{\dots P}_{H,i}\dots P_{H,n-1}{\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{n}} \nonumber \] \[P_{H,1}P_{H,2}P_{H,3}{\dots P}_{H,i}\dots {\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{n}\boldsymbol{-}\boldsymbol{1}}P_{H,n} \nonumber \] \[P_{H,1}P_{H,2}P_{H,3}\dots {\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{i}}\dots P_{H,n-1}P_{H,n} \nonumber \]…\[P_{H,1}P_{H,2}{\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{3}}{\dots P}_{H,i}\dots P_{H,n-1}P_{H,n} \nonumber \] \[P_{H,1}{\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{2}}P_{H,3}{\dots P}_{H,i}\dots P_{H,n-1}P_{H,n} \nonumber \] \[{\boldsymbol{P}}_{\boldsymbol{T},\boldsymbol{1}}P_{H,2}P_{H,3}{\dots P}_{H,i}\dots P_{H,n-1}P_{H,n} \nonumber \]Each of these terms represents the probability that \(n-1\) heads and one tail will occur in the order shown. Each of these terms has the same value. Each of these terms is a distinguishable permutation of \(n-1\) \(P_H\) terms and one \(P_T\) term. Each of these terms corresponds to a combination in which one of n numbered slugs is assigned to Cup \(T\), while the remaining \(n-1\) numbered slugs are assigned to Cup \(H\). It is easy to see that there are \(n\) such terms, because each term is the product of \(n\) probabilities, and the tail can occur at any of the \(n\) positions in the product. If we do not care about the order in which heads and tails occur and are interested only in the value of the sum of these \(n\) terms, we can replace these \(n\) terms by the one term \(nP^{n-1}_HP_T\). We see that \(nP^{n-1}_HP_T\) is the probability of tossing \(n-1\) heads and one tail, irrespective of which toss produces the tail.There is another way to show that there must be \(n\) terms in the total-probability sum in which there are \(n-1\) heads and one tail. This method relies on the fact that the number of such terms is the same as the number of combinations in which n distinguishable things are assigned to two categories, with \(n-1\) of the things in one category and the remaining thing in the other category, \(C\left(n-1,1\right)\). This method is a little more complicated, but it offers the great advantage that it can be generalized.The new method requires that we think about all of the permutations we can create by reordering the results from any particular series of \(n\) tosses. To see what we have in mind when we say all of the permutations, let \(P_{X,k}\) represent the probability of toss number \(k\), where for the moment we do not care whether the outcome was a head or a tail. When we say all of the permutations, we mean the number of different ways we can order (permute) n different values \(P_{X,k}\). It is important to recognize that one and only one of these permutations is a term in the total-probability sum, specifically:\[P_{X,1}P_{X,2}P_{X,3}\dots P_{X,k}\dots P_{X,n} \nonumber \]in which the values of the second subscript are in numerical order. When we set out to construct all of these permutations, we see that there are \(n\) ways to choose the toss to put first and \(n-1\) ways to choose the toss to put second, so there are \(n\left(n-1\right)\) ways to choose the first two tosses. There are \(n-2\) ways to choose the third toss, so there are \(n\left(n-1\right)\left(n-2\right)\) ways to choose the first three tosses. Continuing in this way through all \(n\) tosses, we see that the total number of ways to order the results of n tosses is \(n\left(n-1\right)\left(n-2\right)\left(n-3\right)\dots \left(3\right)\left(2\right)\left(1\right)=n!\)Next, we need to think about the number of ways we can permute \(n\) values \(P_{X,k}\) if \(n-1\) of them are \(P_{H,1}\), \(P_{H,2}\),…, \(P_{H,r-1},\)…, \(P_{H,r+1},\dots ,P_{H,n}\) and one of them is \(P_{T,r}\), and we always keep the one factor \(P_{T,r}\) in the same position. By the argument above, there are \(\left(n-1\right)!\) ways to permute the values \(P_{H,s}\) in a set containing \(n-1\) members. So for every term (product of factors \(P_{X,k}\)) that occurs in the total-probability sum, there are \(\left(n-1\right)!\) other products (other permutations of the same factors) that differ only in the order in which the \(P_{H,s}\) appear. The single tail outcome occupies the same position in each of these permutations. If the \(r^{th}\) factor in the term in the total probability sum is \(P_{T,r}\), then \(P_{T,r}\) is the \(r^{th}\) factor in each of the \(\left(n-1\right)!\) permutations of this term. This is an important point, let us repeat it in slightly different words: For every term that occurs in the total-probability sum, there are \(\left(n-1\right)!\) permutations of the same factors that leave the heads positions occupied by heads and the tails position occupied by tails.Equivalently, for every assignment of \(n-1\) distinguishable objects to one of two categories, there are \(\left(n-1\right)!\) permutations of these objects. There are \(C\left(n-1,1\right)\) such assignments. Accordingly, there are a total of \(\left(n-1\right)!C\left(n-1,1\right)\) permutations of the \(n\) distinguishable objects. Since we also know that the total number of permutations of n distinguishable objects is \(n!\), we have\[n!=\left(n-1\right)!C\left(n-1,1\right) \nonumber \]so that \[C\left(n-1,1\right)=\frac{n!}{\left(n-1\right)!} \nonumber \]which is the same result that we obtained by our first and more obvious method.The distinguishable objects within a category in a particular assignment can be permuted. We give these within-category permutations another name; we call them indistinguishable permutations. (This terminology reflects our intended application, which is to find the number of ways \(n\) identical molecules can be assigned to a set of energy levels. We can tell two isolated molecules of the same substance apart only if they have different energies. We can distinguish molecules in different energy levels from one another. We cannot distinguish two molecules in the same energy level from one another. Two different permutations of the molecules within any one energy level are indistinguishable from one another.) For every term in the expanded representation of the total probability sum, indistinguishable permutations can be obtained by exchanging \(P_H\) factors with one another, or by exchanging \(P_T\) factors with one another, but not by exchanging \(P_H\) factors with \(P_T\) factors. That is, heads are exchanged with heads; tails are exchanged with tails; but heads are not exchanged with tails.Now we can consider the general case. We let \(C\left(n_H,n_T\right)\) be the number of terms in the total-probability sum in which there are \(n_H\) heads and \(n_T\) tails. We want to find the value of \(C\left(n_H,n_T\right)\). Let’s suppose that one of the terms with \(n_H\) heads and \(n_T\) tails is\[\left(P_{H,a}P_{H,b}\dots P_{H,m}\right)\left(P_{T,r}P_{T,s}\dots P_{T,z}\right) \nonumber \]where there are \(n_H\) indices in the set \(\{a,\ b,\ \dots ,m\}\) and \(n_T\) indices in the set \(\{r,s,\dots ,z\}\). There are \(n_H!\) ways to order the heads outcomes and \(n_T!\) ways to order the tails outcomes. So, there are \(n_H!n_T!\) possible ways to order \(n_H\) heads and \(n_T\) tails outcomes. This is true for any sequence in which there are \(n_H\) heads and \(n_T\) tails; there will always be \(n_H!n_T!\) permutations of \(n_H\) heads and \(n_T\) tails, whatever the order in which the heads and tails appear. This is also true for every term in the total-probability sum that contains \(n_H\) heads factors and \(n_T\) tails factors. The number of such terms is \(C\left(n_H,n_T\right)\). For every such term, there are \(n_H!n_T!\) permutations of the same factors that leave the heads positions occupied by heads and the tails positions occupied by tails.Accordingly, there are a total of \(n_H!n_T!C\left(n_H,n_T\right)\) permutations of the \(n\) distinguishable objects. The total number of permutations of n distinguishable objects is \(n!\), so that\[n!=n_H!n_T!C\left(n_H,n_T\right) \nonumber \]and\[C\left(n_H,n_T\right)=\frac{n!}{n_H!n_T!} \nonumber \]Equivalently, we can construct a sum of terms, \(R\), in which the terms are all of the \(n!\) permutations of \(P_{H,i}\) factors for \(n_H\) heads and \(P_{T,j}\) factors for \(n_T\) tails. The value of each term in \(R\) is \(P^{n_H}_HP^{n_T}_T\). So we have\[R=n!P^{n_H}_HP^{n_T}_T \nonumber \]\(R\) contains all \(C\left(n_H,n_T\right)\) of the \(P^{n_H}_HP^{n_T}_T\)-valued terms that appear in the total-probability sum. For each of these \(P^{n_H}_HP^{n_T}_T\)-valued terms there are \(n_H!n_T!\) indistinguishable permutations that leave heads positions occupied by heads and tails positions occupied by tails. \(R\) will also contain all of the \(n_H!n_T!\) permutations of each of these \(P^{n_H}_HP^{n_T}_T\)-valued terms. That is, every term in \(R\) is either a term in the expanded representation of the total probability sum or an indistinguishable permutation of such a term. It follows that \(R\) is also given by\[R=n_H!n_T!C\left(n_H,n_T\right)P^{n_H}_HP^{n_T}_T \nonumber \]Equating these equations for R, we have\[n!P^{n_H}_HP^{n_T}_T=n_H!n_T!C\left(n_H,n_T\right)P^{n_H}_HP^{n_T}_T \nonumber \]and, again,\[C\left(n_H,n_T\right)=\frac{n!}{n_H!n_T!} \nonumber \]In summary: The total number of permutations is \(n!\) The number of combinations of \(n\) distinguishable things in which \(n_H\) of them are assigned to category \(H\) and \(n_T=n-n_H\) are assigned to category \(T\) is \(C\left(n_H,n_T\right)\). (Every combination is a distinguishable permutation.) The number of indistinguishable permutations of the objects in each such combination is \(n_H!n_T!\). The relationship among these quantities istotal number of permutations = (number of distinguishable combinations)\({}_{\ }\)\({}_{\times }\) (number of indistinguishable permutations for each distinguishable combination)We noted earlier that \(C\left(n_H,n_T\right)\) is the formula for the binomial coefficients. If we do not care about the order in which the heads and tails arise, the probability of tossing \(n_T\) tails and \(n_H=n-n_T\) heads is\[C\left(n_H,n_T\right)P^{n_H}_HP^{n_T}_T=\left(\frac{n!}{n_H!n_T!}\right)P^{n_H}_HP^{n_T}_T \nonumber \]and the sum of such terms for all \(n+1\) possible values of \(n_T\) in the interval \(0\le n_T\le n\) is the total probability for all possible outcomes from \(n\) tosses of a coin. This total probability must be unity. That is, we have\[1={\left(P_H+P_T\right)}^n=\sum^n_{n_T=0}{C\left(n_H,n_T\right)P^{n_H}_HP^{n_T}_T}=\sum^n_{n_T=0}{\left(\frac{n!}{n_H!n_T!}\right)P^{n_H}_HP^{n_T}_T} \nonumber \]For an unbiased coin, \(P_H=P_T={1}/{2}\), and \(P^{n_H}_HP^{n_T}_T={\left({1}/{2}\right)}^n\), for all \(n_T\). This means that the probability of tossing \(n_H\) heads and \(n_T\) tails is proportional to \(C\left(n_H,n_T\right)\) where the proportionality constant is \({\left({1}/{2}\right)}^n\). The probability of \(n^{\blacksquare }\) heads and \(n-n^{\blacksquare }\) tails is the same as the probability of \(n-n^{\blacksquare }\) heads and \(n^{\blacksquare }\) tails.Nothing in our development of the equation for the total probability requires that we set \(P_H=P_T\), and in fact, the binomial probability relationship applies to any situation in which there are repeated trials, where each trial has two possible outcomes, and where the probability of each outcome is constant. If \(P_H\neq P_T\), the symmetry observed for tossing coins does not apply, because\[P^{n-n^{\blacksquare }}_HP^{n^{\blacksquare }}_T\neq P^{n^{\blacksquare }}_HP^{n-n^{\blacksquare }}_T \nonumber \]This condition corresponds to a biased coin.Another example is provided by a spinner mounted at the center of a circle painted on a horizontal surface. Suppose that a pie-shaped section accounting for \(25\%\) of the circle’s area is painted white and the rest is painted black. If the spinner’s stopping point is unbiased, it will stop in the white zone with probability \(P_W=0.25\) and in the black zone with probability \(P_B=0.75\). After \(n\) spins, the probability of \(n_W\) white outcomes and \(n_B\) black outcomes is\[\left(\frac{n!}{n_W!n_B!}\right){\left(0.25\right)}^{n_W}{\left(0.75\right)}^{n_B} \nonumber \]After \(n\) spins, the sum of the probabilities for all possible combinations of white and black outcomes is\[\begin{align*} 1 &={\left(P_W+P_B\right)}^n=\sum^n_{n_B=0}{C\left(n_W,n_B\right)P^{n_W}_WP^{n_B}_B} \\[4pt] &=\sum^n_{n_B=0}{\left(\frac{n!}{n_W!n_B!}\right)P^{n_W}_WP^{n_B}_B} \\[4pt] &=\sum^n_{n_{B=0}}{\left(\frac{n!}{n_W!n_B!}\right){\left(0.25\right)}^{n_W}{\left(0.75\right)}^{n_B}} \end{align*} \]This page titled 19.1: Distribution of Results for Multiple Trials with Two Possible Outcomes is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

19.2: Distribution of Results for Multiple Trials with Three Possible Outcomes

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Let us extend the ideas we have developed for binomial probabilities to the case where there are three possible outcomes for any given trial. To be specific, suppose we have a coin-sized object in the shape of a truncated right-circular cone, whose circular faces are parallel to each other. The circular faces have different diameters. When we toss such an object, allowing it to land on a smooth hard surface, it can wind up resting on the big circular face (\(\boldsymbol{H}\)eads), the small circular face (\(\boldsymbol{T}\)ails), or on the conical surface (\(\boldsymbol{C}\)one-side). Let the probabilities of these outcomes in a single toss be \(P_H\), \(P_T\), and \(P_C\), respectively. In general, we expect these probabilities to be different from one another; although, of course, we require \(1=\left(P_H+P_T+P_C\right)\).Following our development for the binomial case, we want to write an equation for the total probability sum after \(n\) tosses. Let \(n_H\), \(n_T\), and \(n_C\) be the number of \(H\), \(T\), and \(C\) outcomes exhibited in \(n_H+n_T+n_C=n\) trials. We let the probability coefficients be \(C\left(n_H,n_T,n_C\right)\). The probability of \(n_H\), \(n_T\), \(n_C\) outcomes in \(n\) trials is\[C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]and the total probability is\[1={\left(P_H+P_T+P_C\right)}^n=\sum_{n_H,n_T,n_C}{C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C} \nonumber \]where the summation is to be carried out over all combinations of integer values for \(n_H\), \(n_T\), and \(n_C\), consistent with \(n_H+n_T+n_C=n\).To find \(C\left(n_H,n_T,n_C\right)\), we proceed as before. We suppose that one of the terms with \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides is\[\left(P_{H,a}P_{H,b}\dots P_{H,f}\right)\left(P_{T,g}P_{T,h}\dots P_{T,m}\right)\left(P_{C,p}P_{C,q}\dots P_{C,z}\right) \nonumber \]where there are \(n_H\) indices in the set \(\{a,\ b,\ \dots ,\ f\}\), \(n_T\) indices in the set \(\{g,\ h,\ \dots ,\ m\}\), and \(n_C\) indices in the set \(\mathrm{\{}\)p, q,…, z\(\mathrm{\}}\). There are \(n_H!\) ways to order the heads outcomes, \(n_T!\) ways to order the tails outcomes, and \(n_C!\) ways to order the cone-sides outcomes. So, there are \(n_H!n_T!n_C!\) possible ways to order \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. There will also be \(n_H!n_T!n_C!\) indistinguishable permutations of any combination (particular assignment) of \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. There are \(n!\) possible permutations of \(n\) probability factors and \(C\left(n_H,n_T,n_C\right)\) distinguishable combinations with \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. As before, we havetotal number of permutations = (number of distinguishable combinations)\({}_{\ }\)\({}_{\times }\) (number of indistinguishable permutations for each distinguishable combination)so that\[n!=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right) \nonumber \]and hence, \[C\left(n_H,n_T,n_C\right)=\frac{n!}{n_H!n_T!n_C!} \nonumber \]Equivalently, we can construct a sum of terms, \(S\), in which the terms are all of the \(n!\) permutations of \(P_{H,r}\) factors for \(n_H\) heads, \(P_{T,s}\) factors for \(n_T\) tails, and \(P_{C,t}\) factors for \(n_C\) cone-sides. The value of each term in \(S\) will be \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\). Thus, we have\[S=n!P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]\(S\) will contain all \(C\left(n_H,n_T,n_C\right)\) of the distinguishable combinations \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides outcomes that give rise to \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms. Moreover, \(S\) will also include all of the \(n_H!n_T!n_C!\) indistinguishable permutations of each of these \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms, and we also have\[S=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]Equating these two expressions for S gives us the number of \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms in the total-probability product,\(\ C\left(n_H,n_T,n_C\right)\). That is,\[S=n!P^{n_H}_HP^{n_T}_TP^{n_C}_C=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]and, again, \[C\left(n_H,n_T,n_C\right)=\frac{n!}{n_H!n_T!n_C!} \nonumber \]In the special case that \(P_H=P_T=P_C={1}/{3}\), all of the products \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\) will have the value \({\left({1}/{3}\right)}^n\). Then the probability of any set of outcomes, \(\{n_H,n_T,n_C,\}\), is proportional to \(C\left(n_H,n_T,n_C\right)\) with the proportionality constant \({\left({1}/{3}\right)}^n\).This page titled 19.2: Distribution of Results for Multiple Trials with Three Possible Outcomes is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

19.3: Distribution of Results for Multiple Trials with Many Possible Outcomes

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It is now easy to extend our results to multiple trials with any number of outcomes. Let the outcomes be \(A\), \(B\), \(C\), …., \(Z\), for which the probabilities in a single trial are \(P_A\), \(P_B\), \(P_C\),…\(P_Z\). We again want to write an equation for the total probability after \(n\) trials. We let \(n_A\), \(n_B\), \(n_C\),…\(n_Z\) be the number of \(A\), \(B\), \(C\),…, \(Z\) outcomes exhibited in \(n_A+n_B+n_C+...+n_Z=n\) trials. If we do not care about the order in which the outcomes are obtained, the probability of \(n_A\), \(n_B\), \(n_C\),…, \(n_Z\) outcomes in \(n\) trials is\[C\left(n_A,n_B,n_C,\dots ,n_Z\right)P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z \nonumber \]and the total probability sum is\[1={\left(P_A+P_B+P_C+\dots +P_Z\right)}^n=\sum_{n_I}{C\left(n_A,n_B,n_C,\dots ,n_Z\right)P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z} \nonumber \]where the summation is to be carried out over all combinations of integer values for \(n_A\), \(n_B\), \(n_C\),…, \(n_Z\) consistent with \(n_A+n_B+n_C+...+n_Z=n\).Let one of the terms for \(n_A\) \(A\)-outcomes, \(n_B\) \(B\)-outcomes, \(n_C\) \(C\)-outcomes, …, \(n_Z\)\({}_{\ }\)\(Z\)-outcomes, be\[\left(P_{A,a}P_{A,b}\dots P_{A,f}\right)\left(P_{B,g}P_{B,h}\dots P_{B,m}\right)\times \left(P_{C,p}P_{C,q}\dots P_{C,t}\right)\dots \left(P_{Z,u}P_{Z,v}\dots P_{Z,z}\right) \nonumber \]where there are \(n_A\) indices in the set \(\{a,\ b,\ \dots ,\ f\}\), \(n_B\) indices in the set \(\{g,\ h,\ \dots ,\ m\}\), \(n_C\) indices in the set \(\{p,\ q,\ \dots ,\ t\}\), …, and \(n_Z\) indices in the set \(\{u,\ v,\ \dots ,\ z\}\). There are \(n_A!\) ways to order the \(A\)-outcomes, \(n_B!\) ways to order the \(B\)-outcomes, \(n_C!\) ways to order the \(C\)-outcomes, …, and \(n_Z!\) ways to order the \(Z\)-outcomes. So, there are \(n_A!n_B!n_C!\dots n_Z!\) ways to order \(n_A\) \(A\)-outcomes, \(n_B\) \(B\)-outcomes, \(n_C\) \(C\)-outcomes, …, and \(n_Z\) \(Z\)-outcomes. The same is true for any other distinguishable combination; for every distinguishable combination belonging to the population set \(\{n_A\), \(n_B\), \(n_C\),…, \(n_Z\}\) there are \(n_A!n_B!n_C!\dots n_Z!\) indistinguishable permutations. Again, we can express this result as the general relationship:total number of permutations = (number of distinguishable combinations)\({}_{\ }\)\({}_{\times }\) (number of indistinguishable permutations for each distinguishable combination)so that\[n!=n_A!n_B!n_C!\dots n_Z!C\left(n_A,n_B,n_C,\dots ,n_Z\right) \nonumber \]and \[C\left(n_A,n_B,n_C,\dots ,n_Z\right)=\frac{n!}{n_A!n_B!n_C!\dots n_Z!} \nonumber \]Equivalently, we can construct a sum, \(T\), in which we add up all of the \(n!\) permutations of \(P_{A,a}\) factors for \(n_A\) \(A\)-outcomes, \(P_{B,b}\) factors for \(n_B\) \(B\)-outcomes, \(P_{C,c}\) factors for \(n_C\) \(C\)-outcomes, …, and \(P_{Z,z}\) factors for \(n_Z\) \(Z\)-outcomes. The value of each term in \(T\) will be \(P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z\). So we have\[T=n!P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z \nonumber \]\(T\) will contain all \(C\left(n_A,n_B,n_C,\dots ,n_Z\right)\) of the \(P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z\)-valued products (distinguishable combinations) that are a part of the total-probability sum. Moreover, \(T\) will also include all of the \(n_A!n_B!n_C!\dots n_Z!\) indistinguishable permutations of each of these \(P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z\)-valued products. Then we also have\[T=n_A!n_B!n_C!\dots n_Z!C\left(n_A,n_B,n_C,\dots ,n_Z\right) \nonumber \] \[\times P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z \nonumber \]Equating these two expressions for\(\ T\) gives us the number of \(P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z\)-valued products\[n!P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z=n_A!n_B!n_C!\dots n_Z! \nonumber \] \[\times C\left(n_A,n_B,n_C,\dots ,n_Z\right)P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z \nonumber \]and hence,\[C\left(n_A,n_B,n_C,\dots ,n_Z\right)=\frac{n!}{n_A!n_B!n_C!\dots n_Z!} \nonumber \]In the special case that \(P_A=P_B=P_C=\dots =P_Z\), all of the products \(P^{n_A}_AP^{n_B}_BP^{n_C}_C\dots P^{n_Z}_Z\) have the same value. Then, the probability of any set of outcomes, \(\{n_A,n_B,n_C,\dots ,n_Z\}\), is proportional to \(C\left(n_A,n_B,n_C,\dots ,n_Z\right)\).This page titled 19.3: Distribution of Results for Multiple Trials with Many Possible Outcomes is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

19.4: Stirling's Approximation

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The polynomial coefficient, \(C\), is a function of the factorials of large numbers. Since \(N!\) quickly becomes very large as \(N\) increases, it is often impractical to evaluate \(N!\) from the definition,\[N!=\left(N\right)\left(N-1\right)\left(N-2\right)\dots \left(3\right)\left(2\right)\left(1\right) \nonumber \]Fortunately, an approximation, known as Stirling’s formula or Stirling’s approximation is available. Stirling’s approximation is a product of factors. Depending on the application and the required accuracy, one or two of these factors can often be taken as unity. Stirling’s approximation is\[N!\approx N^N \left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right)\mathrm{exp}\left(\frac{1}{12N}\right)\approx N^N\left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right)\approx N^N\mathrm{exp}\left(-N\right) \nonumber \]In many statistical thermodynamic arguments, the important quantity is the natural logarithm of \(N!\) or its derivative, \({d ~ { \ln N!\ }}/{dN}\). In such cases, the last version of Stirling’s approximation is usually adequate, even though it affords a rather poor approximation for \(N!\) itself.This page titled 19.4: Stirling's Approximation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

19.5: Problems

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1. Leland got a train set for Christmas. It came with seven rail cars. (We say that all seven cars are “distinguishable.”) Four of the rail cars are box cars and three are tank cars. If we distinguish between permutations in which the box cars are coupled (lined up) differently but not between permutations in which tank cars are coupled differently, how many ways can the seven cars be coupled so that all of the tank cars are together? What are they? What formula can we use to compute this number?(Hint: We can represent one of the possibilities as \(b_1b_2b_3b_4T\). This is one of the possibilities in which the first four cars behind the engine are all box cars. There are \(4!\) such possibilities; that is, there are \(4!\) possible permutations for placing the four box cars.)2. If we don’t care about the order in which the box cars are coupled, and we don’t care about the order in which the tank cars are coupled, how many ways can the rail cars in problem 1 be coupled so that all of the tank cars are together? What are they? What formula can we use to compute this number?3. If we distinguish between permutations in which either the box cars or the tank cars in problem 1 are ordered differently, how many ways can the rail cars be coupled so that all of the tank cars are together? What formula can we use to compute this number?4. How many ways can all seven rail cars in problem 1 be coupled if the tank cars need not be together?5. If, as in the previous problem, we distinguish between permutations in which any of the rail cars are ordered differently, how many ways can the rail cars be coupled so that not all of the tank cars are together?6. If we distinguish between box cars and tank cars, but we do not distinguish one box car from another box car, and we do not distinguish one tank car from another tank car, how many ways can the rail cars in problem 1 be coupled?7. If Leland gets five flat cars for his birthday, he will have four box cars, three tank cars and five flat cars. How many ways will Leland be able to couple (permute) these twelve rail cars?8. If we distinguish between box cars and tank cars, between box cars and flat cars, and between tank cars and flat cars, but we do not distinguish one box car from another box car, and we do not distinguish one tank car from another tank car, and we do not distinguish one flat car from another flat car, how many ways can the rail cars in problem seven be coupled? What formula can we use to compute this number?9. We are given four distinguishable marbles, labeled \(A--D\), and two cups, labeled \(1\) and \(2\). We want to explore the number of ways we can put two marbles in cup \(1\) and two marbles in cup \(2\). This is the number of combinations, \(C\left(2,2\right)\), for the population set \(N_1=2\), \(N_2=2\).(a) One combination is \({\left[AB\right]}_1{\left[CD\right]}_2\). Find the remaining combinations. What is \(C\left(2,2\right)\)?(b) There are four permutations for the combination given in (a):\(\ {\left[AB\right]}_1{\left[CD\right]}_2\); \({\left[BA\right]}_1{\left[CD\right]}_2\); \({\left[AB\right]}_1{\left[DC\right]}_2\); \({\left[BA\right]}_1{\left[DC\right]}_2\). Find all of the permutations for each of the remaining combinations.(c) How many permutations are there for each combination?(d) Write down all of the possible permutations of marbles \(A--D\). Show that there is a one-to-one correspondence with the permutations in (b).(e) Show that the total number of permutations is equal to the number of combinations times the number of permutations possible for each combination.10. We are given seven distinguishable marbles, labeled \(A--G\), and two cups, labeled \(1\) and \(2\). We want to find the number of ways we can put three marbles in cup \(1\) and four marbles in cup\(\ 2\). That is, we seek \(C\left(3,4\right)\), the number of combinations in which \(N_1=3\) and \(N_2=4\). \({\left[ABC\right]}_1{\left[DEFG\right]}_2\) is one such combination.(a) How many different ways can these marbles be placed in different orders without exchanging any marbles between cup \(1\) and cup \(2\)? (This is the number of permutations associated with this combination.)(b) Find a different combination with \(N_1=3\) and \(N_2=4\).(c) How many permutations are possible for the marbles in (b)? How many permutations are possible for any combination with \(N_1=3\) and \(N_2=4\)?(d) If \(C\left(3,4\right)\) is the number of combinations in which \(N_1=3\) and \(N_2=4\), and if \(P\) is the number of permutations for each such combination, what is the total number of permutations possible for 7 marbles?(e) How else can one express the number of permutations possible for 7 marbles?(f) Equate your conclusions in (d) and (e). Find \(C\left(3,4\right)\).11.(a) Calculate the probabilities of 0, 1, 2, 3, and 4 heads in a series of four tosses of an unbiased coin. The event of 2 heads is\(\ 20\%\) of these five events. Note particularly the probability of the event: 2 heads in 4 tosses.(b) Calculate the probabilities of 0, 1, 2, 3,…, 8, and 9 heads in a series of nine tosses of an unbiased coin. The events of 4 heads and 5 heads comprise \(20\%\) of these ten cases. Calculate the probability of 4 heads or 5 heads; i.e., the probability of being in the middle \(20\%\) of the possible events.(c) Calculate the probabilities of 0, 1, 2, 3,…, 13, and 14 heads in a series of fourteen tosses of an unbiased coin. The events of 6 heads, 7 heads, and 8 heads comprise 20% of these fifteen cases. Calculate the probability of 6, 7, or 8 heads; i.e., the probability of being in the middle \(20\%\) of the possible events.(d) What happens to the probabilities for the middle \(20\%\) of possible events as the number of tosses becomes very large? How does this relate to the fraction heads in a series of tosses when the total number of tosses becomes very large?12. Let the value of the outcome heads be one and the value of the outcome tails be zero. Let the “score” from a particular simultaneous toss of \(n\) coins be\[\mathrm{score}=1\times \left(\frac{number\ of\ heads}{number\ of\ coins}\right)\ +0\times \left(\frac{number\ of\ tails}{number\ of\ coins}\right) \nonumber \]Let us refer to the distribution of scores from tosses of \(n\) coins as the “\(S_n\) distribution.”(a) The \(S_1\) distribution comprises two outcomes: \(\mathrm{\{}\)1 head, 0 tail\(\mathrm{\}}\) and \(\mathrm{\{}\)0 head, 1 tail\(\mathrm{\}}\).What is the mean of the \(S_1\) distribution?(b) What is the variance of the \(S_1\) distribution?(c) What is the mean of the \(S_n\) distribution?(d) What is the variance of the \(S_n\) distribution?13. Fifty unbiased coins are tossed simultaneously.(a) Calculate the probability of 25 heads and 25 tails.(b) Calculate the probability of 23 heads and 27 tails.(c) Calculate the probability of 3 heads and 47 tails.(d) Calculate the ratio of your results for parts (a) and (b).(e) Calculate the ratio of your results for parts (a) and (c).14. For \(N=3,\ 6\) and \(10\), calculate\(\)(a) The exact value of \(N!\)(b) The value of \(N!\) according to the approximation \[N!\approx N^N \left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right)\mathrm{exp}\left(\frac{1}{12N}\right) \nonumber \](c) The value of N! according to the approximation \[N!\approx N^N \left(2\pi N\right)^{1/2}\mathrm{exp}\left(-N\right) \nonumber \](d) The value of N! according to the approximation \[N!\approx N^N\mathrm{exp}\left(-N\right) \nonumber \](e) The ratio of the value in (b) to the corresponding value in (a).(f) The ratio of the value in (c) to the corresponding value in (a).(g) The ratio of the value in (d) to the corresponding value in (a).(h) Comment.15. Find , \(d ~ \ln N! /dN\) using each of the approximations \[N!\approx N^N \left(2\pi N\right)^{1/2} \mathrm{exp}\left(-N\right)\mathrm{exp}\left(\frac{1}{12N}\right)\approx N^N \left(2\pi N\right)^{1/2} \mathrm{exp}\left(-N\right)\approx N^N\mathrm{exp}\left(-N\right) \nonumber \]How do the resulting approximations for \(d ~ \ln N! /dN\) compare to one another as \(N\) becomes very large?16. There are three energy levels available to any one molecule in a crystal of the substance. Consider a crystal containing \(1000\) molecules. These molecules are distinguishable because each occupies a unique site in the crystalline lattice. How many combinations (microstates) are associated with the population set \(N_1=800\), \(N_2=150\), \(N_3=50\)?This page titled 19.5: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.1: Boyle's Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/02%3A_Gas_Laws/2.01%3A_Boyle's_Law

Robert Boyle discovered Boyle’s law in 1662. Boyle’s discovery was that the pressure, P, and volume, V, of a gas are inversely proportional to one another if the temperature, T, is held constant. We can imagine rediscovering Boyle’s law by trapping a sample of gas in a tube and then measuring its volume as we change the pressure. We would observe behavior like that in We can represent this behavior mathematically as\[PV={\alpha }^*(n,T) \nonumber \]where we recognize that the “constant”, \({\alpha }^*\), is actually a function of the temperature and of the number of moles, \(n\), of gas in the sample. That is, the product of pressure and volume is constant for a fixed quantity of gas at a fixed temperature.A little thought convinces us that we can be more specific about the dependence on the quantity of gas. Suppose that we have a volume of gas at a fixed pressure and temperature, and imagine that we introduce a very thin barrier that divides the volume into exactly equal halves, without changing anything else. In this case, the pressure and temperature of the gas in each of the new containers will be the same as they were originally. But the volume is half as great, and the number of moles in each of the half-size containers must also be half of the original number. That is, the pressure–volume product must be directly proportional to the number of moles of gas in the sample:\[PV=n\alpha (T) \nonumber \]where \(\alpha (T)\) is now a function only of temperature. When we repeat this experiment using different gaseous substances, we discover a further remarkable fact: Not only do they all obey Boyle’s law, but also the value of \(\alpha (T)\) is the same for any gas.This page titled 2.1: Boyle's Law is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.2: Charles' Law

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Quantitative experiments establishing the law were first published in 1802 by Gay-Lussac, who credited Jacques Charles with having discovered the law earlier. Charles’ law relates the volume and temperature of a gas when measurements are made at constant pressure. We can imagine rediscovering Charles’ law by trapping a sample of gas in a tube and measuring its volume as we change the temperature, while keeping the pressure constant. This presumes that we have a way to measure temperature, perhaps by defining it in terms of the volume of a fixed quantity of some other fluid—like liquid mercury. At a fixed pressure, \(P_1\), we observe a linear relationship between the volume of a sample of gas and its temperature, like that in If we repeat this experiment with the same gas sample at a higher pressure, \(P_2\), we observe a second linear relationship between the volume and the temperature of the gas. If we extend these lines to their intersection with the temperature axis at zero volume, we make a further important discovery: Both lines intersect the temperature axis at the same point.We can represent this behavior mathematically as\[V={\beta }^*\left(n,P\right)T^*+{\gamma }^*(n,P) \nonumber \]where we recognize that both the slope and the V-axis intercept of the graph depend on the pressure of the gas and on the number of moles of gas in the sample. A little reflection shows that here too the slope and intercept must be directly proportional to the number of moles of gas, so that we can rewrite our equation as\[V=n\beta \left(P\right)T^*+n\gamma (P) \nonumber \]When we repeat these experiments with different gaseous substances, we discover an additional important fact: \(\beta (P)\) and \(\gamma (P)\) are the same for any gas. This means that the temperature at which the volume extrapolates to zero is the same for any gas and is independent of the constant pressure we maintain as we vary the temperature.This page titled 2.2: Charles' Law is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.3: Avogadro's Hypothesis

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Avogadro’s hypothesis is another classical gas law. It can be stated: At the same temperature and pressure, equal volumes of different gases contain the same number of molecules.When the mass, in grams, of an ideal gas sample is equal to the gram molar mass (traditionally called the molecular weight) of the gas, the number of molecules in the sample is equal to Avogadro’s number, \(\overline{N}\)\({}^{1}\). Avogadro’s number is the number of molecules in a mole. In the modern definition, one mole is the number of atoms of \(C^{12}\) in exactly 12 g of \(C^{12}\). That is, the number of atoms of \(C^{12}\) in exactly 12 g of \(C^{12}\) is Avogadro’s number. The currently accepted value is \(\mathrm{6.02214199\times }{\mathrm{10}}^{\mathrm{23}}\) molecules per mole. We can find the gram atomic mass of any other element by finding the mass of that element that combines with exactly 12 g of \(C^{12}\) in a compound whose molecular formula is known.The validity of Avogadro’s hypothesis follows immediately either from the fact that the Boyle’s law constant, \(\alpha (T)\), is the same for any gas or from the fact that the Charles’ law constants, \(\beta (P)\) and \(\gamma \left(P\right)\), are the same for any gas. However, this entails a significant circularity; these experiments can show that \(\alpha (T)\), \(\beta (P)\), and \(\gamma \left(P\right)\) are the same for any gas only if we know how to find the number of moles of each gas that we use. To do so, we must know the molar mass of each gas. Avogadro’s hypothesis is crucially important in the history of chemistry: Avogadro’s hypothesis made it possible to determine relative molar masses. This made it possible to determine molecular formulas for gaseous substances and to create the atomic mass scale.This page titled 2.3: Avogadro's Hypothesis is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.4: Finding Avogadro's Number

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This use of Avogadro’s number raises the question of how we know its value. There are numerous ways to measure Avogadro’s number. One such method is to divide the charge of one mole of electrons by the charge of a single electron. We can obtain the charge of a mole of electrons from electrolysis experiments. The charge of one electron can be determined in a famous experiment devised by Robert Millikan, the “Millikan oil-drop experiment”. The charge on a mole of electrons is called the faraday. Experimentally, it has the value \(96,485\ \mathrm{C\ }{\mathrm{mol}}^{\mathrm{-1}}\)\({}^{\ }\)(coulombs per mole). As determined by Millikan’s experiment, the charge on one electron is \(1.6022\times {10}^{-19}\ \mathrm{C}\). Then\[ \left(\frac{96,485\ C}{\mathrm{mole\ electrons}}\right)\left(\frac{1\ \mathrm{electron}}{1.6022\times {10}^{-19\ }\ C}\right) =6.022\times {10}^{23\ \ }\frac{\mathrm{electrons}}{\mathrm{mole\ electrons}} \nonumber \]This page titled 2.4: Finding Avogadro's Number is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.5: The Kelvin Temperature Scale

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Thus far, we have assumed nothing about the value of the temperature corresponding to any particular volume of our standard fluid. We could define one unit of temperature to be any particular change in the volume of our standard fluid. Historically, Fahrenheit defined one unit (degree) of temperature to be one one-hundredth of the increase in volume of a fixed quantity of standard fluid as he warmed it from the lowest temperature he could achieve, which he elected to call 0 degrees, to the temperature of his body, which he elected to call 100 degrees. Fahrenheit’s zero of temperature was achieved by mixing salt with ice and water. This is not a very reproducible condition, so the temperature of melting ice (with no salt present), soon became the calibration standard. Fahrenheit’s experiments put the melting point of ice at 32 F. The normal temperature for a healthy person is now taken to be 98.6 F; possibly Fahrenheit had a slight fever when he was doing his calibration experiments. In any case, human temperatures vary enough so that Fahrenheit’s 100-degree point was not very practical either. The boiling point of water, which Fahrenheit’s experiments put at 212 F, became the calibration standard. Later, the centigrade scale was developed with fixed points at 0 degrees and 100 degrees at the melting point of ice and the boiling point of water, respectively. The centigrade scale is now called the Celsius scale after Anders Celsius, Anders, a Swedish astronomer. In 1742, Celsius proposed a scale on which the temperature interval between the boiling point and the freezing point of water was divided into 100 degrees; however, a more positive number corresponded to a colder condition.Further reflection convinces us that the Charles’ law equation can be simplified by defining a new temperature scale. When we extend the straight line in any of our volume-versus-temperature plots, it always intersects the zero-volume horizontal line at the same temperature. Since we cannot associate any meaning with a negative volume, we infer that the temperature at zero volume represents a natural minimum point for our temperature scale. Let the value of \(T^*\) at this intersection be \(T^*_0\). Substituting into our volume-temperature relationship, we have\[0=n\beta \left(P\right)T^*_0+n\gamma (P) \nonumber \]or\[\gamma \left(P\right)=-\beta (P)T^*_0 \nonumber \]So that\[\begin{align} V&= n\beta \left(P\right)T^*-n\beta (P)T^*_0 \\[4pt] &=n\beta \left(P\right)[T^*-T^*_0] \\[4pt] &=n\beta \left(P\right)T \end{align} \nonumber \]where we have created a new temperature scale. Temperature values on our new temperature scale, T, are related to temperature values on the old temperature scale, \(T^*\), by the equation\[T=T^*-T^*_0 \nonumber \]When the size of one unit of temperature is defined using the Celsius scale (i.e., \(T^*\) is the temperature in degrees Celsius), this is the origin of the Kelvin temperature scale \({}^{2}\). Then, on the Kelvin temperature scale, \(T^*_0\) is -273.15 degrees. (That is, \(T=0\) when \(T^*_0\) = 273.15; 0 K is \(-\)273.15 degrees Celsius.) The temperature at which the volume extrapolates to zero is called the absolute zero of temperature. When the size of one unit of temperature is defined using the Fahrenheit scale and the zero of temperature is set at absolute zero, the resulting temperature scale is called the Rankine scale, after William Rankine, a Scottish engineer who proposed it in 1859.This page titled 2.5: The Kelvin Temperature Scale is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.6: Deriving the Ideal Gas Law from Boyle's and Charles' Laws

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We can solve Boyle’s law and Charles’ law for the volume. Equating the two, we have\[\dfrac{n\alpha (T)}{P}=n\beta \left(P\right)T \nonumber \]The number of moles, \(n\), cancels. Rearranging gives\[\dfrac{\alpha (T)}{T}=P\beta (P) \nonumber \]In this equation, the left side is a function only of temperature, the right side only of pressure. Since pressure and temperature are independent of one another, this can be true only if each side is in fact constant. If we let this constant be \(R\), we have\[\alpha \left(T\right)=RT \nonumber \]and\[\beta \left(P\right)={R}/{P} \nonumber \]Since the values of \(\alpha (T)\) and \(\beta (P)\) are independent of the gas being studied, the value of \(R\) is also the same for any gas. \(R\) is called the gas constant, the ideal gas constant, or the universal gas constant. Substituting the appropriate relationship into either Boyle’s law or Charles’ law gives the ideal gas equation\[PV=nRT \nonumber \]The product of pressure and volume has the units of work or energy, so the gas constant has units of energy per mole per degree. (Remember that we simplified the form of Charles’s law by defining the Kelvin temperature scale; temperature in the ideal gas equation is in degrees Kelvin.)This page titled 2.6: Deriving the Ideal Gas Law from Boyle's and Charles' Laws is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.7: The Ideal Gas Constant and Boltzmann's Constant

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Having developed the ideal gas equation and analyzed experimental results for a variety of gases, we will have found the value of R. It is useful to have R expressed using a number of different energy units. Frequently useful values are\[ \begin{aligned} R & = 8.314 \text{ Pa m}^{3} \text{ K}^{-1} \text{ mol}^{-1} \\ ~ & = 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \\ ~ & = 0.08314 \text{ L bar K}^{-1} \text{ mol}^{-1} \\ ~ & = 1.987 \text{ cal K}^{-1} \text{ mol}^{-1} \\ ~ & = 0.08205 \text{ L atm K}^{-1} \text{ mol}^{-1} \end{aligned} \nonumber \]We also need the gas constant expressed per molecule rather than per mole. Since there is Avogadro’s number of molecules per mole, we can divide any of the values above by \(\overline{N}\) to get \(R\) on a per-molecule basis. Traditionally, however, this constant is given a different name; it is Boltzmann’s constant, usually given the symbol \(k\).\[k={R}/{\overline{N}}=1.381\times {10}^{-23}\ \mathrm{J}\ {\mathrm{K}}^{-1}\ {\mathrm{molecule}}^{-1} \nonumber \]This means that we can also write the ideal gas equation as \(PV=nRT=n\overline{N}kT\). Because the number of molecules in the sample, \(N\), is \(N=n\overline{N}\), we have\[PV=NkT. \nonumber \]This page titled 2.7: The Ideal Gas Constant and Boltzmann's Constant is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.8: Real Gases Versus Ideal Gases

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Now, we need to expand on the qualifications with which we begin this chapter. We imagine that the results of a large number of experiments are available for our analysis. Our characterization of these results has been that all gases obey the same equations—Boyle’s law, Charles’ law, and the ideal gas equation—and do so exactly. This is an oversimplification. In fact they are always approximations. They are approximately true for all gases under all “reasonable” conditions, but they are not exactly true for any real gas under any condition. It is useful to introduce the idea of hypothetical gases that obey the classical gas equations exactly. In the previous section, we call the combination of Boyle’s law and Charles’ law the ideal gas equation. We call the hypothetical substances that obey this equation ideal gases. Sometimes we refer to the classical gas laws collectively as the ideal gas laws.At very high gas densities, the classical gas laws can be very poor approximations. As we have noted, they are better approximations the lower the density of the gas. In fact, experiments show that the pressure—volume—temperature behavior of any real gasreal gas becomes arbitrarily close to that predicted by the ideal gas equation in the limit as the pressure goes to zero. This is an important observation that we use extensively.At any given pressure and temperature, the ideal gas laws are better approximations for a compound that has a lower boiling point than they are for a compound with a higher boiling point. Another way of saying this is that they are better approximations for molecules that are weakly attracted to one another than they are for molecules that are strongly attracted to one another.Forces between molecules cause them to both attract and repel one another. The net effect depends on the distance between them. If we assume that there are no intermolecular forcesintermolecular forces acting between gas molecules, we can develop exact theories for the behavior of macroscopic amounts of the gas. In particular, we can show that such substances obey the ideal gas equation. (We shall see that a complete absence of repulsive forces implies that the molecules behave as point masses.) Evidently, the difference between the behavior of a real gas and the behavior it would exhibit if it were an ideal gas is just a measure of the effects of intermolecular forces.The ideal gas equation is not the only equation that gives a useful representation for the interrelation of gas pressure–volume–temperature data. There are many such equations of state. They are all approximations, but each can be a particularly useful approximation in particular circumstances. We discuss van der Waal’s equation equation and the virial equations later in this chapter. Nevertheless, we use the ideal gas equation extensively.We will see that much of chemical thermodynamics is based on the behavior of ideal gases. Since there are no ideal gases, this may seem odd, at best. If there are no ideal gases, why do we waste time talking about them? After all, we don’t want to slog through tedious, long-winded, pointless digressions. We want to understand how real stuff behaves! Unfortunately, this is more difficult. The charm of ideal gases is that we can understand their behavior; the ideal gas equation expresses this understanding in a mathematical model. Real gases are another story. We can reasonably say that we can best understand the behavior of a real gas by understanding how and why it is different from the behavior of a (hypothetical) ideal gas that has the same molecular structure.This page titled 2.8: Real Gases Versus Ideal Gases is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.9: Temperature and the Ideal Gas Thermometer

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In Section 2.2 we suppose that we have a thermometer that we can use to measure the temperature of a gas. We suppose that this thermometer uses a liquid, and we define an increase in temperature by the increase in the volume of this liquid. Our statement of Charles’ law asserts that the volume of a gas is a linear function of the volume of the liquid in our thermometer, and that the same linear function is observed for any gas. As we note in Section 2.8, there is a problem with this statement. Careful experiments with such thermometers produce results that deviate from Charles’ law. With sufficiently accurate volume measurements, this occurs to some extent for any choice of the liquid in the thermometer. If we make sufficiently accurate measurements, the volume of a gas is not exactly proportional to the volume of any liquid (or solid) that we might choose as the working substance in our thermometer. That is, if we base our temperature scale on a liquid or solid substance, we observe deviations from Charles’ law. There is a further difficulty with using a liquid as the standard fluid on which to base our temperature measurements: temperatures outside the liquid range of the chosen substance have to be measured in some other way.Evidently, we can choose to use a gas as the working fluid in our thermometer. That is, our gas-volume measuring device is itself a thermometer. This fact proves to be very useful because of a further experimental observation. To a very good approximation, we find: If we keep the pressures in the thermometer and in some other gaseous system constant at low enough values, both gases behave as ideal gases, and we find that the volumes of the two gases are proportional to each other over any range of temperature. Moreover, this proportionality is observed for any choice of either gas. This means that we can define temperature in terms of the expansion of any constant-pressure gas that behaves ideally. In principle, we can measure the same temperature using any gas, so long as the constant operating pressure is low enough. When we do so, our device is called the ideal gas thermometer. In so far as any gas behaves as an ideal gas at a sufficiently low pressure, any real gas can be used in an ideal gas thermometer and to measure any temperature accurately. Of course, practical problems emerge when we attempt to make such measurements at very high and very low temperatures.The (very nearly) direct proportionality of two low-pressure real gas volumes contrasts with what we observe for liquids and solids. In general, the volume of a given liquid (or solid) substance is not exactly proportional to the volume of a second liquid (or solid) substance over a wide range of temperatures.In practice, the ideal-gas thermometer is not as convenient to use as other thermometers—like the mercury-in-glass thermometer. However, the ideal-gas thermometer is used to calibrate other thermometers. Of course, we have to calibrate the ideal-gas thermometer itself before we can use it.We do this by assigning a temperature of 273.16 K to the triple point of water. (It turns out that the melting point of ice isn’t sufficiently reproducible for the most precise work. Recall that the triple point is the temperature and pressure at which all three phases of water are at equilibrium with one another, with no air or other substances present. The triple-point pressure is 611 Pa or \(\mathrm{6.03\times }{\mathrm{10}}^{\mathrm{-3\ }}\)atm. See Section 6.3.) From both theoretical considerations and experimental observations, we are confident that no system can attain a temperature below absolute zero. Thus, the size\({}^{3}\) of the kelvin (one degree on the Kelvin scale) is fixed by the difference in temperature between a system at the triple point of water and one at absolute zero. If our ideal gas thermometer has volume \(V\) at thermal equilibrium with some other constant-temperature system, the proportionality of \(V\) and \(T\) means that\[\frac{T}{V}=\frac{273.16}{V_{273.16}} \nonumber \]With the triple point fixed at 273.16 K, experiments find the freezing point of air-saturated water to be 273.15 K when the system pressure is 1 atmosphere. (So the melting point of ice is 273.15 K, and the triple-point is 0.10 C. We will find two reasons for the fact that the melting point is lower than the triple point: In Section 6.3 we find that the melting point of ice decreases as the pressure increases. In Section 16.10 we find that solutes usually decrease the temperature at which the liquid and solid states of a substance are in phase equilibrium.)If we could use an ideal gas in our ideal-gas thermometer, we could be confident that we had a rigorous operational definition of temperature. However, we note in Section 2.8 that any real gas will exhibit departures from ideal gas behavior if we make sufficiently accurate measurements. For extremely accurate work, we need a way to correct the temperature value that we associate with a given real-gas volume. The issue here is the value of the partial derivative\[{\left(\frac{\partial V}{\partial T}\right)}_P \nonumber \]For one mole of an ideal gas,\[{\left(\frac{\partial V}{\partial T}\right)}_P=\frac{R}{P}=\frac{V}{T} \nonumber \]is a constant. For a real gas, it is a function of temperature. Let us assume that we know this function. Let the molar volume of the real gas at the triple point of water be \(V_{273.16}\) and its volume at thermal equilibrium with a system whose true temperature is \(V\) be \(V_T\). We have\[ \int_{273.16}^T \left( \frac{ \partial V}{ \partial T} \right)_P dT = \int_{V_{273.16}}^{V_T} dV = V_T - V_{273.16} \nonumber \]When we know the integrand on the left as a function of temperature, we can do the integration and find the temperature corresponding to any measured volume, \(V_T\).When the working fluid in our thermometer is a real gas we make measurements to find \({\left({\partial V}/{\partial T}\right)}_P\) as a function of temperature. Here we encounter a circularity: To find \({\left({\partial V}/{\partial T}\right)}_P\) from pressure-volume-temperature data we must have a way to measure temperature; however, this is the very thing that we are trying to find.In principle, we can surmount this difficulty by iteratively correcting the temperature that we associate with a given real-gas volume. As a first approximation, we use the temperatures that we measure with an uncorrected real-gas thermometer. These temperatures are a first approximation to the ideal-gas temperature scale. Using this scale, we make non-pressure-volume-temperature measurements that establish \({\left({\partial V}/{\partial T}\right)}_P\) as a function of temperature for the real gas. [This function is\[{\left(\frac{\partial V}{\partial T}\right)}_P=\frac{V+{\mu }_{JT}C_P}{T} \nonumber \]where \(C_P\) is the constant-pressure heat capacity and \({\mu }_{JT}\) is the Joule-Thomson coefficient. Both are functions of temperature. We introduce \(C_P\) in Section 7.9. We discuss the Joule-Thomson coefficient further in Section 2.10 below, and in detail in Section 10.14. Typically \(V\gg C_P\), and the value of \({\left({\partial V}/{\partial T}\right)}_P\) is well approximated by \({V}/{T}={R}/{P}\). With \({\left({\partial V}/{\partial T}\right)}_P\) established using this scale, integration yields a second-approximation to the ideal-gas temperatures. We could repeat this process until successive temperature scales converge at the number of significant figures that our experimental accuracy can support.In practice, there are several kinds of ideal-gas thermometers, and numerous corrections are required for very accurate measurements. There are also numerous other ways to measure temperature, each of which has its own complications. Our development has considered some of the ideas that have given rise to the concept\({}^{4}\) that temperature is fundamental property of nature that can be measured using a thermodynamic-temperature scale on which values begin at zero and increase to arbitrarily high values. This thermodynamic temperature scale is a creature of theory, whose real-world counterpart would be the scale established by an ideal-gas thermometer whose gas actually obeyed \(PV=nRT\) at all conditions. We have seen that such an ideal-gas thermometer is itself a creature of theory.The current real-world standard temperature scale is the International Temperature Scale of 1990 (ITS-90). This defines temperature over a wide range in terms of the pressure-volume relationships of helium isotopes and the triple points of several selected elements. The triple points fix the temperature at each of several conditions up to 1357.77 K (the freezing point of copper). Needless to say, the temperatures assigned at the fixed points are the results of painstaking experiments designed to give the closest possible match to the thermodynamic scale. A variety of measuring devices—thermometers—can be used to interpolate temperature values between different pairs of fixed points.This page titled 2.9: Temperature and the Ideal Gas Thermometer is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.10: Deriving Boyle's Law from Newtonian Mechanics

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We can derive Boyle’s law from Newtonian mechanics. This derivation assumes that gas molecules behave like point masses that do not interact with one another. The pressure of the gas results from collisions of the gas molecules with the walls of the container. The contribution of one collision to the force on the wall is equal to the change in the molecule’s momentum divided by the time between collisions. The magnitude of this force depends on the molecule’s speed and the angle at which it strikes the wall. Each such collision makes a contribution to the pressure that is equal to the force divided by the area of the wall. To find the pressure from this model, it is necessary to average over all possible molecular speeds and all possible collision angles. In Chapter 4, we derive Boyle’s law in this way.We can do a simplified derivation by making a number of assumptions. We assume that all of the molecules in a sample of gas have the same speed. Let us call it \(u\). As sketched in . If we consider all of the collisions between molecules and walls, it is clear that each wall will experience \({1}/{6}\) of the collisions; or, each pair of opposing walls will experience \({\mathrm{1}}/{\mathrm{3}}\) of the collisions. Instead of averaging over all of the possible angles at which a molecule could strike a wall and all of the possible times between collisions, we assume that the molecules travel at constant speed back and forth between opposite faces of the box. Since they are point masses, they never collide with one another. If we suppose that \({\mathrm{1}}/{\mathrm{3}}\) of the molecules go back and forth between each pair of opposite walls, we can expect to accomplish the same kind of averaging in setting up our artificial model that we achieve by averaging over the real distribution of angles and speeds. In fact, this turns out to be the case; the derivation below gets the same result as the rigorous treatment we develop in Chapter 4.Since each molecule goes back and forth between opposite walls, it collides with each wall once during each round trip. At each collision, the molecule’s speed remains constant, but its direction changes by 180\({}^{o}\); that is, the molecule’s velocity changes from \(\mathop{u}\limits^{\rightharpoonup}\) to \(-\mathop{u}\limits^{\rightharpoonup}\). Letting \(\Delta t\) be the time required for a round trip, the distance traversed in a round trip is\[\begin{aligned} 2d & =\left|\mathop{u}\limits^{\rightharpoonup}\right|\Delta t \\ ~ & =u\Delta t \end{aligned} \nonumber \]The magnitude of the momentum change for a molecule in one collision is\[\begin{align*} \left|\Delta (m\mathop{u}\limits^{\rightharpoonup})\right| &=\left|m{\mathop{u}\limits^{\rightharpoonup}}_{final}-m{\mathop{u}\limits^{\rightharpoonup}}_{initial}\right| \\[4pt] &=\left|m{\mathop{u}\limits^{\rightharpoonup}}_{final}-\left({-m\mathop{u}\limits^{\rightharpoonup}}_{final}\right)\right| \\[4pt] &=2mu \end{align*} \]The magnitude of the force on the wall from one collision is\[F=\frac{\left|\Delta \left(m\mathop{u}\limits^{\rightharpoonup}\right)\right|}{\Delta t}=\frac{2mu}{\left({2d}/{u}\right)}=\frac{mu^2}{d} \nonumber \]and the pressure contribution from one collision on the wall, of area \(d^2\), is\[P=\frac{F}{A}=\frac{mu^2}{d\bullet d^2}=\frac{{mu}^2}{d^3}=\frac{{mu}^2}{V} \nonumber \]so that we have\[PV=mu^2 \nonumber \]from the collision of one molecule with one wall.If the number of molecules in the box is \(N\), \(N/3\) of them make collisions with this wall, so that the total pressure on one wall attributable to all \(N\) molecules in the box is\[P=\frac{mu^2}{V}\frac{N}{3} \nonumber \]or\[PV=\frac{Nmu^2}{3} \nonumber \]Since the ideal gas equation can be written as \(PV=NkT\) we see that \({Nmu^2}/{3}=NkT\) so that \(mu^2=3kT\) and\[u=\sqrt{\frac{3kT}{m}} \nonumber \]Thus we have found a relationship between the molecular speed and the temperature of the gas. (The actual speed of a molecule, \(v\), can have any value between zero and—for present purposes—infinity. When we average the values of \(v^2\) for many molecules, we find the average value of the squared speeds, \(\overline{v^2}\). In Chapter 4, we find that \(u^2=\overline{v^2}\). That is, the average speed we use in our derivation turns out to be a quantity called the root-mean-square speed, \(v_{rms}=u=\sqrt{\overline{v^2}}\).) This result also gives us the (average) kinetic energy of a single gas molecule:\[KE=\frac{mu^2}{2}=\frac{3kT}{2} \nonumber \]From this derivation, we have a simple mechanical model that explains Boyle’s law as the logical consequence of point-mass molecules colliding with the walls of their container. By combining this result with the ideal gas equation, we find that the average speed of ideal gas molecules depends only on the temperature. From this we have the very important result that the translational kinetic energy of an ideal gas depends only on temperature.Since our non-interacting point-mass molecules have no potential energy arising from their interactions with one another, their translational kinetic energy is the whole of their energy. (Because two such molecules neither attract nor repel one another, no work is required to change the distance between them. The work associated with changing the volume of a confined sample of an ideal gas arises because of the pressure the molecules exert on the walls of the container; the pressure arises because of the molecules’ kinetic energy.) The energy of one mole of monatomic ideal gas molecules is\[KE=\left({3}/{2}\right)RT \nonumber \]When we expand our concept of ideal gases to include molecules that have rotational or vibrational energy, but which neither attract nor repel one another, it remains true that the energy of a macroscopic sample depends only on temperature. However, the molar energy of such a gas is greater than \(\left({3}/{2}\right)RT\), because of the energy associated with these additional motions.We make extensive use of the conclusion that the energy of an ideal gas depends only on temperature. As it turns out, this conclusion follows rigorously from the second law of thermodynamics. In Chapter 10, we show that\[{\left(\frac{\partial E}{\partial V}\right)}_T={\left(\frac{\partial E}{\partial P}\right)}_T=0 \nonumber \]for a substance that obeys the ideal gas equation; at constant temperature, the energy of an ideal gas is independent of the volume and independent of the pressure. So long as pressure, volume, and temperature are the only variables needed to specify its state, the laws of thermodynamics imply that the energy of an ideal gas depends only on temperature.While the energy of an ideal gas is independent of pressure, the energy of a real gas is a function of pressure at a given temperature. At ordinary pressures and temperatures, this dependence is weak and can often be neglected. The first experimental investigation of this issue was made by James Prescott Joule, for whom the SI unit of energy is named. Beginning in 1838, Joule did a long series of careful measurements of the mechanical equivalent of heat. These measurements formed the original experimental basis for the kinetic theory of heat. Among Joule’s early experiments was an attempt to measure the heat absorbed by a gas as it expanded into an evacuated container, a process known as a free expansion. No absorption of heat was observed, which implied that the energy of the gas was unaffected by the volume change. However, it is difficult to do this experiment with meaningful accuracy.Subsequently, Joule collaborated with William Thomson (Lord Kelvin) on a somewhat different experimental approach to essentially the same question. The Joule-Thomson experiment provides a much more sensitive measure of the effects of intermolecular forces of attraction and repulsion on the energy of a gas during its expansion. Since our definition of an ideal gas includes the stipulation that there are no intermolecular forces, the Joule-Thomson experiment is consistent with the conclusion that the energy of an ideal gas depends only on temperature. However, since intermolecular forces are not zero for any real gas, our analysis reaches this conclusion in a somewhat indirect way. The complication arises because the Joule-Thomson results are not entirely consistent with the idea that all properties of a real gas approach those of an ideal gas at a sufficiently low pressure. (The best of models can have limitations.) We discuss the Joule-Thomson experiment in Section 10.14.This page titled 2.10: Deriving Boyle's Law from Newtonian Mechanics is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.11: The Barometric Formula

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We can measure the pressure of the atmosphere at any location by using a barometer. A mercury barometer is a sealed tube that contains a vertical column of liquid mercury. The space in the tube above the liquid mercury is occupied by mercury vapor. Since the vapor pressure of liquid mercury at ordinary temperatures is very low, the pressure at the top of the mercury column is very low and can usually be ignored. The pressure at the bottom of the column of mercury is equal to the pressure of a column of air extending from the elevation of the barometer all the way to the top of the earth’s atmosphere. As we take the barometer to higher altitudes, we find that the height of the mercury column decreases, because less and less of the atmosphere is above the barometer.If we assume that the atmosphere is composed of an ideal gas and that its temperature is constant, we can derive an equation for atmospheric pressure as a function of altitude. Imagine a cylindrical column of air extending from the earth’s surface to the top of the atmosphere. The force exerted by this column at its base is the weight of the air in the column; the pressure is this weight divided by the cross-sectional area of the column. Let the cross-sectional area of the column be \(A\).Consider a short section of this column. Let the bottom of this section be a distance \(h\) from the earth’s surface, while its top is a distance \(h+\Delta h\) from the earth’s surface. The volume of this cylindrical section is then \(V_S=A\Delta h\). Let the mass of the gas in this section be \(M_S\). The pressure at \(h+\Delta h\) is less than the pressure at \(h\) by the weight of this gas divided by the cross-sectional area. The weight of the gas is \(M_Sg\). The pressure difference is \(\Delta P=-{M_Sg}/{A}\). We have\[\frac{P\left(h+\Delta h\right)-P\left(h\right)}{\Delta h}=\frac{\Delta P}{\Delta h}=\frac{-M_Sg}{A\Delta h}=\frac{-M_Sg}{V_S} \nonumber \]Since we are assuming that the sample of gas in the cylindrical section behaves ideally, we have \(V_S={n_SRT}/{P}\). Substituting for \(V_S\) and taking the limit as \(\Delta h\to 0\), we find\[\frac{dP}{dh}=\left(\frac{{-M}_Sg}{n_SRT}\right)P=\left(\frac{{-n}_S\overline{M}g}{n_SRT}\right)P=\left(\frac{-mg}{kT}\right)P \nonumber \]where we introduce \(n_S\) as the number of moles of gas in the sample, \(\overline{M}\) as the molar mass of this gas, and \(m\) as the mass of an individual atmosphere molecule. The last equality on the right makes use of the identities \(\overline{M}=m\overline{N}\) and \(R=\overline{N}k\). Separating variables and integrating between limits \(P\left(0\right)=P_0\) and \(P\left(h\right)=P\), we find\[\int^P_{P_0}{\frac{dP}{P}}=\left(\frac{-mg}{kT}\right)\int^h_0{dh} \nonumber \]so that \[{ \ln \left(\frac{P}{P_0}\right)\ }=\frac{-mgh}{kT} \nonumber \]and\[P=P_0\mathrm{exp}\left(\frac{-mgh}{kT}\right) \nonumber \]Either of the latter relationships is frequently called the barometric formula.If we let \(\eta\) be the number of molecules per unit volume, \(\eta ={N}/{V}\), we can write \(P={NkT}/{V}=\eta kT\) and \(P_0={\eta }_0kT\) so that the barometric formula can be expressed in terms of these number densities as\[\eta ={\eta }_0\mathrm{exp}\left(\frac{-mgh}{kT}\right) \nonumber \]This page titled 2.11: The Barometric Formula is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.12: Van der Waals' Equation

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An equation due to van der Waals extends the ideal gas equation in a straightforward way. Van der Waals’ equation is\[\left(P+\frac{an^2}{V^2}\right)\left(V-nb\right)=nRT \nonumber \]It fits pressure-volume-temperature data for a real gas better than the ideal gas equation does. The improved fit is obtained by introducing two parameters (designated “\(a\)” and “\(b\)”) that must be determined experimentally for each gas. Van der Waals’ equation is particularly useful in our effort to understand the behavior of real gases, because it embodies a simple physical picture for the difference between a real gas and an ideal gas.In deriving Boyle’s law from Newton’s laws, we assume that the gas molecules do not interact with one another. Simple arguments show that this can be only approximately true. Real gas molecules must interact with one another. At short distances they repel one another. At somewhat longer distances, they attract one another. The ideal gas equation can also be derived from the basic assumptions that we make in §10 by an application of the theory of statistical thermodynamics. By making different assumptions about molecular properties, we can apply statistical thermodynamics to derive\({}^{5}\) van der Waals’ equation. The required assumptions are that the molecules occupy a finite volume and that they attract one another with a force that varies as the inverse of a power of the distance between them. (The attractive force is usually assumed to be proportional to \(r^{-6}\).)To recognize that real gas molecules both attract and repel one another, we need only remember that any gas can be liquefied by reducing its temperature and increasing the pressure applied to it. If we cool the liquid further, it freezes to a solid. Now, two distinguishing features of a solid are that it retains its shape and that it is almost incompressible. We attribute the incompressibility of a solid to repulsive forces between its constituent molecules; they have come so close to one another that repulsive forces between them have become important. To compress the solid, the molecules must be pushed still closer together, which requires inordinate force. On the other hand, if we throw an ice cube across the room, all of its constituent water molecules fly across the room together. Evidently, the water molecules in the solid are attracted to one another, otherwise they would all go their separate ways—throwing the ice cube would be like throwing a handful of dry sand. But water molecules are the same molecules whatever the temperature or pressure, so if there are forces of attraction and repulsion between them in the solid, these forces must be present in the liquid and gas phases also.In the gas phase, molecules are far apart; in the liquid or the solid phase, they are packed together. At its boiling point, the volume of a liquid is much less than the volume of the gas from which it is condensed. At the freezing point, the volume of a solid is only slightly different from the volume of the liquid from which it is frozen, and it is certainly greater than zero. These commonplace observations are readily explained by supposing that any molecule has a characteristic volume. We can understand this, in turn, to be a consequence of the nature of the intermolecular forces; evidently, these forces become stronger as the distance between a pair of molecules decreases. Since a liquid or a solid occupies a definite volume, the repulsive force must increase more rapidly than the attractive force when the intermolecular distance is small. Often it is useful to talk about the molar volume of a condensed phase. By molar volume, we mean the volume of one mole of a pure substance. The molar volume of a condensed phase is determined by the intermolecular distance at which there is a balance between intermolecular forces of attraction and repulsion.Evidently molecules are very close to one another in condensed phases. If we suppose that the empty spaces between molecules are negligible, the volume of a condensed phase is approximately equal to the number of molecules in the sample multiplied by the volume of a single molecule. Then the molar volume is Avogadro’s number times the volume occupied by one molecule. If we know the density, D, and the molar mass, \(\overline{M}\), we can find the molar volume, \(\overline{V}\), as\[\overline{V}=\frac{\overline{M}}{D} \nonumber \]The volume occupied by a molecule, V\({}_{molecule}\), becomes\[V_{molecule}=\frac{\overline{V}}{\overline{N}} \nonumber \]The pressure and volume appearing in van der Waals’ equation are the pressure and volume of the real gas. We can relate the terms in van der Waals’ equation to the ideal gas equation: It is useful to think of the terms \(\left(P+{{an}^2}/{V^2}\right)\) and \(\left(V-nb\right)\) as the pressure and volume of a hypothetical ideal gas. That is\[ \begin{align*} P_{ideal\ gas}V_{ideal\ gas} &=\left(P_{real\ gas}+\frac{an^2}{V^2_{real\ gas}}\right)\left(V_{real\ gas}-nb\right) \\[4pt] &=nRT \end{align*} \]Then we have\[V_{real\ gas}=V_{ideal\ gas}+nb \nonumber \]We derive the ideal gas equation from a model in which the molecules are non-interacting point masses. So the volume of an ideal gas is the volume occupied by a gas whose individual molecules have zero volume. If the individual molecules of a real gas effectively occupy a volume \({b}/{\overline{N}}\), then \(n\) moles of them effectively occupy a volume\[\left({b}/{\overline{N}}\right)\left(n\overline{N}\right)=nb. \nonumber \]Van der Waals’ equation says that the volume of a real gas is the volume that would be occupied by non-interacting point masses, \(V_{ideal\ gas}\), plus the effective volume of the gas molecules themselves. (When data for real gas molecules are fit to the van der Waals’ equation, the value of \(b\) is usually somewhat greater than the volume estimated from the liquid density and molecular weight. See problem 24.)Similarly, we have\[P_{\text{real gas}}=P_{\text{ideal gas}}-\frac{an^2}{V^2_{\text{real gas}}} \nonumber \]We can understand this as a logical consequence of attractive interactions between the molecules of the real gas. With \(a>0\), it says that the pressure of the real gas is less than the pressure of the hypothetical ideal gas, by an amount that is proportional to \({\left({n}/{V}\right)}^2\). The proportionality constant is \(a\). Since \({n}/{V}\) is the molar density (moles per unit volume) of the gas molecules, it is a measure of concentration. The number of collisions between molecules of the same kind is proportional to the square of their concentration. (We consider this point in more detail in Chapters 4 and 5.) So \({\left({n}/{V}\right)}^2\) is a measure of the frequency with which the real gas molecules come into close contact with one another. If they attract one another when they come close to one another, the effect of this attraction should be proportional to \({\left({n}/{V}\right)}^2\). So van der Waals’ equation is consistent with the idea that the pressure of a real gas is different from the pressure of the hypothetical ideal gas by an amount that is proportional to the frequency and strength of attractive interactions.But why should attractive interactions have this effect; why should the pressure of the real gas be less than that of the hypothetical ideal gas? Perhaps the best way to develop a qualitative picture is to recognize that attractive intermolecular forces tend to cause the gas molecules to clump up. After all, it is these attractive forcesattractive force that cause the molecules to aggregate to a liquid at low temperatures. Above the boiling point, the ability of gas molecules to go their separate ways limits the effects of this tendency; however, even in the gas, the attractive forces must act in a way that tends to reduce the volume occupied by the molecules. Since the volume occupied by the gas is dictated by the size of the container—not by the properties of the gas itself—this clumping-up tendency finds expression as a decrease in pressure.It is frequently useful to describe the interaction between particles or chemical moieties in terms of a potential energy versus distance diagram. The van der Waals’ equation corresponds to the case that the repulsive interaction between molecules is non-existent until the molecules come into contact. Once they come into contact, the energy required to move them still closer together becomes arbitrarily large. Often this is described by saying that they behave like “hard spheres”. The attractive force between two molecules decreases as the distance between them increases. When they are very far apart the attractive interaction is very small. We say that the energy of interaction is zero when the molecules are infinitely far apart. If we initially have two widely separated, stationary, mutually attracting molecules, they will spontaneously move toward one another, gaining kinetic energy as they go. Their potential energy decreases as they approach one another, reaching its smallest value when the molecules come into contact. Thus, van der Waals’ equation implies the potential energy versus distance diagram sketched in Figure 5.This page titled 2.12: Van der Waals' Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.13: Virial Equations

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It is often useful to fit accurate pressure-volume-temperature data to polynomial equations. The experimental data can be used to compute a quantity called the compressibility factor, \(Z\), which is defined as the pressure–volume product for the real gas divided by the pressure–volume product for an ideal gas at the same temperature.We have\[{\left(PV\right)}_{ideal\ gas}=nRT \nonumber \]Letting P and V represent the pressure and volume of the real gas, and introducing the molar volume, \(\overline{V}={V}/{n}\), we have\[Z=\frac{\left(PV\right)_{real\ gas}}{\left(PV\right)_{ideal\ gas}}=\frac{PV}{nRT}=\frac{P\overline{V}}{RT} \nonumber \]Since \(Z=1\) if the real gas behaves exactly like an ideal gas, experimental values of Z will tend toward unity under conditions in which the density of the real gas becomes low and its behavior approaches that of an ideal gas. At a given temperature, we can conveniently ensure that this condition is met by fitting the Z values to a polynomial in P or a polynomial in \({\overline{V}}^{-1}\). The coefficients are functions of temperature. If the data are fit to a polynomial in the pressure, the equation is\[Z=1+B^*\left(T\right)P+C^*\left(T\right)P^2+D^*\left(T\right)P^3+\dots \nonumber \]For a polynomial in \({\overline{V}}^{-1}\), the equation is\[Z=1+\frac{B\left(T\right)}{\overline{V}}+\frac{C\left(T\right)}{\overline{V}^2}+\frac{D\left(T\right)}{\overline{V}^3}+\dots \nonumber \]These empirical equations are called virial equations. As indicated, the parameters are functions of temperature. The values of \(B^*\left(T\right)\), \(C^*\left(T\right)\), \(D^*\left(T\right)\), …, and \(B\left(T\right)\), \(C\left(T\right)\), \(D\left(T\right)\),…, must be determined for each real gas at every temperature. (Note also that \(B^*\left(T\right)\neq B\left(T\right)\), \(C^*\left(T\right)\neq C\left(T\right)\), \(D^*\left(T\right)\neq D\left(T\right)\), etc. However, it is true that \(B^*={B}/{RT}\).) Values for these parameters are tabulated in various compilations of physical data. In these tabulations, \(B\left(T\right)\) and \(C\left(T\right)\) are called the second virial coefficient and third virial coefficient, respectively.This page titled 2.13: Virial Equations is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.14: Gas Mixtures - Dalton's Law of Partial Pressures

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Thus far, our discussion of the properties of a gas has implicitly assumed that the gas is pure. We turn our attention now to mixtures of gases—gas samples that contain molecules of more than one compound. Mixtures of gases are common, and it is important to understand their behavior in terms of the properties of the individual gases that make it up. The ideal-gas laws we have for mixtures are approximations. Fortunately, these approximations are often very good. When we think about it, this is not surprising. After all, the distinguishing feature of a gas is that its molecules do not interact with one another very much. Even if the gas is composed of molecules of different kinds, the unimportance of molecule—molecule interactions means that the properties of one kind of molecules should be nearly independent of the properties of the other kinds.Consider a sample of gas that contains a fixed number of moles of each of two or more compounds. This sample has a pressure, a volume, a temperature, and a specified composition. Evidently, the challenge here is to describe the pressure, volume, and temperature of the mixture in terms of measurable properties of the component compounds.There is no ambiguity about what we mean by the pressure, volume, and temperature of the mixture; we can measure these properties without difficulty. Given the nature of temperature, it is both reasonable and unambiguous to say that the temperature of the sample and the temperature of its components are the same. However, we cannot measure the pressure or volume of an individual component in the mixture. If we hope to describe the properties of the mixture in terms of properties of the components, we must first define some related quantities that we can measure. The concepts of a component partial pressure and a component partial volume meet this need.We define the partial pressure of a component of a gas mixture as the pressure exerted by the same number of moles of the pure component when present in the volume occupied by the mixture, \(V_{mixture}\), at the temperature of the mixture. In a mixture of \(n_A\) moles of component \(A\), \(n_B\) moles of component \(B\), etc., it is customary to designate the partial pressure of component \(A\) as \(P_A\). It is important to appreciate that the partial pressure of a real gas can only be determined by experiment.We define the partial volume of a component of a gas mixture as the volume occupied by the same number of moles of the pure component when the pressure is the same as the pressure of the mixture, \(P_{mixture}\), at the temperature of the mixture. In a mixture of components \(A\), \(B\), etc., it is customary to designate the partial volume of component \(A\) as \(V_A\). The partial volume of a real gas can only be determined by experiment.Dalton’s law of partial pressures asserts that the pressure of a mixture is equal to the sum of the partial pressures of its components. That is, for a mixture of components A, B, C, etc., the pressure of the mixture is\[P_{mixture}=P_A+P_B+P_C+\dots \label{Dalton} \]Under conditions in which the ideal gas law is a good approximation to the behavior of the individual components, Dalton’s law is usually a good approximation to the behavior of real gas mixtures. For mixtures of ideal gases, it is exact. To see this, we recognize that, for an ideal gas, the definition of partial pressure becomes\[P_A=\frac{n_ART}{V_{mixture}} \nonumber \]The ideal-gas mixture contains \(n_{mixture}=n_A+n_B+n_C+\dots \text{moles}\), so that\[ \begin{align*} P_{mixture} &=\frac{n_{mixture}RT}{V_{mixture}} \\[4pt] &=\frac{\left(n_A+n_B+n_C+\dots \right)RT}{V_{mixture}} \\[4pt] &=\frac{n_ART}{V_{mixture}}+\frac{n_BRT}{V_{mixture}}+\frac{n_CRT}{V_{mixture}}+\dots \\[4pt] &=P_A+P_B+P_C+\dots \end{align*} \]Applied to the mixture, the ideal-gas equation yields Dalton’s law (Equation \ref{Dalton}). When \(x_A\) is the mole fraction of A in a mixture of ideal gases,\[P_A=x_AP_{mixture}. \nonumber \]This page titled 2.14: Gas Mixtures - Dalton's Law of Partial Pressures is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.15: Gas Mixtures - Amagat's Law of Partial Volums

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Amagat’s law of partial volumes asserts that the volume of a mixture is equal to the sum of the partial volumes of its components. For a mixture of components \(A\), \(B\), \(C\), etc., Amagat’s law gives the volume as\[V_{mixture}=V_A+V_B+V_C+\dots \nonumber \]For real gases, Amagat’s law is usually an even better approximation than Dalton’s law\({}^{6}\). Again, for mixtures of ideal gases, it is exact. For an ideal gas, the partial volume is\[V_A=\frac{n_ART}{P_{mixture}} \nonumber \]Since \(n_{mixture}=n_A+n_B+n_C+\dots\), we have, for a mixture of ideal gases,\[ \begin{align*} V_{mixture}&=\frac{n_{mixture}RT}{P_{mixture}} \\[4pt] &=\frac{\left(n_A+n_B+n_C+\dots \right)RT}{P_{mixture}} \\[4pt] &=V_A+V_B+V_C+\dots \end{align*} \]Applied to the mixture, the ideal-gas equation yields Amagat’s law. Also, we have \(V_A=x_AV_{mixture}\).This page titled 2.15: Gas Mixtures - Amagat's Law of Partial Volums is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

2.16: Problems

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\[ \begin{array}{|c|c|c|c|} \hline \text{ Compound } & \text{ Mol Mass, g mol}^{–1} ~ & \text{ Density, g mL}^{–1} & \text{ Van der Waals } b, \text{ L mol}^{–1} \\ \hline \text{Acetic acid} & 60.05 & 1.0491 & 0.10680 \\ \hline \text{Acetone} & 58.08 & 0.7908 & 0.09940 \\ \hline \text{Acetonitrile} & 41.05 & 0.7856 & 0.11680 \\ \hline \text{Ammonia} & 17.03 & 0.7710 & 0.03707 \\ \hline \text{Aniline} & 93.13 & 1.0216 & 0.13690 \\ \hline \text{Benzene} & 78.11 & 0.8787 & 0.11540 \\ \hline \text{Benzonitrile} & 103.12 & 1.0102 & 0.17240 \\ \hline \text{iso-Butylbenzene } & 134.21 & 0.8621 & 0.21440 \\ \hline \text{Chlorine} & 70.91 & 3.2140 & 0.05622 \\ \hline \text{Durene} & 134.21 & 0.8380 & 0.24240 \\ \hline \text{Ethane} & 30.07 & 0.5720 & 0.06380 \\ \hline \text{Hydrogen chloride} & 36.46 & 1.1870 & 0.04081 \\ \hline \text{Mercury} & 200.59 & 13.5939 & 0.01696 \\ \hline \text{Methane} & 16.04 & 0.4150 & 0.04278 \\ \hline \text{Nitrogen dioxide} & 46.01 & 1.4494 & 0.04424 \\ \hline \text{Silicon tetrafluoride} & 104.08 & 1.6600 & 0.05571 \\ \hline \text{Water} & 18.02 & 1.0000 & 0.03049 \\ \hline \end{array} \nonumber \]Notes1We use the over-bar to indicate that the quantity is per mole of substance. Thus, we write \(\overline{N}\) to indicate the number of particles per mole. We write \(\overline{M}\) to represent the gram molar mass. In Chapter 14, we introduce the use of the over-bar to denote a partial molar quantity; this is consistent with the usage introduced here, but carries the further qualification that temperature and pressure are constant at specified values. We also use the over-bar to indicate the arithmetic average; such instances will be clear from the context.2The unit of temperature is named the kelvin, which is abbreviated as K.3A redefinition of the size of the unit of temperature, the kelvin, is under consideration. The practical effect will be inconsequential for any but the most exacting of measurements.4For a thorough discussion of the development of the concept of temperature, the evolution of our means to measure it, and the philosophical considerations involved, see Hasok Chang, Inventing Temperature, Oxford University Press, 2004.5See T. L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley Publishing Company, 1960, p 286.6See S. M. Blinder, Advanced Physical Chemistry, The Macmillan Company, Collier-Macmillan Canada, Ltd., Toronto, 1969, pp 185-189This page titled 2.16: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.1: The Independent-Molecule Approximation

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In Chapter 18, our survey of quantum mechanics introduces the idea that a molecule can have any of an infinite number of discrete energies, which we can put in order starting with the smallest. We now turn our attention to the properties of a system composed of a large number of molecules. This multi-molecule system must obey the laws of quantum mechanics. Therefore, there exists a Schrödinger equation, whose variables include all of the inter-nucleus, inter-electron, and electron-nucleus distance and potential terms in the entire multi-molecule system. The relevant boundary conditions apply at the physical boundaries of the macroscopic system. The solutions of this equation include a set of infinitely many wavefunctions, \({\mathit{\Psi}}_{i,j}\), each describing a quantum mechanical state of the entire multi-molecule system. In general, the collection of elementary particles that can be assembled into a particular multi-molecule system can also be assembled into many other multi-molecule systems. For example, an equimolar mixture of \(CO\) and \(H_2O\) can be reassembled into a system comprised of equimolar \(CO_2\) and \(H_2\), or into many other systems containing mixtures of \(CO\), \(H_2O\), \(CO_2\), and \(H_2\). Infinitely many quantum-mechanical states are available to each of these multi-molecule systems.For every such multi-molecule wavefunction, \({\psi }_{i,j}\), there is a corresponding system energy, \(E_i\). In general, the system energy, \(E_i\), is \({\mathit{\Omega}}_i\)-fold degenerate; there are \({\mathit{\Omega}}_i\) wavefunctions, \({\mathit{\Psi}}_{i,1}\), \({\mathit{\Psi}}_{i,2}\), …, \({\mathit{\Psi}}_{i,{\mathit{\Omega}}_i}\), whose energy is \(E_i\). The wavefunctions include all of the interactions among the molecules of the system, and the energy levels of the system reflect all of these interactions. While generating and solving this multi-molecule Schrödinger equation is straightforward in principle, it is completely impossible in practice.Fortunately, we can model multi-molecule systems in another way. The primary focus of chemistry is the study of the properties and reactions of molecules. Indeed, the science of chemistry exists, as we know it, only because the atoms comprising a molecule stick together more tenaciously than molecules stick to one another. (Where this is not true, we get macromolecular materials like metals, crystalline salts, etc.) This occurs because the energies that characterize the interactions of atoms within a molecule are much greater than the energies that characterize the interaction of one molecule with another. Consequently, the energy of the system can be viewed as the sum of two terms. One term is a sum of the energies that the component molecules would have if they were all infinitely far apart. The other term is a sum of the energies of all of the intermolecular interactions, which is the energy change that would occur if the molecules were brought from a state of infinite separation to the state of interest.In principle, we can describe a multi-molecule system in this way with complete accuracy. This description has the advantage that it breaks a very large and complex problem into two smaller problems, one of which we have already solved: In Chapter 18, we see that we can approximate the quantum-mechanical description of a molecule and its energy levels by factoring molecular motions into translational, rotational, vibrational, and electronic components. It remains only to describe the intermolecular interactions. When intramolecular energies are much greater than intermolecular-interaction energies, it may be a good approximation to ignore the intermolecular interactions altogether. This occurs when we describe ideal gas molecules; in the limit that a gas behaves ideally, the force between any two of its molecules is nil.In Chapter 23, we return to the idea of multi-molecule wavefunctions and energy levels. Meanwhile we assume that intermolecular interactions can be ignored. This is a poor approximation for many systems. However, it is a good approximation for many others, and it enables us to keep our description of the system simple while we use molecular properties in our development of the essential ideas of statistical thermodynamics.We focus on developing a theory that gives the macroscopic thermodynamic properties of a pure substance in terms of the energy levels available to its individual molecules. To begin, we suppose that we solve the Schrödinger equation for an isolated molecule. In this Schrödinger equation, the variables include the inter-nucleus, inter-electron, and electron-nucleus distance and potential terms that are necessary to describe the molecule. The solutions are a set of infinitely many wavefunctions, \({\psi }_{i,j}\), each describing a different quantum-mechanical state of an isolated molecule. We refer to each of the possible wavefunctions as quantum state of the molecule. For every such wavefunction, there is a corresponding molecular energy, \({\epsilon }_i\). Every unique molecular energy, \({\epsilon }_i\), is called an energy level. Several quantum states can have the same energy. When two or more quantum states have the same energy, we say that they belong to the same energy level, and the energy level is said to be degenerate. In general, there are \(g_i\) quantum states that we can represent by the \(g_i\) wavefunctions, \({\psi }_{i,1}\), , \({\psi }_{i,2}\), ..., , \({\psi }_{i,g_i}\), each of whose energy is \({\epsilon }_i\). The number of quantum states that have the same energy is called the degeneracy of the energy level. illustrates the terms we use to describe the quantum states and energy levels available to a molecule.In our development of classical thermodynamics, we find it convenient to express the value of a thermodynamic property of a pure substance as the change that occurs during a formal process that forms one mole of the substance, in its standard state, from its unmixed constituent elements, in their standard states. In developing statistical thermodynamics, we find it convenient to express the value of a molecular energy, \({\epsilon }_i\), as the change that occurs during a formal process that forms a molecule of the substance, in one of its quantum states, \({\psi }_{i,j}\), from its infinitely separated, stationary, constituent atoms. That is, we let the isolated constituent atoms be the reference state for the thermodynamic properties of a pure substance.This page titled 20.1: The Independent-Molecule Approximation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.2: The Probability of An Energy Level at Constant N, V, and T

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If only pressure–volume work is possible, the state of a closed, reversible system can be specified by specifying its volume and temperature. Since the system is closed, the number, \(N\), of molecules is constant. Let us consider a closed, equilibrated, constant-volume, constant-temperature system in which the total number of molecules is very large. Let us imagine that we can monitor the quantum state of one particular molecule over a very long time. Eventually, we are able to calculate the fraction of the elapsed time that the molecule spends in each of the quantum states. We label the available quantum states with the wavefunction symbols, \({\psi }_{i,j}\).We assume that the fraction of the time that a molecule spends in the quantum state \({\psi }_{i,j}\) is the same thing as the probability of finding the molecule in quantum state \({\psi }_{i,j}\). We denote this probability as \(\rho \left({\psi }_{i,j}\right)\). To develop the theory of statistical thermodynamics, we assume that this probability depends on the energy, and only on the energy, of the quantum state \({\psi }_{i,j}\). Consequently, any two quantum states whose energies are the same have the same probability, and the \(g_i\)-fold degenerate quantum states, \({\psi }_{i,j}\), whose energies are \({\epsilon }_i\), all have the same probability. In our imaginary monitoring of the state of a particular molecule, we observe that the probabilities of two quantum states are the same if and only if their energies are the same; that is, we observe \(\rho \left({\psi }_{i,j}\right)=\mathrm{\ }\rho \left({\psi }_{k,m}\right)\) if and only if \(i=k\).The justification for this assumption is that the resulting theory successfully models experimental observations. We can ask, however, why we might be led to make this assumption in the first place. We can reason as follows: The fact that we observe a definite value for the energy of the macroscopic system implies that quantum states whose energies are much greater than the average molecular energy must be less probable than quantum states whose energies are smaller. Otherwise, the sum of the energies of high-energy molecules would exceed the energy of the system. Therefore, we can reasonably infer that the probability of a quantum state depends on its energy. On the other hand, we can think of no plausible reason for a given molecule to prefer one quantum state to another quantum state that has the same energy.This assumption means that a single function suffices to specify the probability of finding a given molecule in any quantum state, \({\psi }_{i,j}\), and the only independent variable is the quantum-state energy, \({\epsilon }_i\). We denote the probability of a single quantum state, \({\psi }_{i,j}\), whose energy is \({\epsilon }_i\), as \(\rho \left({\epsilon }_i\right)\). Since this is the probability of each of the \(g_i\)-fold degenerate quantum states, \({\psi }_{i,j}\), that have energy \({\epsilon }_i\), the probability of finding a given molecule in any energy level, \({\epsilon }_i\), is \(P\left({\epsilon }_i\right)=g_i\rho \left({\epsilon }_i\right)\). We find it convenient to introduce “\(P_i\)” to abbreviate this probability; that is, we let\[P_i=\sum^{g_i}_{j=1}{\rho \left({\psi }_{i,j}\right)}=P\left({\epsilon }_i\right)=g_i\rho \left({\epsilon }_i\right) \nonumber \](the probability of energy level \(\mathrm{\epsilon}_{\mathrm{i}}\))There is a \(P_i\) for every energy level \({\epsilon }_i\). \(P_i\) must be the same for any molecule, since every molecule has the same properties. If the population set \(\{N^{\textrm{⦁}}_1,\ N^{\textrm{⦁}}_2,\dots ,N^{\textrm{⦁}}_i,\dots \}\) characterizes the equilibrium system, the fraction of the molecules that have energy \({\epsilon }_i\) is \({N^{\textrm{⦁}}_i}/{N}\). (Elsewhere, an energy-level population set is often called a “distribution.” Since we define a distribution somewhat differently, we avoid this usage.) Since the fraction of the molecules in an energy level at any instant of time is the same as the fraction of the time that one molecule spends in that energy level, we have\[P_i=P\left({\epsilon }_i\right)=g_i\rho \left({\epsilon }_i\right)=\frac{N^{\textrm{⦁}}_i}{N} \nonumber \]As long as the system is at equilibrium, this fraction is constant. In Chapter 21, we find an explicit equation for the probability function, \(\rho \left({\epsilon }_i\right)\).The energy levels, \({\epsilon }_i\), depend on the properties of the molecules. In developing Boltzmann statistics for non-interacting molecules, we assume that the probability of finding a molecule in a particular energy level is independent of the number of molecules present in the system. While \(P_i\) and \(\rho \left({\epsilon }_i\right)\) depend on the energy level, \({\epsilon }_i\), neither depends on the number of molecules, \(N\). If we imagine inserting a barrier that converts an equilibrated collection of molecules into two half-size collections, each of the new collections is still at equilibrium. Each contains half as many molecules and has half the total energy of the original. In our model, the fraction of the molecules in any given energy level remains constant. Consequently, the probabilities associated with each energy level remain constant. (In Chapter 25, we introduce Fermi-Dirac and Bose-Einstein statistics. When we must use either of these models to describe the system, \(P_i\) is affected by rules for the number of molecules that can occupy an energy level.)The number of molecules and the total energy are extensive properties and vary in direct proportion to the size of the system. The probability, \(P_i\), is an intensive variable that is a characteristic property of the macroscopic system. \(P_i\) is a state function. \(P_i\) depends on \({\epsilon }_i\). So long as the thermodynamic variables that determine the state of the system remain constant, the \({\epsilon }_i\) are constant. For a given macroscopic system in which only pressure–volume work is possible, the quantum mechanical energy levels, \({\epsilon }_i\), are constant so long as the system volume and temperature are constant. However, the \({\epsilon }_i\) are quantum-mechanical quantities that depend on our specification of the molecule and on the boundary values in our specification of the system. If we change any molecular properties or the dimensions of the system, the probabilities, \(P_i\), change.This page titled 20.2: The Probability of An Energy Level at Constant N, V, and T is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.3: The Population Sets of a System at Equilibrium at Constant N, V, and T

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In developing Boltzmann statistics, we assume that we can tell different molecules of the same substance apart. We say that the molecules are distinguishable. This assumption is valid for molecules that occupy lattice sites in a crystal. In a crystal, we can specify a particular molecule by specifying its position in the lattice. In other systems, we may be unable to distinguish between different molecules of the same substance. Most notably, we cannot distinguish between two molecules of the same substance in the gas phase. The fact that gas molecules are indistinguishable, while we assume otherwise in developing Boltzmann statistics, turns out to be a problem that is readily overcome. We discuss this in Section 24.2.We want to model properties of a system that contains \(N\), identical, distinguishable, non-interacting molecules. The solutions of the Schrödinger equation presume fixed boundary conditions. This means that the volume of this \(N\)-molecule system is constant. We assume also that the temperature of the \(N\)-molecule system is constant. Thus, our goal is a theory that predicts the properties of a system when \(N\), \(V\), and \(T\) are specified. When there are no intermolecular interactions, the energy of the system is just the sum of the energies of the individual molecules. If we know how the molecules are allocated among the energy levels, we can find the energy of the system. Letting \(N_i\) be the population of the energy level \({\epsilon }_i\), any such allocation is a population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). We have\[N=\sum^{\infty }_{i=1}{N_i} \nonumber \]and the system energy is\[E=\sum^{\infty }_{i=1}{N_i}{\epsilon }_i \nonumber \]Let us imagine that we can assemble a system with the molecules allocated among the energy levels in any way we please. Let \(\{N^o_1,\ N^o_2,\dots ,N^o_i,\dots \}\) represent an initial population set that describes a system that we assemble in this way. This population set corresponds to a well-defined system energy. We imagine immersing the container in a constant-temperature bath. Since the system can exchange energy with the bath, the molecules of the system gain or lose energy until the system attains the temperature of the bath in which it is immersed. As this occurs, the populations of the energy levels change. A series of different population sets characterizes the state of the system as it evolves toward thermal equilibrium. When the system reaches equilibrium, the population sets that characterize it are different from the initial one, \(\{N^o_1,\ N^o_2,\dots ,N^o_i,\dots \}\).Evidently, the macroscopic properties of such a system also change with time. The changes in the macroscopic properties of the system parallel the changing energy-level populations. At thermal equilibrium, macroscopic properties of the system cease to undergo any further change. In Section 3.9, we introduce the idea that the most probable population set, which we denote as\[\left\{N^{\textrm{⦁}}_1,\ N^{\textrm{⦁}}_2,\dots ,N^{\textrm{⦁}}_i,\dots \right\} \nonumber \]or its proxy,\[\left\{NP\left({\epsilon }_1\right),NP\left({\epsilon }_2\right),\dots ,NP\left({\epsilon }_i\right),\dots \right\} \nonumber \](where \(N=N^{\textrm{⦁}}_1+N^{\textrm{⦁}}_2+...+N^{\textrm{⦁}}_i+...\)), is the best prediction we can make about the outcomes in a future set of experiments in which we find the energy of each of \(N\) different molecules at a particular instant. We hypothesize that the most probable population set specifies all of the properties of the macroscopic system in its equilibrium state. When we develop the logical consequences of this hypothesis, we find a theory that expresses macroscopic thermodynamic properties in terms of the energy levels available to individual molecules. In the end, the justification of this hypothesis is that it enables us to calculate thermodynamic properties that agree with experimental measurements made on macroscopic systems.Our hypothesis asserts that the properties of the equilibrium state are the same as the properties of the system when it is described by the most probable population set. Evidently, we can predict the system’s equilibrium state if we can find the equilibrium \(N^{\textrm{⦁}}_i\) values, and vice versa. To within an arbitrary factor representing its size, an equilibrated system can be completely described by its intensive properties. In the present instance, the fractions \({N^{\textrm{⦁}}_1}/{N}\), \({N^{\textrm{⦁}}_2}/{N}\), …, \({N^{\textrm{⦁}}_i}/{N},\dots\) describe the equilibrated system to within the factor, \(N\), that specifies its size. Since we infer that \(P_i=P\left({\epsilon }_i\right)={N^{\textrm{⦁}}_i}/{N}\), the equilibrated system is also described by the probabilities \(\left(P_1,P_2,\dots ,\ P_i,\dots \right)\).Our hypothesis does not assert that the most-probable population set is the only population set possible at equilibrium. A very large number of other population sets may describe an equilibrium system at different instants of time. However, when its state is specified by any such population set, the macroscopic properties of the system are indistinguishable from the macroscopic properties of the system when its state is specified by the most-probable population set. The most-probable population set characterizes the equilibrium state of the system in the sense that we can calculate the properties of the equilibrium state of the macroscopic system by using the single-molecule energy levels and the most probable population set—or its proxy. The relationship between a molecular energy level, \({\epsilon }_i\), and its equilibrium population, \(N^{\textrm{⦁}}_i\), is called the Boltzmann equation. From \(P_i={N^{\textrm{⦁}}_i}/{N}\), we see that the Boltzmann equation specifies the probability of finding a given molecule in energy level \({\epsilon }_i\).Although we calculate thermodynamic properties from the most probable population set, the population set that describes the system can vary from instant to instant while the system remains at equilibrium. The central limit theorem enables us to characterize the amount of variation that can occur. When \(N\) is comparable to the number of molecules in a macroscopic system, the probability that variation among population sets can result in a macroscopically observable effect is vanishingly small. The hypothesis is successful because the most probable population set is an excellent proxy for any other population set that the equilibrium system is remotely likely to attain.We develop the theory of statistical thermodynamics for \(N\)-molecule systems by considering the energy levels, \({\epsilon }_i\), available to a single molecule that does not interact with other molecules. Thereafter, we develop a parallel set of statistical thermodynamic results by considering the energy levels, \({\hat{E}}_i\), available to a system of \(N\) molecules. These \(N\)-molecule-system energies can reflect the effects of any amount of intermolecular interaction. We can apply the same arguments to find that the Boltzmann equation also describes the equilibrium properties of systems in which intermolecular interactions are important. That is, the probability, \(P_i\left({\hat{E}}_i\right)\), that an \(N\)-molecule system has energy \({\hat{E}}_i\) is the same function of \({\hat{E}}_i\) as the molecular-energy probability, \(P_i=P\left({\epsilon }_i\right)\), is of \({\epsilon }_i\).When we finish our development based on single-molecule energy levels, we understand nearly all of the ideas that we need in order to complete the development for the energies of an \(N\)-molecule system. This development is an elegant augmentation of the basic argument called the ensemble treatment or the ensemble method. The ensemble treatment is due to J. Willard Gibbs; we discuss it in Chapter 23. For now, we simply note that our approach involves no wasted effort. When we discuss the ensemble method, we use all of the ideas that we develop in this chapter and the next. The extension of these arguments that is required for the ensemble treatment is so straightforward as to be (almost) painless.This page titled 20.3: The Population Sets of a System at Equilibrium at Constant N, V, and T is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.4: How can Infinitely Many Probabilities Sum to Unity?

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There are an infinite number of successively greater energies for a quantum mechanical system. We infer that the probability that a given energy level is occupied is a property of the energy level. Each of the probabilities must be between 0 and 1. When we sum the fixed probabilities associated with the energy levels, the sum contains an infinite number of terms. By the nature of probability, the sum of this infinite number of terms must be one:\[\begin{align*} 1 &=P_1+P_2+\dots +P_i+\dots \\[4pt] &=P\left({\epsilon }_1\right)+P\left({\epsilon }_2\right)+\dots +P\left({\epsilon }_i\right)+\dots \\[4pt] &=\sum^{\infty }_{i=1}{P\left({\epsilon }_i\right)} \end{align*} \]That is, the sum of the probabilities is an infinite series, which must converge: The sum of all of the occupancy probabilities must be unity. This can happen only if all later members of the series are very small. In the remainder of this chapter, we explore some of the thermodynamic ramifications of these facts. In the next chapter, we use this relationship to find the functional dependence of the \(P_i\) on the energy levels, \({\epsilon }_i\). To obtain these results, we need to think further about the probabilities associated with the various population sets that can occur. Also, we need to introduce a new fundamental postulate.To focus on the implications of this sum of probabilities, let us review geometric series. A geometric series is a sum of terms, in which each successive term is a multiple of its predecessor. A geometric series is an infinite sum that can converge:\[T=a+ar+ar^2+\dots +ar^i\dots =a\left(1+r+r^2+\dots +r^i+\dots \right)=a+a\sum^{\infty }_{i=1}{r^i} \nonumber \]Successive terms approach zero if \(\left|r\right|<1\). If \(\left|r\right|\ge 1\), successive terms do not become smaller, and the sum does not have a finite limit. If \(\left|r\right|\ge 1\), we say that the infinite series diverges.We can multiply an infinite geometric series by its constant factor to obtain\[ \begin{align*} rT &=ar+ar^2+ar^3+\dots +ar^i+\dots \\[4pt] &=a\left(r+r^2+r^3+\dots +r^i+\dots \right) \\[4pt] &=a\sum^{\infty }_{i=1}{r^i} \end{align*} \]If \(\left|r\right|<1\), we can subtract and find the value of the infinite sum: \[T-rT=a \nonumber \] so that \[T={a}/{\left(1-r\right)} \nonumber \]In a geometric series, the ratio of two successive terms is \({r^{n+1}}/{r^n}=r\) The condition of convergence for a geometric series can also be written as \[\left|\frac{r^{n+1}}{r^n}\right|<1 \nonumber \]We might anticipate that any other series also converges if its successive terms become smaller at least as fast as those of a geometric series. In fact, this is true and is the basis for the ratio test for convergence of an infinite series. If we represent successive terms in an infinite series as \(t_i\), their sum is \[T=\sum^{\infty }_{i=0}{t_i} \nonumber \]The ratio test is a theorem which states that the series converges, and \(T\) has a finite value, if\[{\mathop{\mathrm{lim}}_{n\to \infty } \left|\frac{t_{n+1}}{t_n}\right|<1\ } \nonumber \]One of our goals is to discover the relationship between the energy, \({\epsilon }_i\), of a quantum state and the probability that a molecule will occupy one of the quantum states that have this energy, \(P_i=g_i\rho \left({\epsilon }_i\right)\). When we do so, we find that the probabilities for all of the quantum mechanical systems that we discuss in Chapter 18 satisfy the ratio test.This page titled 20.4: How can Infinitely Many Probabilities Sum to Unity? is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.5: The Total Probability Sum at Constant N, V, and T

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In a collection of distinguishable independent molecules at constant \(N\), \(V\), and \(T\), the probability that a randomly selected molecule has energy \({\epsilon }_i\) is \(P_i\); we have \(1=P_1+P_2+\dots +P_i+\dots\). At any instant, every molecule in the \(N\)-molecule system has a specific energy, and the state of the system is described by a population set, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), wherein \(N_i\) can have any value in the range \(0\le N_i\le N\), subject to the condition that\[N=\sum^{\infty }_{i=1}{N_i} \nonumber \]The probabilities that we assume for this system of molecules have the properties we assume in Chapter 19 where we find the total probability sum by raising the sum of the energy-level probabilities to the \(N^{th}\) power.\[1={\left(P_1+P_2+\dots +P_i+\dots \right)}^N=\sum_{\{N_i\}}{\frac{N!}{N_1!N_2!\dots N_i!\dots }}P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots \nonumber \]The total-probability sum is over all possible population sets, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), which we abbreviate to \(\{N_i\}\), in indicating the range of the summation. Each term in this sum represents the probability of the corresponding population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\),. At any given instant, one of the possible population sets describes the way that the molecules of the physical system are apportioned among the energy levels. The corresponding term in the total probability sum represents the probability of this apportionment. It is not necessary that all of the energy levels be occupied. We can have \(N_k=0\), in which case \(P^{N_k}_k=P^0_k=1\) and \(N_k!=1\). Energy levels that are not occupied have no effect on the probability of a population set. The unique population set\[\{N^{\textrm{⦁}}_1,\ N^{\textrm{⦁}}_2,\dots ,N^{\textrm{⦁}}_i,\dots .\} \nonumber \]that we conjecture to characterize the equilibrium state is represented by one of the terms in this total probability sum. We want to focus on the relationship between a term in the total probability sum and the corresponding state of the physical system.Each term in the total probability sum includes a probability factor, \(P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots\) This factor is the probability that \(N_i\) molecules occupy each of the energy levels \({\epsilon }_i\). This term is not affected by our assumption that the molecules are distinguishable. The probability factor is multiplied by the polynomial coefficient\[\frac{N!}{N_1!N_2!\dots N_i!\dots } \nonumber \]This factor is the number of combinations of distinguishable molecules that arise from the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). It is the number of ways that the \(N\) distinguishable molecules can be assigned to the available energy levels so that \(N_1\) of them are in energy level, \({\epsilon }_1\), etc.The combinations for the population set {3,2} are shown in =\frac{N!}{N_1!N_2!\dots N_i!\dots } \nonumber \]Because there are infinitely many energy levels and probabilities, \(P_i\), there are infinitely many terms in the total-probability sum. Every energy available to the macroscopic system is represented by one or more terms in this total-probability sum. Since there is no restriction on the energy levels that can be occupied, there are an infinite number of such system energies. There is an enormously large number of terms each of which corresponds to an enormously large system energy. Nevertheless, the sum of all of these terms must be one. The \(P_i\) form a convergent series, and the total probability sum must sum to unity.Just as the \(P_i\) series can converge only if the probabilities of high molecular energies become very small, so the total probability sum can converge only if the probabilities of high system energies become very small. If a population set has \(N_i\) molecules in the \(i^{th}\) energy level, the probability of that population set is proportional to \(P^{N_i}_i\). We see therefore, that the probability of a population set in which there are many molecules in high energy levels must be very small. Terms in the total probability sum that correspond to population sets with many molecules in high energy levels must be negligible. Equivalently, at a particular temperature, macroscopic states in which the system energy is anomalously great must be exceedingly improbable.What terms in the total probability sum do we need to consider? Evidently from among the infinitely many terms that occur, we can select a finite subset whose sum is very nearly one. If there are many terms that are small and nearly equal to one another, the number of terms in this finite subset could be large. Nevertheless, we can see that terms in this subset must involve the largest possible \(P_i\) values raised to the smallest possible powers, \(N_i\), consistent with the requirement that the \(N_i\) sum to \(N\).If an equilibrium macroscopic system could have only one population set, the probability of that population set would be unity. Could an equilibrium system be characterized by two or more population sets for appreciable fractions of an observation period? Would this require that the macroscopic system change its properties with time as it jumps from one population set to another? Evidently, it would not, since our observations of macroscopic systems show that the equilibrium properties are unique. A system that wanders between two (or more) macroscopically distinguishable states cannot be at equilibrium. We are forced to the conclusion that, if a macroscopic equilibrium system has multiple population sets with non-negligible probabilities, the macroscopic properties associated with each of these population sets must be indistinguishably similar. (The alternative is to abandon the theory, which is useful only if its microscopic description of a system makes useful predictions about the system’s macroscopic behavior.)To be a bit more precise about this, we recognize that our theory also rests on another premise: Any intensive macroscopic property of many independent molecules depends on the energy levels available to an individual molecule and the fraction of the molecules that populate each energy level. The average energy is a prime example. For the population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), the average molecular energy is\[\overline{\epsilon }=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}{\epsilon }_i \nonumber \]We recognize that many population sets may contribute to the total probability sum at equilibrium. If we calculate essentially the same \(\overline{\epsilon }\) from each of these contributing population sets, then all of the contributing population sets correspond to indistinguishably different macroscopic energies. We see in the next section that the central limit theorem guarantees that this happens whenever \(N\) is as large as the number of molecules in a macroscopic system.This page titled 20.5: The Total Probability Sum at Constant N, V, and T is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.6: The Most Probable Population Set at Constant N, V, and T

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We are imagining that we can examine a collection of \(N\) distinguishable molecules and determine the energy of each molecule in the collection at any particular instant. If we do so, we find the population set, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), that characterizes the system at that instant. In Section 3.9, we introduce the idea that the most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,.,,,\}\), or its proxy, \(\{NP\left({\epsilon }_1\right),NP\left({\epsilon }_2\right),\dots ,NP\left({\epsilon }_i\right),\dots .\}\), is the best prediction we can make about the outcome of a future replication of this measurement. In Section 20.2, we hypothesize that the properties of the system when it is characterized by the most probable population set are indistinguishable from the properties of the system at equilibrium.Now let us show that this hypothesis is implied by the central limit theorem. We suppose that the population set that characterizes the system varies from instant to instant and that we can find this population set at any given instant. The population set that we find at a particular instant comprises a random sample of \(N\) molecular energies. For this sample, we can find the average energy from\[\overline{\epsilon }=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}{\epsilon }_i \nonumber \]The expected value of the molecular energy is \[\left\langle \epsilon \right\rangle =\sum^{\infty }_{i=1}{P_i{\epsilon }_i} \nonumber \]It is important that we remember that \(\overline{\epsilon }\) and \(\left\langle \epsilon \right\rangle\) are not the same thing. There is a distribution of \(\overline{\epsilon }\) values, one \(\overline{\epsilon }\) value for each of the possible population sets \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\). In contrast, when \(N\), \(V\), and \(T\) are fixed, the expected value, \(\left\langle \epsilon \right\rangle\), is a constant; the value of \(\left\langle \epsilon \right\rangle\) is completely determined by the values of the variables that determine the state of the system and fix the probabilities \(P_i\). If our theory is to be useful, the value of \(\left\langle \epsilon \right\rangle\) must be the per-molecule energy that we observe for the macroscopic system we are modeling.According to the central limit theorem, the average energy of a randomly selected sample, \(\overline{\epsilon }\), approaches the expected value for the distribution, \(\left\langle \epsilon \right\rangle\), as the number of molecules in the sample becomes arbitrarily large. In the present instance, we hypothesize that the most probable population set, or its proxy, characterizes the equilibrium system. When \(N\) is sufficiently large, this hypothesis implies that the probability of the \(i^{th}\) energy level is given by \(P_i={N^{\textrm{⦁}}_i}/{N}\). Then the expected value of a molecular energy is\[\left\langle \epsilon \right\rangle =\sum^{\infty }_{i=1}{P_i{\epsilon }_i}=\sum^{\infty }_{i=1}{\left(\frac{N^{\textrm{⦁}}_i}{N}\right){\epsilon }_i} \nonumber \]Since the central limit theorem asserts that \(\overline{\epsilon }\) approaches \(\left\langle \epsilon \right\rangle\) as \(N\) becomes arbitrarily large:\[0={\mathop{\mathrm{lim}}_{N\to \infty } \left(\overline{\epsilon }-\left\langle \epsilon \right\rangle \ \right)\ }={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-P_i\right)}\ }{\varepsilon }_i={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-\frac{N^{\textrm{⦁}}_i}{N}\right)}{\epsilon }_i\ } \nonumber \]One way for the limit of this sum to be zero is for the limit of every individual term to be zero. If the \({\epsilon }_i\) were arbitrary, this would be the only way that the sum could always be zero. However, the \({\epsilon }_i\) and the \(P_i\) are related, so we might think that the sum is zero because of these relationships.To see that the limit of every individual term must in fact be zero, we devise a new distribution. We assign a completely arbitrary number, \(X_i\), to each energy level. Now the \(i^{th}\) energy level is associated with an \(X_i\) as well as an \({\epsilon }_i\). We have an \(X\) distribution as well as an energy distribution. We can immediately calculate the expected value of \(X\). It is\[\left\langle X\right\rangle =\sum^{\infty }_{i=1}{P_iX_i} \nonumber \]When we find the population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), we can calculate the corresponding average value of \(X\). It is \[\overline{X}=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}X_i \nonumber \]The central limit theorem applies to any distribution. So, it certainly applies to the \(X\) distribution; the average value of \(X\) approaches the expected value of \(X\) as \(N\) becomes arbitrarily large:\[0={\mathop{\mathrm{lim}}_{N\to \infty } \left(\overline{X}-\left\langle X\right\rangle \ \right)\ }={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-P_i\right)}\ }X_i={\mathop{\mathrm{lim}}_{N\to \infty } \sum^{\infty }_{i=1}{\left(\frac{N_i}{N}-\frac{N^{\textrm{⦁}}_i}{N}\right)}X_i\ } \nonumber \]Now, because the \(X_i\) can be chosen completely arbitrarily, the only way that the limit of this sum can always be zero is that every individual term becomes zero.In the limit as \(N\to \infty\), we find that\[{N_i}/{N}\to {N^{\textrm{⦁}}_i}/{N} \nonumber \]As the number of molecules in the equilibrium system becomes arbitrarily large, the fraction of the molecules in each energy level at an arbitrarily selected instant approaches the fraction in that energy level in the equilibrium-characterizing most-probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i\dots \}\). In other words, the only population sets that we have any significant chance of observing in a large equilibrium system are population sets whose occupation fractions, \({N_i}/{N}\), are all very close to those, \({N^{\textrm{⦁}}_i}/{N}\), in the equilibrium-characterizing population set. Estimating \(P_i\) as the ratio \({N_i}/{N}\) gives essentially the same result whichever of these population sets we use. Below, we see that the \({\epsilon }_i\) and the \(P_i\) determine the thermodynamic properties of the system. Consequently, when we calculate any observable property of the macroscopic system, each of these population sets gives the same result.Since the only population sets that we have a significant chance of observing are those for which\[{N_i}/{N}\approx {N^{\textrm{⦁}}_i}/{N} \nonumber \]we frequently say that we can ignore all but the most probable population set. What we have in mind is that the most probable population set is the only one we need in order to calculate the macroscopic properties of the equilibrium system. We are incorrect, however, if we allow ourselves to think that the most probable population set is necessarily much more probable than any of the others. Nor does the fact that the \({N_i}/{N}\) are all very close to the \({N^{\textrm{⦁}}_i}/{N}\) mean that the \(N_i\) are all very close to the \(N^{\textrm{⦁}}_i\). Suppose that the difference between the two ratios is \({10}^{-10}\). If \(N={10}^{20}\), the difference between \(N_i\) and \(N^{\textrm{⦁}}_i\) is \({10}^{10}\), which probably falls outside the range of values that we usually understand by the words “very close.”We develop a theory that includes a mathematical model for the probability that a molecule has any one of its quantum-mechanically possible energies. It turns out that we are frequently interested in macroscopic systems in which the number of energy levels greatly exceeds the number of molecules. For such systems, we find \(NP_i\ll 1\), and it is no longer possible to say that a single most-probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,\dots \}\), describes the equilibrium state of the system. When it is very unlikely that any energy level is occupied by more than one molecule, the probability of any population set in which any \(N_i\) is greater than one becomes negligibly small. We can approximate the total probability sum as\[1={\left(P_1+P_2+\dots +P_i+\dots \right)}^N\approx \sum_{\{N_i\}}{N!}P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots \nonumber \]However, the idea that the proxy, \(\{NP\left({\epsilon }_1\right),NP\left({\epsilon }_2\right),\dots ,NP\left({\epsilon }_i\right),\dots .\}\), describes the equilibrium state of the system remains valid. In these circumstances, a great many population sets can have essentially identical properties; the properties calculated from any of these are indistinguishable from each other and indistinguishable from the properties calculated from the proxy. Since the equilibrium properties are fixed, the value of these extended products is fixed. For any of the population sets available to such a system at equilibrium, we have\[P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots =P^{{NP}_1}_1P^{{NP}_2}_2\dots P^{{NP}_i}_i\dots =\mathrm{constant} \nonumber \]It follows that, for some constant, \(c\), we have\[c=\sum^{\infty }_{i=1}{NP_i{ \ln P_i\ }}=N\sum^{\infty }_{i=1}{P_i{ \ln P_i\ }} \nonumber \]As it evolves, we see that the probability of finding a molecule in an energy level is the central feature of our theory.This page titled 20.6: The Most Probable Population Set at Constant N, V, and T is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.7: The Microstates of a Given Population Set

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/20%3A_Boltzmann_Statistics/20.07%3A_The_Microstates_of_a_Given_Population_Set

Thus far, we have considered only the probabilities associated with the assignments of distinguishable molecules to the allowed energy levels. In Section 20.2, we introduce the hypothesis that all of the \(g_i\) degenerate quantum states with energy \(\epsilon_i\) are equally probable, so that the probability that a molecule has energy \(\epsilon _i\) is \(P_i=P\left(\epsilon _i\right)=g_i\rho \left(\epsilon _i\right)\). Making this substitution, the total probability sum becomes\[\begin{align*} 1&=\left(P_1+P_2+\dots +P_i+\dots \right)^N \\[4pt] &=\sum_{\left\{N_i\right\}}{\frac{N!}{N_1!N_2!\dots N_i!\dots }}P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots \\[4pt] &=\sum_{\left\{N_i\right\}}{\frac{N!g^{N_1}_1g^{N_2}_2\dots g^{N_i}_i\dots }{N_1!N_2!\dots N_i!\dots }}{\rho \left(\epsilon _1\right)}^{N_1}{\rho \left(\epsilon _2\right)}^{N_2}\dots {\rho \left(\epsilon _i\right)}^{N_i}\dots . \\[4pt] &=\sum_{\left\{N_i\right\}}{N!} \prod^{\infty}_{i=1} \left(\frac{g^{N_i}_i}{N_i!}\right) \rho \left(\epsilon_i\right)^{N_i} \\[4pt] &=\sum_{\left\{N_i\right\}} W\prod^{\infty }_{i=1}\rho \left(\epsilon _i\right)^{N_i} \end{align*} \]where we use the notation\[a_1\times a_2\times \dots a_i\times \dots a_{\omega }\times =\prod^{\omega }_{i=1}{a_i} \nonumber \]for extended products and introduce the function\[\begin{align*} W &= W\left(N_i,g_i\right) \\[4pt] &=W\left(N_1,g_1,N_2,g_2,\dots ,N_i,g_i,\dots .\right) \\[4pt] &=N!\prod^{\infty }_{i=1}{\left(\frac{g^{N_i}_i}{N_i!}\right)} \\[4pt] &= C\left(N_1,N_2,\dots ,N_i,\dots \right)\prod^{\infty }_{i=1}{g^{N_i}_i} \end{align*} \]For reasons that become clear later, \(W\) is traditionally called the thermodynamic probability. This name is somewhat unfortunate, because \(W\) is distinctly different from an ordinary probability.In Section 20.5, we note that \(P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\) is the probability that \(N_i\) molecules occupy each of the energy levels \(\epsilon _i\) and that \({N!}/{\left(N_1!N_2!\dots N_i!\dots \right)}\) is the number of combinations of distinguishable molecules that arise from the population set \(\{N_1,N_2,\dots ,N_i,\dots \}\). Now we observe that the extended product\[{\rho \left(\epsilon _1\right)}^{N_1}{\rho \left(\epsilon _2\right)}^{N_2}\dots {\rho \left(\epsilon _i\right)}^{N_i}\dots . \nonumber \]is the probability of any one assignment of the distinguishable molecules to quantum states such that \(N_i\) molecules are in quantum states whose energies are \(\epsilon _i\). Since a given molecule of energy \(\epsilon _i\) can be in any of the \(g_i\) degenerate quantum states, the probability that it is in the energy level \(\epsilon _i\) is \(g_i\)-fold greater that the probability that it is in any one of these quantum states.We call a particular assignment of distinguishable molecules to the available quantum states a microstate. For any population set, there are many combinations. When energy levels are degenerate, each combination gives rise to many microstates. The factor \({\rho \left(\epsilon _1\right)}^{N_1}{\rho \left(\epsilon _2\right)}^{N_2}\dots {\rho \left(\epsilon _i\right)}^{N_i}\dots .\) is the probability of any one microstate of the population set \(\{N_1,N_2,\dots ,N_i,\dots \}\). Evidently, the thermodynamic probability\[W=N!\prod^{\infty }_{i=1}{\left(\frac{g^{N_i}_i}{N_i!}\right)} \label{micro} \]is the total number of microstates of that population set.To see directly that the number of microstates is dictated by Equation \ref{micro}, let us consider the number of ways we can assign \(N\) distinguishable molecules to the quantum states when the population set is \(\{N_1,N_2,\dots ,N_i,\dots \}\) and energy level \(\epsilon _i\) is \(g_i\)-fold degenerate. We begin by assigning the \(N_1\) molecules in energy level \(\epsilon _1\). We can choose the first molecule from among any of the \(N\) distinguishable molecules and can choose to place it in any of the \(g_1\) quantum states whose energy is \(\epsilon _1\). The number of ways we can make these choices is \({Ng}_1\). We can choose the second molecule from among the \(N-1\) remaining distinguishable molecules. In Boltzmann statistics, we can place any number of molecules in any quantum state, so there are again \(g_1\) quantum states in which we can place the second molecule. The total number of ways we can place the second molecule is \(\left(N-1\right)g_1\).The number of ways the first and second molecules can be chosen and placed is therefore \(N\left(N-1\right)g^2_1\). We find the number of ways that successive molecules can be placed in the quantum states of energy \(\epsilon _1\) by the same argument. The last molecule whose energy is \(\epsilon _1\) can be chosen from among the \(\left(N-N_1+1\right)\) remaining molecules and placed in any of the \(g_1\) quantum states. The total number of ways of placing the \(N_1\) molecules in energy level \(\epsilon _1\) is \(N\left(N-1\right)\left(N-2\right)\dots \left(N-N_1+1\right)g^{N_1}_1\).This total includes all possible orders for placing every set of \(N_1\) distinguishable molecules into every possible set of quantum states. However, the order doesn’t matter; the only thing that affects the state of the system is which molecules go into which quantum state. (When we consider all of the ways our procedure puts all of the molecules into any of the quantum states, we find that any assignment of molecules \(A\), \(B\), and \(C\) to any particular set of quantum states occurs six times. Selections in the orders \(A\),\(B\),\(C\); \(A\),\(C\),\(B\); \(B\),\(A\),\(C\); \(B\),\(C\),\(A\); \(C\),\(A\),\(B\); and \(C\),\(B\),\(A\) all put the same molecules in the same quantum states.) There are \(N_1!\) orders in which our procedure chooses the \(N_1\) molecules; to correct for this, we must divide by \(N_1!\), so that the total number of assignments we want to include in our count is\[N\left(N-1\right)\left(N-2\right)\dots \left(N-N_1+1\right)g^{N_1}_1/N_1! \nonumber \]The first molecule that we assign to the second energy level can be chosen from among the \(N-N_1\) remaining molecules and placed into any of the \(g_2\) quantum states whose energy is \(\epsilon _2\). The last one can be chosen from among the remaining \(\left(N-N_1-N_2+1\right)\) molecules. The number of assignments of the \(N_2\) molecules to \(g_2\)-fold degenerate quantum states whose energy is \(\epsilon _2\) is\[\left(N-N_1\right)\left(N-N_1-1\right)\dots \left(N-N_1-N_2+1\right)g^{N_2}_2/N_2! \nonumber \]When we consider the number of assignments of molecules to quantum states with energies \(\epsilon _1\) and \(\epsilon _2\) we have\[N\left(N-1\right)\dots \left(N-N_1+1\right)\left(N-N_1\right)\left(N-N_1-1\right)\dots \nonumber \] \[\times \left(N-N_1-N_2+1\right)\left(\frac{g^{N_1}_1}{N_1!}\right)\left(\frac{g^{N_2}_2}{N_2!}\right) \nonumber \]Let the last energy level to contain any molecules be \(\epsilon _{\omega }\). The number of ways that the \(N_{\omega }\) molecules can be assigned to the quantum states with energy \(\epsilon _{\omega }\) is \(N_{\omega }\left(N_{\omega }-1\right)\dots \left(1\right)g^{N_{\omega }}_{\omega }/N_{\omega }!\) The total number of microstates for the population set \(\{N_1,N_2,\dots ,N_i,\dots \}\) becomes\[N\left(N-1\right)\dots \left(N-N_1\right)\left(N-N_1-1\right)\dots \nonumber \] \[\times \left(N_{\omega }\right)\left(N_{\omega }-1\right)\dots \left(1\right)\prod^{\infty }_{i=1}{\left(\frac{g^{N_i}_i}{N_i!}\right)}=N!\prod^{\infty }_{i=1}{\left(\frac{g^{N_i}_i}{N_i!}\right)} \nonumber \]When we consider Fermi-Dirac and Bose-Einstein statistics, it is no longer true that the molecules are distinguishable. For Fermi-Dirac statistics, no more than one molecule can be assigned to a particular quantum state. For a given population set, Boltzmann, Fermi-Dirac, and Bose-Einstein statistics produce different numbers of microstates.It is helpful to have notation that enables us to specify different combinations and different microstates. If \(\epsilon _i\) is the energy associated with the wave equation that describes a particular molecule, it is convenient to say that the molecule is in energy level \(\epsilon _i\); that is, its quantum state is one of those that has energy \(\epsilon _i\). Using capital letters to represent molecules, we indicate that molecule \(A\) is in energy level \(\epsilon _i\) by writing \(\epsilon _i\left(A\right)\). To indicate that \(A\), \(B\), and \(C\) are in \(\epsilon _i\), we write \(\epsilon _i\left(A,B,C\right)\). Similarly, to indicate that molecules \(D\) and \(E\) are in \(\epsilon _k\), we write \(\epsilon _k\left(D,E\right)\). For this system of five molecules, the assignment \(\epsilon _i\left(A,B,C\right)\epsilon _k\left(D,E\right)\) represents one of the possible combinations. The order in which we present the molecules that have a given energy is immaterial: \(\epsilon _i\left(A,B,C\right)\epsilon _k\left(D,E\right)\) and \(\epsilon _i\left(C,B,A\right)\epsilon _k\left(E,D\right)\) represent the same combination. When any one molecule is distinguishable from others of the same substance, assignments in which a given molecule has different energies are physically different and represent different combinations. The assignments \(\epsilon _i\left(A,B,C\right)\epsilon _k\left(D,E\right)\) and \(\epsilon _i\left(D,B,C\right)\epsilon _k\left(A,E\right)\) represent different combinations. In energy level is three-fold degenerate, a molecule in any of the quantum states \({\psi }_{i,1}\), \({\psi }_{i,2}\), or \({\psi }_{i,3}\) has energy \(\epsilon _i\). Let us write\[{\psi }_{i,1}\left(A,B\right){\psi }_{i,2}\left(C\right){\psi }_{k,1}\left(DE\right) \nonumber \]to indicate the microstate arising from the combination \(\epsilon _i\left(A,B,C\right)\epsilon _k\left(D,E\right)\) in which molecules \(A\) and \(B\) occupy \({\psi }_{i,1}\), molecule \(C\) occupies \({\psi }_{i,2}\), and molecules \(D\) and \(E\) occupy \({\psi }_{k,1}\). Then,\[{\psi }_{i,1}\left(A,B\right){\psi }_{i,2}\left(C\right){\psi }_{k,1}\left(DE\right) \nonumber \] \[{\psi }_{i,1}\left(B,C\right){\psi }_{i,2}\left(A\right){\psi }_{k,1}\left(DE\right) \nonumber \] \[{\psi }_{i,1}\left(A\right){\psi }_{i,2}\left(B,C\right){\psi }_{k,1}\left(DE\right) \nonumber \]are three of the many microstates arising from the combination \(\epsilon _i\left(A,B,C\right)\epsilon _k\left(D,E\right)\). shows all of the microstates possible for the population set \(\{2,1\}\) when the quantum states of a molecule are \({\psi }_{1,1}\), \({\psi }_{1,2}\), and \({\psi }_{2,1}\).This page titled 20.7: The Microstates of a Given Population Set is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.8: The Probabilities of Microstates that Have the Same Energy

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In Section 20.2, we introduce the assumption that, for a molecule in a constant-N-V-T system, for which the \(g_i\) and \({\epsilon }_i\) are fixed, the probability of a quantum state, \(\rho \left({\epsilon }_i\right)\), depends only on its energy. It follows that two or more quantum states that have the same energy must have equal probabilities. We accept the idea that the probability depends only on energy primarily because we cannot see any reason for a molecule to prefer one state to another if both states have the same energy.We extend this thinking to multi-molecule systems. If two microstates have the same energy, we cannot see any reason for the system to prefer one rather than the other. In a constant-N-V-T system, in which the total energy is not otherwise restricted, each microstate of \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\) occurs with probability \({\rho \left({\epsilon }_1\right)}^{N_1}{\rho \left({\epsilon }_2\right)}^{N_2}\dots {\rho \left({\epsilon }_i\right)}^{N_i}\dots .\), and each microstate of \(\{N^{\#}_1,N^{\#}_2,\ \dots ,N^{\#}_I,\dots \ \}\) occurs with probability \({\rho \left({\epsilon }_1\right)}^{N^{\#}_1}{\rho \left({\epsilon }_2\right)}^{N^{\#}_2}\dots {\rho \left({\epsilon }_i\right)}^{N^{\#}_i}\dots .\) When the energies of these population sets are equal, we infer that these probabilities are equal, and their value is a constant of the system. That is,\[{\rho \left({\epsilon }_1\right)}^{N_1}{\rho \left({\epsilon }_2\right)}^{N_2}\dots {\rho \left({\epsilon }_i\right)}^{N_i}\dots . \nonumber \] \[={\rho \left({\epsilon }_1\right)}^{N^{\#}_1}{\rho \left({\epsilon }_2\right)}^{N^{\#}_2}\dots {\rho \left({\epsilon }_i\right)}^{N^{\#}_i}\dots .\ \ \ \ ={\rho }_{MS,N,E}=\mathrm{constant} \nonumber \]where we introduce \({\rho }_{MS,N,E}\) to represent the probability of a microstate of a system of \(N\) molecules that has total energy \(E\). If \(E=E^{\#}\), then \({\rho }_{MS,N,E}={\rho }_{MS,N,E^{\#}}\).When we think about it critically, the logical basis for this equal-probability idea is not very impressive. While the idea is plausible, it is not securely rooted in any particular empirical observation or prior postulate. The equal-probability idea is useful only if it leads us to theoretical models that successfully mirror the behavior of real macroscopic systems. This it does. Accordingly, we recognize that the equal-probability idea is really a fundamental postulate about the behavior of quantum-mechanical systems. It is often called the principle of equal a priori probabilities:Definition: principle of equal a priori probabilitiesFor a particular system, all microstates that have the same energy have the same probability.Our development of statistical thermodynamics relies on the principle of equal a priori probabilities. For now, let us summarize the important relationships that the principle of equal a priori probabilities imposes on our microscopic model for the probabilities of two population sets of a constant-N-V-T system that have the same energy:This page titled 20.8: The Probabilities of Microstates that Have the Same Energy is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.9: The Probabilities of the Population Sets of an Isolated System

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In principle, the energy of an equilibrium system that is in contact with a constant-temperature heat reservoir can vary slightly with time. In contrast, the energy of an isolated system is constant. A more traditional and less general statement of the equal a priori probability principle focuses on isolated systems, for which all possible microstates necessarily have the same energy:All microstates of an isolated (constant energy) system occur with equal probability.If we look at the fraction of the molecules of an isolated system that are in each microstate, we expect to find that these fractions are approximately equal. In consequence, for an isolated system, the probability of a population set, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), is proportional to the number of microstates, \(W\left(N_i,g_i\right)\), to which that population set gives rise.In principle, the population sets of a constant-N-V-T system can be significantly different from those of a constant-N-V-E system. That is, if we move an isolated system, whose temperature is T, into thermal contact with a heat reservoir at constant-temperature T, the population sets that characterize the system can change. In practice, however, for a system containing a large number of molecules, the population sets that contribute to the macroscopic properties of the system must be essentially the same.The fact that the same population sets are important in both systems enables us to make two further assumptions that become important in our development. We assume that the proportionality between the probability of a population set and \(W\left(N_i,g_i\right)\), which is strictly true only for a constant-N-V-E system, is also true for the corresponding constant-N-V-T system. We also assume that the probabilities of a quantum state, \(\rho \left({\epsilon }_i\right)\), and a microstate, \({\rho }_{MS,N,E}\), which we defined for the constant-N-V-T system, are the same for the corresponding constant-N-V-E system.Let us see why we expect the same population sets to dominate the macroscopic properties of otherwise identical constant-energy and constant-temperature systems. Suppose that we isolate a constant-N-V-T system in such a way that the total energy, \(E=\sum^{\infty }_{i=1}{N_i{\epsilon }_i}\), of the isolated system is exactly equal to the expected value, \(\left\langle E\right\rangle =N\sum^{\infty }_{i=1}{P_i{\epsilon }_i}\), of the energy of the system when its temperature is constant. What we have in mind is a gedanken experiment, in which we monitor the energy of the thermostatted system as a function of time, waiting for an instant in which the system energy, \(E=\sum^{\infty }_{i=1}{N_i{\epsilon }_i}\), is equal to the expected value of the system energy, \(\left\langle E\right\rangle\). When this occurs, we instantaneously isolate the system.We suppose that the isolation process is accomplished before any molecule can experience an energy change, so that the population set that characterizes the system immediately afterwards is the same as the one that characterizes it before. After isolation, of course, the molecules can exchange energy with one another, and many population sets may be available to the system.Clearly, the value of every macroscopic property of the isolated system must be the same as its observable value in the original constant-temperature system. Our microscopic description of it is different. Every population set that is available to the isolated system has energy \(E=\left\langle E\right\rangle\), and gives rise to\[W\left(N_i,g_i\right)=N!\prod^{\infty }_{i=1}{\left(\frac{g^{N_i}_i}{N_i!}\right)} \nonumber \]microstates. At the same temperature, each of these microstates occurs with the same probability. Since the isolated-system energy is \(\left\langle E\right\rangle\), this probability is \({\rho }_{MS,N,\left\langle E\right\rangle }\). The probability of an available population set is \(W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }\).Since the temperature can span a range of values centered on \(\left\langle T\right\rangle\), where \(\left\langle T\right\rangle\) is equal to the temperature of the original constant-N-V-T system, there is a range of \({\rho }_{MS,N,\left\langle E\right\rangle }\) values spanning the (small) range of temperatures available to the constant-energy system. Summing over all of the population sets that are available to the isolated system, we find\[1=\sum_{\left\{N_i\right\},\ \ E=\left\langle E\right\rangle ,T=\left\langle T\right\rangle }{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }}+\sum_{\left\{N_i\right\},\ \ E=\left\langle E\right\rangle ,T\neq \left\langle T\right\rangle }{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }} \nonumber \]The addition of “\(E=\left\langle E\right\rangle\)” beneath the summation sign emphasizes that the summation is to be carried out over the population sets that are consistent with both the molecule-number and total-energy constraints and no others. The total probability sum breaks into two terms, one spanning population sets whose temperature is exactly \(\left\langle T\right\rangle\) and another spanning all of the other population sets. (Remember that the \(\rho \left({\epsilon }_i\right)\) are temperature dependent.)The population sets available to the isolated system are slightly different from those available to the constant-temperature system. In our microscopic model, only population sets that have exactly the right total energy can occur in the isolated system. Only population sets that have exactly the right temperature can occur in the constant-temperature system.Summing over all of the population sets that are available to the constant-temperature system, we partition the total probability sum into two terms:\[1=\sum_{\left\{N_i\right\},E=\left\langle E\right\rangle ,T=\left\langle T\right\rangle }{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }}+\sum_{\left\{N_i\right\},\ \ E\neq \left\langle E\right\rangle ,T=\left\langle T\right\rangle }{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }} \nonumber \]From the central limit theorem, we expect the constant-energy system to have (relatively) few population that fail to meet the condition \(E=\left\langle E\right\rangle\). Likewise, we expect the constant temperature system to have (relatively) few population sets that fail to meet the condition \(T=\left\langle T\right\rangle\). The population sets that satisfy both of these criteria must dominate both sums. For the number of molecules in macroscopic systems, we expect the approximation to the total probability sum\[1=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }}\approx \sum_{\left\{N_i\right\},E=\left\langle E\right\rangle ,T=\left\langle T\right\rangle }{W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }} \nonumber \]to be very good. The same population sets dominate both the constant-temperature and constant-energy systems. Each system must have a most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,.,,,\}\). If these are not identically the same set, they must be so close that the same macroscopic properties are calculated using either one.Thus, the central limit theorem implies that the total probability sum, which we develop for the constant-temperature system, also describes the constant-energy system, so long as the number of molecules in the system is sufficiently large.Now, two aspects of this development warrant elaboration. The first is that the probability of population sets that have energies and temperature that satisfy \(E=\left\langle E\right\rangle\) and \(T=\left\langle T\right\rangle\) exactly may actually be much less than one. The second is that constant-energy and constant-temperature systems are creatures of theory. No real system can actually have an absolutely constant energy or temperature.Recognizing these facts, we see that when we stipulate \(E=\left\langle E\right\rangle\) or \(T=\left\langle T\right\rangle\), what we really mean is that \(E=\left\langle E\right\rangle \pm \delta E\) and \(T=\left\langle T\right\rangle \pm \delta T\), where the intervals \(\pm \delta E\) and \(\pm \delta T\) are vastly smaller than any differences we could actually measure experimentally. When we write \(E\neq \left\langle E\right\rangle\) and \(T\neq \left\langle T\right\rangle\), we really intend to specify energies and temperatures that fall outside the intervals \(E=\left\langle E\right\rangle \pm \delta E\) and \(T=\left\langle T\right\rangle \pm \delta T\). If the system contains sufficiently many molecules, the population sets whose energies and temperatures fall within the intervals \(E=\left\langle E\right\rangle \pm \delta E\) and \(T=\left\langle T\right\rangle \pm \delta T\) account for nearly all of the probability—no matter how small we choose \(\delta E\) and \(\delta T\). All of the population sets whose energies and temperatures fall within the intervals \(E=\left\langle E\right\rangle \pm \delta E\) and \(T=\left\langle T\right\rangle \pm \delta T\) correspond to the same macroscopically observable properties.This page titled 20.9: The Probabilities of the Population Sets of an Isolated System is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.10: Entropy and Equilibrium in an Isolated System

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In an isolated system, the probability of population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) is \(W\left(N_i,g_i\right){\rho }_{MS,N,\left\langle E\right\rangle }\), where \({\rho }_{MS,N,\left\langle E\right\rangle }\) is a constant. It follows that \(W=W\left(N_i,g_i\right)\) is proportional to the probability that the system is in one of the microstates associated with the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). Likewise, \(W^{\#}=W\left(N^{\#}_i,g_i\right)\) is proportional to the probability that the system is in one of the microstates associated with the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\). Suppose that we observe the isolated system for a long time. Let \(F\) be the fraction of the time that the system is in microstates of population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) and \(F^{\#}\) be the fraction of the time that the system is in microstates of the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\). The principle of equal a priori probabilities implies that we would find\[\frac{F^{\#}}{F}=\frac{W^{\#}}{W} \nonumber \]Suppose that \(W^{\#}\) is much larger than \(W\). This means there are many more microstates for \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) than there are for \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). The fraction of the time that the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) characterizes the system is much greater than the fraction of the time \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) characterizes it. Alternatively, if we examine the system at an arbitrary instant, we are much more likely to find the population set \(\{N^{\#}_1,N^{\#}_2,\dots N^{\#}_i,\dots \}\) than the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). The larger \(W\left(N_1,g_1,\ N_2,g_2,\dots ,N_i,g_i,\dots \right)\), the more likely it is that the system will be in one of the microstates associated with the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). In short, \(W\) predicts the state of the system; it is a measure of the probability that the macroscopic properties of the system are those of the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\).If an isolated system can undergo change, and we re-examine it at after a few molecules have moved to different energy levels, we expect to find it in one of the microstates of a more-probable population set; that is, in one of the microstates of a population set for which \(W\) is larger. At still later times, we expect to see a more-or-less smooth progression: the system is in microstates of population sets for which the values of \(W\) are increasingly larger. This can continue only until the system occupies one of the microstates of the population set for which \(W\) is a maximum or a microstate of one of the population sets whose macroscopic properties are essentially the same as those of the constant-\(N\)-\(V\)-\(E\) population set for which \(W\) is a maximum.Once this occurs, later inspection may find the system in other microstates, but it is overwhelmingly probable that the new microstate will still be one of those belonging to the largest-\(W\) population set or one of those that are macroscopically indistinguishable from it. Any of these microstates will belong to a population set for which \(W\) is very well approximated by \(W\left(\ N^{\textrm{⦁}}_1,g_1,\ N^{\textrm{⦁}}_2,g_2,\dots ,N^{\textrm{⦁}}_i,g_i,\dots \right)\). Evidently, the largest-\(W\) population set characterizes the equilibrium state of the either the constant-\(N\)-\(V\)-\(T\) system or the constant–\(N\)-\(V\)-\(E\) system. Either system can undergo change until \(W\) reaches a maximum. Thereafter, it is at equilibrium and can undergo no further macroscopically observable change.Boltzmann recognized this relationship between \(W\), the thermodynamic probability, and equilibrium. He noted that the unidirectional behavior of \(W\) in an isolated system undergoing spontaneous change is like the behavior we found for the entropy function. Boltzmann proposed that, for an isolated (constant energy) system, \(S\) and \(W\) are related by the equation \(S=k{ \ln W\ }\), where \(k\) is Boltzmann’s constant. This relationship associates an entropy value with every population set. For an isolated macroscopic system, equilibrium corresponds to a state of maximum entropy. In our microscopic model, equilibrium corresponds to the population set for which \(W\) is a maximum. By the argument we make in §6, this population set must be well approximated by the most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,.,,,\}\). That is, the entropy of the equilibrium state of the macroscopic system is\[ \begin{align*} S &= k ~ { \ln W_{max}\ } \\[4pt] &=k ~ { \ln \frac{N!}{N^{\textrm{⦁}}_i!N^{\textrm{⦁}}_i!\dots N^{\textrm{⦁}}_i!\dots }\ }+k ~ \sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i{ \ln g_i\ }} \end{align*} \]This equation can be taken as the definition of entropy. Clearly, this definition is different from the thermochemical definition, \(S={q^{rev}}/{T}\). We can characterize—imperfectly—the situation by saying that the two definitions provide alternative scales for measuring the same physical property. As we see below, our statistical theory enables us to define entropy in still more ways, all of which prove to be functionally equivalent. Gibbs characterized these alternatives as “entropy analogues;” that is, functions whose properties parallel those of the thermochemically defined entropy.We infer that the most probable population set characterizes the equilibrium state of either the constant-temperature or the constant-energy system. Since our procedure for isolating the constant-temperature system affects only the thermal interaction between the system and its surroundings, the entropy of the constant-temperature system must be the same as that of the constant-energy system. Using \(N^{\textrm{⦁}}_i=NP_i=Ng_i\rho \left({\epsilon }_i\right)\) and assuming that the approximation \({ \ln N^{\textrm{⦁}}_i!\ }=N^{\textrm{⦁}}_i{ \ln N^{\textrm{⦁}}_i\ }-N^{\textrm{⦁}}_i\) is adequate for all of the energy levels that make a significant contribution to \(S\), substitution shows that the entropy of either system depends only on probabilities:\[ \begin{align*} S &= kN ~ { \ln N - kN - k\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{\left[NP_i{ \ln \left(NP_i\right)\ } - NP_i\right]}\ } + k\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{NP_i{ \ln g_i\ }} \\[4pt]&= kN ~ { \ln N\ }\mathrm{-kN} -kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{\left[P_i{ \ln \left(N\right)\ } + P_i{ \ln P_i\ } - P_i - P_i{ \ln g_i\ }\right]} \\[4pt] &= k ~ \left(N{ \ln N\ } - N\right) - k\left(N{ \ln N\ } - N\right)\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i} - kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i}\left[{ \ln P_i - { \ln g_i\ }\ }\right]\mathrm{=-}kN\sum^{\mathrm{\infty }}_{i\mathrm{=1}}{P_i{ \ln \rho \left({\epsilon }_i\right)\ }} \end{align*} \]The entropy per molecule, \({S}/{N}\), is proportional to the expected value of \({ \ln \rho \left({\epsilon }_i\right)\ }\); Boltzmann’s constant is the proportionality constant. At constant temperature, \(\rho \left({\epsilon }_i\right)\) depends only on \({\epsilon }_i\). The entropy per molecule depends only on the quantum state properties, \(g_i\) and \({\epsilon }_i\).This page titled 20.10: Entropy and Equilibrium in an Isolated System is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.11: Thermodynamic Probability and Equilibrium in an Isomerization Reaction

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To relate these ideas to a change in a more specific macroscopic system, let us consider isomeric substances \(A\) and \(B\). (We consider this example further in Chapter 21.) In principle, we can solve the Schrödinger equation for a molecule of isomer \(A\) and for a molecule of isomer \(B\). We obtain all possible energy levels for a molecule of each isomer.\({}^{1}\) If we list these energy levels in order, beginning with the lowest, some of these levels belong to isomer \(A\) and the others belong to isomer \(B\).Now let us consider a mixture of \(N_A\) molecules of \(A\) and \(N_B\) molecules of \(B\). We suppose that individual molecules are distinguishable and that intermolecular interactions can be ignored. Since a group of atoms that can form an \(A\) molecule can also form a \(B\) molecule, every energy level is accessible to this group of atoms; that is, we can view both sets of energy levels as being available to the atoms that make up the molecules. For a given system energy, there will be many population sets in which only the energy levels belonging to isomer \(A\) are occupied. For each of these population sets, there is a corresponding thermodynamic probability, \(W\). Let \(W^{max}_A\) be the largest of these thermodynamic probabilities. Similarly, there will be many population sets in which only the energy levels corresponding to isomer \(B\) are occupied. Let \(W^{max}_B\) be the largest of the thermodynamic probabilities associated with these population sets. Finally, there will be many population sets in which the occupied energy levels belong to both isomer \(A\) and isomer \(B\). Let \(W^{max}_{A,B}\) be the largest of the thermodynamic probabilities associated with this group of population sets.Now, \(W^{max}_A\) is a good approximation to the number of ways that the atoms of the system can come together to form isomer \(A\). \(W^{max}_B\) is a good approximation to the the number of ways that the atoms of the system can come together to form isomer \(B\). At equilibrium, therefore, we expect\[K=\frac{N_B}{N_A}=\frac{W^{max}_B}{W^{max}_A} \nonumber \]If we consider the illustrative—if somewhat unrealistic—case of isomeric molecules whose energy levels all have the same degeneracy (\(g_i=g\) for all \(i\)), we can readily see that the equilibrium system must contain some amount of each isomer. For a system containing \(N\) molecules, \(N!g^N\) is the numerator in each of the thermodynamic probabilities \(W^{max}_A\), \(W^{max}_B\), and \(W^{max}_{A,B}\). The denominators are different. The denominator of \(W^{max}_{A,B}\) must contain terms, \(N_i!\), for essentially all of the levels represented in the denominator of \(W^{max}_A\). Likewise, it must contain terms, \(N_j!\), for essentially all of the energy levels represented in the denominator of \(W^{max}_B\). Then the denominator of \(W^{max}_{A,B}\) is a product of \(N_k!\) terms that are generally smaller than the corresponding factorial terms in the denominators of \(W^{max}_A\) and \(W^{max}_B\). As a result, the denominators of \(W^{max}_A\) and \(W^{max}_B\) are larger than the denominator of \(W^{max}_{A,B}\). In consequence, \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\). (See problems 5 and 6.)If we create the system as a collection of \(A\) molecules, or as a collection of \(B\) molecules, redistribution of the sets of atoms among all of the available energy levels must eventually produce a mixture of \(A\) molecules and \(B\) molecules. Viewed as a consequence of the principle of equal a priori probabilities, this occurs because there are necessarily more microstates of the same energy available to some mixture of \(A\) and \(B\) molecules than there are microstates available to either \(A\) molecules alone or \(B\) molecules alone. Viewed as a consequence of the tendency of the isolated system to attain the state of maximum entropy, this occurs because \(k{ \ln W^{max}_{A,B}>k{ \ln W^{max}_A\ }\ }\) and \(k{ \ln W^{max}_{A,B}>k{ \ln W^{max}_B\ }\ }\).This page titled 20.11: Thermodynamic Probability and Equilibrium in an Isomerization Reaction is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.12: The Degeneracy of an Isolated System and Its Entropy

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In Section 20.9, we find that the sum of the probabilities of the population sets of an isolated system is\[1=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right){\rho }_{MS,N,E}}. \nonumber \]By the principle of equal a priori probabilities, \({\rho }_{MS,N,E}\) is a constant, and it can be factored out of the sum. We have\[1={\rho }_{MS,N,E}\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)} \nonumber \]Moreover, the sum of the thermodynamic probabilities over all allowed population sets is just the number of microstates that have energy \(E\). This sum is just the degeneracy of the system energy, \(E\). The symbol \(\mathit{\Omega}_E\) is often given to this system-energy degeneracy. That is,\[\mathit{\Omega}_E=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)} \nonumber \]The sum of the probabilities of the population sets of an isolated system becomes\[1={\rho }_{MS,N,E}{\mathit{\Omega}}_E \nonumber \]In Section 20.9, we infer that\[\rho_{MS,N,E}=\prod^{\infty }_{i=1}{\rho \left({\epsilon }_i\right)^{N_i}} \nonumber \]so we have\[1={\mathit{\Omega}}_E\prod^{\infty }_{i=1}\rho \left(\epsilon_i\right)^{N_i} \nonumber \]This page titled 20.12: The Degeneracy of an Isolated System and Its Entropy is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.13: The Degeneracy of an Isolated System and its Entropy

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/20%3A_Boltzmann_Statistics/20.13%3A_The_Degeneracy_of_an_Isolated_System_and_its_Entropy

In Section 20.10, we observe that the entropy of an isolated equilibrium system can be defined as \(S=k{\ln W_{max}}\). In Section 20.12, we see that the system-energy degeneracy is a sum of terms, one of which is \(W_{max}=W\left(N^{\textrm{⦁}}_i,g_i\right)\). That is, we have \[{\Omega}_E=W_{max}+\sum_{\left\{N_i\right\}\neq \left\{N^{\textrm{⦁}}_i\right\},E_{total}}{W\left(N_i,g_i\right)} \nonumber \]where the last sum is taken over all energy-qualifying population sets other than the most-probable population set.Let us now consider the relative magnitude of \(\Omega_E\) and \(W_{max}\). Clearly, \(\Omega_E\ge W_{max}\). If only one population set is consistent with the total-molecule and total-energy constraints of the isolated system, then \(\Omega_E=W_{max}\). In general, however, we must expect that there will be many, possibly an enormous number, of other population sets that meet the constraints. Ultimately, the relative magnitude of \({\Omega}_E\) and \(W_{max}\) depends on the energy levels available to the molecules and the number of molecules in the system and so could be almost anything. However, rather simple considerations lead us to expect that, for most macroscopic collections of molecules, the ratio \(\alpha ={\Omega_E}/W_{max}\) will be much less than \(W_{max}\). That is, although the value of \(\alpha\) may be very large, for macroscopic systems we expect to find \(\alpha \ll W_{max}\). If \(\Omega_E=W_{max}\), then \(\alpha =1\), and \({\ln \alpha }=0\).Because \(W\) for any population set that contributes to \({\Omega}_E\) must be less than or equal to \(W_{max}\), the maximum value of \(\alpha\) must be less than the number of population sets which satisfy the system constraints. For macroscopic systems whose molecules have even a modest number of accessible energy levels, calculations show that \(W_{max}\) is a very large number indeed. Calculation of \(\alpha\) for even a small collection of molecules is intractable unless the number of accessible molecular energy levels is small. Numerical experimentation on small systems, with small numbers of energy levels, shows that the number of qualifying population sets increases much less rapidly than \(W_{max}\) as the total number of molecules increases. Moreover, the contribution that most qualifying population sets make to \({\Omega}_E\) is much less than \(W_{max}\).For macroscopic systems, we can be confident that \(W_{max}\) is enormously greater than \(\alpha\). Hence \(\Omega_E\) is enormously greater than \(\alpha\). When we substitute for \(W_{max}\) in the isolated-system entropy equation, we find\[ \begin{align*} S &=k \ln W_{max} \\[4pt] &=k \ln \left(\Omega_E/\alpha \right) \\[4pt] &=k \mathrm{l}\mathrm{n} \Omega_E -k \ln \alpha \\[4pt] &\approx k \ln \Omega_E \end{align*} \]where the last approximation is usually very good.In many developments, the entropy of an isolated system is defined by the equation \(S=k{\ln {\Omega}_E}\) rather than the equation we introduced first, \(S=k{\ln W_{max}}\). From the considerations above, we expect the practical consequences to be the same. In Section 20.14, we see that the approximate equality of \({\ln W_{max}}\) and \({\ln {\Omega}_E}\) is a mathematical consequence of our other assumptions and approximations.This page titled 20.13: The Degeneracy of an Isolated System and its Entropy is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.14: Effective Equivalence of the Isothermal and Constant-energy Conditions

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In principle, an isolated system is different from a system with identical macroscopic properties that is in equilibrium with its surroundings. We emphasize this point, because this distinction is important in the logic of our development. However, our development also depends on the assumption that, when \(N\) is a number that approximates the number molecules in a macroscopic system, the constant-temperature and constant-energy systems are functionally equivalent.In Section 20.9, we find that any calculation of macroscopic properties must produce the same result whether we consider the constant-temperature or the constant-energy system. The most probable population set, \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,\dots N^{\textrm{⦁}}_i,\dots \}\), provides an adequate description of the macroscopic state of the constant-temperature system precisely because it is representative of all the population sets that contribute significantly to the total probability of the constant-temperature system. The effective equivalence of the constant-temperature and constant-energy systems ensures that the most probable population set is also representative of all the population sets that contribute significantly to the total probability of the constant-energy system.In Section 20.12, we see that the essential equivalence of the isothermal and constant-energy systems means that we have\[1={\Omega}_E\prod^{\infty }_{i=1}{\rho {\left({\epsilon }_i\right)}^{N^{\textrm{⦁}}_i}} \nonumber \]Taking logarithms of both sides, we find\[{ \ln {\Omega}_E=-\sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i}}{ \ln \rho \left({\epsilon }_i\right)} \nonumber \]From \(S=k{ \ln {\Omega}_E}\), it follows that\[\mathrm{S} =-k\sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i} \ln \rho \left(\epsilon_i\right) \nonumber \]For the constant-temperature system, we have \(N^{\textrm{⦁}}_i=NP_i\). When we assume that the equilibrium constant-temperature and constant-energy systems are essentially equivalent, the entropy of the N-molecule system becomes\[\begin{align*} S &=-k\sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i}{\mathrm{l}\mathrm{n} \rho \left({\epsilon }_i\right)} \\[4pt] &=-kN\sum^{\infty }_{i=1}{P_i}{ \ln \rho \left({\epsilon }_i\right)} \end{align*} \]so that we obtain the same result from assuming that \(S=k{ \ln {\Omega}_E}\) as we do in Section 20.10 from assuming that \(S=k{ \ln W_{max}}\). Under the approximations we introduce, \({ \ln {\Omega}_E}\) and \({ \ln W_{max}}\) evaluate to the same thing.This page titled 20.14: Effective Equivalence of the Isothermal and Constant-energy Conditions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

20.15: Problems

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1. Three non-degenerate energy levels are available to a set of five distinguishable molecules, \(\{A,\ B,\ C,\ D,\ E\}\). The energies of these levels are \(1\), \(2\), and \(3\), in arbitrary units. Find all of the population sets that are possible in this system. For each population set, find the system energy, \(E\), and the number of microstates, \(W\). For each system energy, \(E\), list the associated population sets and the total number of microstates. How many population sets are there? What is \(W_{max}\)? If this system is isolated with \(E=10\), how many population sets are possible? What is \({\mathit{\Omega}}_E\) for \(E=10\)?2. For the particle in a box, the allowed energies are proportional to the squares of the successive integers. What population sets are possible for the distinguishable molecules, \(\{A,\ B,\ C,\ D,\ E\}\), if they can occupy three quantum states whose energies are \(1\), \(4\), and \(9\)? For each population set, find the system energy, \(E\), and the number of microstates. For each system energy, \(E\), list the associated population sets and the total number of microstates. How many population sets are there? What is \(W_{max}\)? If this system is isolated with \(E=24\), how many population sets are possible? What is \({\mathit{\Omega}}_E\) for \(E=24\)?3. Consider the results you obtained in problem 2. In general, when the allowed energies are proportional to the squares of successive integers, how many population sets do you think will be associated with each system energy?4.(a) Compare \(W\) for the population set \(\{3,3,3\}\) to \(W\) for the population set \(\{2,5,2\}\). The energy levels are non-degenerate.(b) Consider an \(N\)-molecule system that has a finite number, \(M\), of quantum states. Show that \(W\) is (at least locally) a maximum when \(N_1=N_2=\dots =N_M={N}/{M}\). (Hint: Let \(U={N}/{M}\), and assume that \(N\) can be chosen so that \(U\) is an integer. Let \[W_U={N!}/{\left[U!U!\prod^{i=M-2}_{i=1}{U!}\right]} \nonumber \]and let \[W_O={N!}/{\left[\left(U+1\right)!\left(U-1\right)!\prod^{i=M-2}_{i=1}{U!}\right]} \nonumber \]Show that \({W_O}/{W_U}<1\).)5. The energy levels available to isomer \(A\) are \({\epsilon }_0=1\), \({\epsilon }_2=2\), and \({\epsilon }_4=3\), in arbitrary units. The energy levels available to isomer B are \({\epsilon }_1=2\), \({\epsilon }_3=3\), and \({\epsilon }_5=4\). The energy levels are non-degenerate.(a) A system contains five molecules. The energy of the system is \(10\). List the population sets that are consistent with \(N=5\) and \(E=10\). Find \(W\) for each of these population sets. What are \(W^{max}_{A,B}\), \(W^{max}_A\), and \(W^{max}_B\)? What is the total number of microstates, \(=\mathit{\Omega}_{A,B}\), available to the system in all of the cases in which \(A\) and \(B\) molecules are present? What is the ratio \(\mathit{\Omega}_{A,B}/W^{max}_{A,B}\)?(b) Repeat this analysis for a system that contains six molecules and whose energy is \(12\).(c) Would the ratio \(\mathit{\Omega}_{A,B}/W^{max}_{A,B}\) be larger or smaller for a system with \(N=50\) and \(E=100\)?(d) What would happen to this ratio if the number of molecules became very large, while the average energy per molecule remained the same?6. In Section 20.11, we assume that all of the energy levels available to an isomeric pair of molecules have the same degeneracy. We then argue that the thermodynamic probabilities of a mixture of the isomers must be greater than the thermodynamic probability of either pure isomer: \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\). Implicitly, we assume that many energy levels are multiply occupied: \(N_i>1\) for many energy levels \({\epsilon }_i\). Now consider the case that \(g_i>1\) for most \({\epsilon }_i\), but that nearly all energy levels are either unoccupied or contain only one molecule: \(N_i=0\) or \(N_i=1\). Show that under this assumption also, we must have \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\).Notes\({}^{1}\)The statistical-mechanical procedures that have been developed for finding the energy levels available to a molecule express molecular energies as the difference between the molecule energy and the energy that its constituent atoms have when they are motionless. This is usually effected in two steps. The molecular energy levels are first expressed relative to the energy of the molecule’s own lowest energy state. The energy released when the molecules is formed in its lowest energy state from the isolated constituent atoms is then added. The energy of each level is then equal to the work done on the component atoms when they are brought together from infinite separation to form the molecule in that energy level. (Since energy is released in the formation of a stable molecule, the work done on the atoms and the energy of the resulting molecule are less than zero.) In our present discussion, we suppose that we can solve the Schrödinger equation to find the energies of the allowed quantum states. This corresponds to choosing the isolated constituent electrons and nuclei as the zero of energy for both isomers.This page titled 20.15: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.1: Finding the Boltzmann Equation

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The probabilities of the energy levels of a constant-temperature system at equilibrium must depend only on the intensive variables that serve to characterize the equilibrium state. In Section 20.8, we introduce the principle of equal a priori probabilities, which asserts that any two microstates of an isolated system have the same probability. From the central limit theorem, we infer that an isolated system is functionally equivalent to a constant-temperature system when the system contains a sufficiently large number of molecules. From these ideas, we can now find the relationship between the energy values, \({\epsilon }_i\), and the corresponding probabilities,\[P_i=P\left({\epsilon }_i\right)=g_i\rho \left({\epsilon }_i\right).\nonumber \]Let us consider the microstates of an isolated system whose energy is \(E^{\#}\). For any population set, \(\{N_1,\ N_2,\dots ,N_i,\dots \}\), that has energy \(E^{\#}\), the following relationships apply.We want to find a function, \(\rho \left(\epsilon \right)\), that satisfies all four of these conditions. One way is to keep trying functions that look like they might work until we find one that does. A slightly more sophisticated version of this approach is to try the most general possible version of each such function and see if any set of restrictions will make it work. We could even try an infinite series. Suppose that we are clever (or lucky) enough to try the series solution\[{ \ln \rho \left(\epsilon \right)\ }=c_0+c_1\epsilon +\dots +c_i{\epsilon }^i+\dots =\displaystyle \sum^{\infty }_{k=0}{c_k}{\epsilon }^k\nonumber \]Then the third condition becomes\[\begin{align*} {\ln \textrm{ĸ}\ } &=\displaystyle \sum^{\infty }_{i=1}{N_i}{ \ln \rho \ }\left({\epsilon }_i\right) \\[4pt]&=\displaystyle \sum^{\infty }_{i=1}{N_i}\displaystyle \sum^{\infty }_{k=0}{\left[c_k{\epsilon }^k_i\right]}\\[4pt]&=\displaystyle \sum^{\infty }_{k=0}{\displaystyle \sum^{\infty }_{i=1}{c_kN_i{\epsilon }^k_i}}=c_0\displaystyle \sum^{\infty }_{i=1}{N_i}{\epsilon }^0_i+c_1\displaystyle \sum^{\infty }_{i=1}{N_i}{\epsilon }^1_i+\dots +c_k\displaystyle \sum^{\infty }_{k=2}{\displaystyle \sum^{\infty }_{i=1}{N_i{\epsilon }^k_i}}+\dots \\[4pt]&=c_0N+c_1E^{\#}+\dots +c_k\displaystyle \sum^{\infty }_{k=2}{\displaystyle \sum^{\infty }_{i=1}{N_i{\epsilon }^k_i}}+\dots \end{align*}\nonumber \]We see that the coefficient of \(c_0\) is \(N\) and the coefficient of \(c_1\) is the total energy, \(E^{\#}\). Therefore, the sum of the first two terms is a constant. We can make the trial function satisfy the third condition if we set \(c_k=0\) for all \(k>1\). We find\[{ \ln \textrm{ĸ}\ }=\displaystyle \sum^{\infty }_{i=1}{N_i}{ \ln \rho \ }\left({\epsilon }_i\right)=\displaystyle \sum^{\infty }_{i=1}{N_i}\left(c_0+c_1{\epsilon }_i\right)\nonumber \]The last equality is satisfied if, for each quantum state, we have\[{ \ln \rho \ }\left({\epsilon }_i\right)=c_0+c_1{\epsilon }_i\nonumber \] or \[\rho \left({\epsilon }_i\right)=\alpha \ \mathrm{exp}\left(c_1{\epsilon }_i\right)\nonumber \]where \(\alpha =\mathrm{exp}\left(c_0\right)\). Since the \({\epsilon }_i\) are positive and the probabilities \(\rho \left({\epsilon }_i\right)\) lie in the interval \(0<\rho \left({\epsilon }_i\right)<1\), we must have \(c_1<0\). Following custom, we let \(c_1=-\beta\), where \(\beta\) is a constant, and \(\beta >0\). Then,\[\rho \left({\epsilon }_i\right)=\alpha \ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \] and \[P_i=g_i\rho \left({\epsilon }_i\right)=\alpha g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \]The fourth condition is that the energy-level probabilities sum to one. Using this, we have\[1=\displaystyle \sum^{\infty }_{i=1}{P\left({\epsilon }_i\right)}=\alpha \displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}\nonumber \]The sum of exponential terms is so important that it is given a name. It is called the molecular partition function. It is often represented by the letter “\(z\).” Letting\[z=\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}\nonumber \] we have\[\alpha =\frac{1}{\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}}=z^{-1}\nonumber \]Thus, we have the Boltzmann probability:\[\begin{align*} P\left({\epsilon }_i\right) &=g_i\rho \left({\epsilon }_i\right) \\[4pt] &=\frac{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}{\displaystyle \sum^{\infty }_{i=1}{g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)}} \\[4pt] &=\frac{g_i}{z}\ \mathrm{exp}\left(-\beta {\epsilon }_i\right) \end{align*} \]The probability of an energy level depends only on its degeneracy, \(g_i\), its energy, \({\epsilon }_i\), and the constant \(\beta\). Since the equilibrium-characterizing population set is determined by the probabilities, we have \(P_i={N^{\textrm{⦁}}_i}/{N}\), and\[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\ \mathrm{exp}\left(-\beta {\epsilon }_i\right)\nonumber \]In Section 21.2, we develop Lagrange’s method of undetermined multipliers. In Section 21.3, we develop the same result by applying Lagrange’s method to our model for the probabilities of the microstates of an isolated system. That is, we find the Boltzmann probability equation by applying Lagrange’s method to the entropy relationship,\[S=-Nk\displaystyle \sum^{\infty }_{i=1}{P_i}{ \ln \rho \left({\epsilon }_i\right)\ }\nonumber \]that we first develop in §20-11. In §4, we find the Boltzmann probability equation by using Lagrange’s method to find the values of \(N^{\textrm{⦁}}_i\) that produce the largest possible value for \(W_{max}\) in an isolated system. This argument requires us to assume that there is a very large number of molecules in each of the occupied energy levels of the most probable population set. Since our other arguments do not assume anything about the magnitude of the various \(N^{\textrm{⦁}}_i\), it is evident that some of the assumptions we make when we apply Lagrange’s method to find the \(N^{\textrm{⦁}}_i\) are not inherent characteristics of our microscopic model.This page titled 21.1: Finding the Boltzmann Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.2: Lagrange's Method of Undetermined Multipliers

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Lagrange’s method of undetermined multipliers is a method for finding the minimum or maximum value of a function subject to one or more constraints. A simple example serves to clarify the general problem. Consider the function\[z=z_0\ \mathrm{exp}\left(x^2+y^2\right)\nonumber \]where \(z_0\) is a constant. This function is a surface of revolution, which is tangent to the plane \(z=z_0\) at \(\left(0,0,z_0\right)\). The point of tangency is the minimum value of \(z\). At any other point in the \(xy\)-plane, \(z\left(x,y\right)\) is greater than \(z_0\). If either \(x\) or \(y\) becomes arbitrarily large, \(z\) does also. If we project a contour of constant \(z\) onto the \(xy\)-plane, the projection is a circle of radius\[r=\left(x^2+y^2\right)^{1/2}. \nonumber \]Suppose that we introduce an additional condition; we require \(y=1-x\). Then we ask for the smallest value of \(z\) consistent with this constraint. In the \(xy\)-plane the constraint is a line of slope \(-1\) and intercept \(1\). A plane that includes this line and is parallel to the \(z\)-axis intersects the function \(z\). As sketched in at which the intersection occurs is large. Nearer the origin, the value of \(z\) is smaller, and there is some \(\left(x,y\right)\) at which it is a minimum. Our objective is to find this minimum.There is a straightforward solution of this problem; we can substitute the constraint equation for \(y\) into the equation for\(\ z\), making \(z\) a function of only one variable, \(x\). We have\[ \begin{align*} z&=z_0\ \mathrm{exp} \left(x^2+{\left(1-x\right)}^2\right) \\[4pt] &=z_0\ \mathrm{exp} \left(2x^2-2x+1\right)\end{align*} \nonumber \]To find the minimum, we equate the derivative to zero, giving\[0=\frac{dz}{dx}=\left(4x-2\right)z_0\ \mathrm{exp} \left(2x^2-2x+1\right)\nonumber \]so that the minimum occurs at \(x={1}/{2}\), y\(={1}/{2}\), and\[z=z_0\ \mathrm{exp}\left({1}/{2}\right)\nonumber \]Solving such problems by elimination of variables can become difficult. Lagrange’s method of undetermined multipliers is a general method, which is usually easy to apply and which is readily extended to cases in which there are multiple constraints. We can see how Lagrange’s method arises by thinking further about our particular example. We can imagine that we “walk” along the constraint line in the \(xy\)-plane and measure the \(z\) that is directly overhead as we progress. The problem is to find the minimum value of \(z\) that we encounter as we proceed along the line. This perspective highlights the central feature of the problem: While it is formally a problem in three dimensions (\(x\), \(y\), and \(z\)), the introduction of the constraint makes it a two-dimensional problem. We can think of one dimension as a displacement along the line \(y=1-x\), from some arbitrary starting point on the line. The other dimension is the perpendicular distance from the \(xy\)-plane to the intersection with the surface \(z\).The relevant part of the \(xy\)-plane is just the one-dimensional constraint line. We can recognize this by parameterizing the line. Let \(t\) measure location on the line relative to some initial point at which \(t\ =\ 0\). Then we have \(x=x\left(t\right)\) and \(y=y\left(t\right)\) and\[z\left(x,y\right)=z\left(x\left(t\right),y\left(t\right)\right)=z\left(t\right).\nonumber \]The point we seek is the one at which \({dz}/{dt}=0\).Now let us examine a somewhat more general problem. We want a general way to find the values \(\left(x,y\right)\) that minimize (or maximize) a function \(h=h\left(x,y\right)\) subject to a constraint of the form \(c=g\left(x,y\right)\), where \(c\) is a constant. As in our example, this constraint requires a solution in which \(\left(x,y\right)\) are on a particular line. If we parameterize this problem, we have\[h=h\left(x,y\right)=h\left(x\left(t\right),y\left(t\right)\right)=h\left(t\right)\nonumber \]and\[c=g\left(x,y\right)=g\left(x\left(t\right),y\left(t\right)\right)=g\left(t\right)\nonumber \]Because \(c\) is a constant, \({dc}/{dt}={dg}/{dt}=0\). The solution we seek is the point at which \(h\) is an extremum. At this point, \({dh}/{dt}=0\). Therefore, at the point we seek, we have\[\frac{dh}{dt}={\left(\frac{\partial h}{\partial x}\right)}_y\frac{dx}{dt}+{\left(\frac{\partial h}{\partial y}\right)}_x\frac{dy}{dt}=0\nonumber \] and \[\frac{dg}{dt}={\left(\frac{\partial g}{\partial x}\right)}_y\frac{dx}{dt}+{\left(\frac{\partial g}{\partial y}\right)}_x\frac{dy}{dt}=0\nonumber \]We can multiply either of these equations by any factor, and the product will be zero. We multiply \({dg}/{dt}\) by \(\lambda\) (where \(\lambda \neq 0\)) and subtract the result from \({dh}/{dt}\). Then, at the point we seek,\[0=\frac{dh}{dt}-\lambda \frac{dg}{dt}={\left(\frac{\partial h}{\partial x}-\lambda \frac{\partial g}{\partial x}\right)}_y\frac{dx}{dt}+{\left(\frac{\partial h}{\partial y}-\lambda \frac{\partial g}{\partial y}\right)}_x\frac{dy}{dt}\nonumber \]Since we can choose \(x\left(t\right)\) and \(y\left(t\right)\) any way we please, we can insure that \({dx}/{dt}\neq 0\) and \({dy}/{dt}\neq 0\) at the solution point. If we do so, the terms in parentheses must be zero at the solution point.Conversely, setting\[{\left(\frac{\partial h}{\partial x}-\lambda \frac{\partial g}{\partial x}\right)}_y=0\nonumber \] and \[{\left(\frac{\partial h}{\partial y}-\lambda \frac{\partial g}{\partial y}\right)}_x=0\nonumber \]is sufficient to insure that\[\frac{dh}{dt}=\lambda \frac{dg}{dt}\nonumber \]Since \({dg}/{dt}=0\), these conditions insure that \({dh}/{dt}=0\). This means that, if we can find a set \(\{x,y,\lambda \}\) satisfying\[{\left(\frac{\partial h}{\partial x}-\lambda \frac{\partial g}{\partial x}\right)}_y=0\nonumber \] and \[{\left(\frac{\partial h}{\partial y}-\lambda \frac{\partial g}{\partial y}\right)}_x=0\nonumber \] and \[c-g\left(x,y\right)=0\nonumber \]then the values of \(x\) and \(y\) must be those make \(h\left(x,y\right)\) an extremum, subject to the constraint that \(c=g\left(x,y\right)\). We have not shown that the set \(\{x,y,\lambda \}\) exists, but we have shown that if it exists, it is the desired solution.A useful mnemonic simplifies the task of generating the family of equations that we need to use Lagrange’s method. The mnemonic calls upon us to form a new function, which is a sum of the function whose extremum we seek and a series of additional terms. There is one additional term for each constraint equation. We generate this term by putting the constraint equation in the form \(c-g\left(x,y\right)=0\) and multiplying by an undetermined parameter. For the case we just considered, the mnemonic function is\[F_{mn}=h\left(x,y\right)+\lambda \left(c-g\left(x,y\right)\right)\nonumber \]We can generate the set of equations that describe the solution set, \(\{x,y,\lambda \}\), by equating the partial derivatives of \(F_{mn}\) with respect to \(x\), \(y\), and \(\lambda\) to zero. That is, the solution set satisfies the simultaneous equations\[\frac{\partial F_{mn}}{\partial x}=0\nonumber \]\[\frac{\partial F_{mn}}{\partial y}=0\nonumber \] and \[\frac{\partial F_{mn}}{\partial \lambda }=0\nonumber \]If there are multiple constraint equations, \(c_{\lambda }-g_{\lambda }\left(x,y\right)=0\), \(c_{\alpha }-g_{\alpha }\left(x,y\right)=0\), and \(c_{\beta }-g_{\beta }\left(x,y\right)=0\), then the mnemonic function is\[F_{mn}=h\left(x,y\right)+\lambda \left(c_{\lambda }-g_{\lambda }\left(x,y\right)\right)+\alpha \left(c_{\alpha }-g_{\alpha }\left(x,y\right)\right)+\beta \left(c_{\beta }-g_{\beta }\left(x,y\right)\right)\nonumber \]and the simultaneous equations that represent the constrained extremum areTo illustrate the use of the mnemonic, let us return to the example with which we began. The mnemonic equation is\[F_{mn}=z_0\ \mathrm{exp} \left(x^2+y^2\right)+\lambda \left(1-x-y\right)\nonumber \]so that\[\frac{\partial F_{mn}}{\partial x}=2xz_0\ \mathrm{exp} \left(x^2+y^2\right)-\lambda =0, \nonumber \]\[\frac{\partial F_{mn}}{\partial y}=2yz_0\ \mathrm{exp} \left(x^2+y^2\right)-\lambda =0\nonumber \]and\[\frac{\partial F_{mn}}{\partial \lambda }=1-x-y=0 \nonumber \]which yield \(x={1}/{2}\), y\(={1}/{2}\), and \(\lambda =z_0\ \mathrm{exp} \left({1}/{2}\right)\).This page titled 21.2: Lagrange's Method of Undetermined Multipliers is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.3: Deriving the Boltzmann Equation I

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/21%3A_The_Boltzmann_Distribution_Function/21.03%3A_Deriving_the_Boltzmann_Equation_I

In Sections 20-10 and 20-14, we develop the relationship between the system entropy and the probabilities of a microstate, \(\rho \left({\epsilon }_i\right)\), and an energy level, \(P_i=g_i\rho \left({\epsilon }_i\right)\), in our microscopic model. We find\[\begin{align*} S &=-Nk\sum^{\infty }_{i=1}{P_i}{ \ln \rho \left({\epsilon }_i\right)\ } \\[4pt] &=-Nk\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}{ \ln \rho \left({\epsilon }_i\right)\ } \end{align*} \]For an isolated system at equilibrium, the entropy must be a maximum, and hence\[-\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}{ \ln \rho \left({\epsilon }_i\right)} \label{maxentropy} \]must be a maximum. We can use Lagrange’s method to find the dependence of the quantum-state probability on its energy. The \(\rho \left({\epsilon }_i\right)\) must be such as to maximize entropy (Equation \ref{maxentropy}) subject to the constraints\[1=\sum^{\infty }_{i=1}{P_i}=\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)} \nonumber \]and\[\left\langle \epsilon \right\rangle =\sum^{\infty }_{i=1}{P_i{\epsilon }_i}=\sum^{\infty }_{i=1}{g_i{\varepsilon }_i\rho \left({\epsilon }_i\right)} \nonumber \]where \(\left\langle \epsilon \right\rangle\) is the expected value of the energy of one molecule. The mnemonic function becomes\[F_{mn}=-\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}{ \ln \rho \left({\epsilon }_i\right)\ }+{\alpha }^*\left(1-\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}\right)+\beta \left(\left\langle \epsilon \right\rangle -\sum^{\infty }_{i=1}{g_i{\varepsilon }_i\rho \left({\epsilon }_i\right)}\right) \nonumber \]Equating the partial derivative with respect to \(\rho \left({\epsilon }_i\right)\) to zero, \[\frac{\partial F_{mn}}{\partial \rho \left({\epsilon }_i\right)}=-g_i{ \ln \rho \left({\epsilon }_i\right)\ }-g_i-{\alpha }^*g_i-\beta g_i{\epsilon }_i=0 \nonumber \]so that\[\rho \left({\epsilon }_i\right)={\mathrm{exp} \left(-{\alpha }^*-1\right)\ }{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ } \nonumber \]From\[1=\sum^{\infty }_{i=1}{P_i}=\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)} \nonumber \]the argument we use in Section 21.1 again leads to the partition function, \(z\), and the Boltzmann equation\[P_i=g_i\rho \left({\epsilon }_i\right)=z^{-1}g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right) \nonumber \]This page titled 21.3: Deriving the Boltzmann Equation I is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.4: Deriving the Boltzmann Equation II

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/21%3A_The_Boltzmann_Distribution_Function/21.04%3A_Deriving_the_Boltzmann_Equation_II

In Section 20-9, we find that the probability of the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\) in an isolated system is\[{\rho }_{MS,N,E} = N!\prod^{\infty }_{i=1}{\frac{g^{N_i}_i}{N_i!}} \nonumber \]The thermodynamic probability\[W\left(N_i,g_i\right)=N!\prod^{\infty }_{i=1}{\frac{g^{N_i}_i}{N_i!}} \nonumber \]is the number of microstates of the population set. \(\rho_{MS,N,E}\) is the constant probability of any one microstate. In consequence, as we see in Section 20.10, the probability of a population set is proportional to its thermodynamic probability, \(W\left(N_i,g_i\right)\). It follows that the most probable population set is that for which \(W\left(N_i,g_i\right)\) is a maximum. Our microscopic model asserts that the most probable population set, \(\{N^{\textrm{⦁}}_1,\ N^{\textrm{⦁}}_2,\dots ,N^{\textrm{⦁}}_i,\dots \}\), characterizes the equilibrium state, because the equilibrium system always occupies the either the most probable population set or another population set whose macroscopic properties are indistinguishable from those of the most probable one.Evidently, the equilibrium-characterizing population set is the one for which \(W\left(N_i,g_i\right)\), or \({ \ln W\left(N_i,g_i\right)\ }\), is a maximum. Let us assume that the \(N_i\) are very large so that we can treat them as continuous variables, and we can use Stirling’s approximation for \(N_i!\). Then we can use Lagrange’s method of undetermined multipliers to find the most probable population set by finding the set, \(N_1,\ N_2,\dots ,N_i,\dots\), for which \( \ln W\left(N_i,g_i\right)\) is a maximum, subject to the constraints\[N=\sum^{\infty }_{i=1}{N_i} \nonumber \]and\[E=\sum^{\infty }_{i=1}{N_i}{\epsilon }_i. \nonumber \]From our definition of the system, both \(N\) and \(E\) are constant. The mnemonic function is\[ \begin{align*} F_{mn} &={ \ln \left(\frac{N!g^{N_1}_1g^{N_2}_2\dots g^{N_i}_i\dots }{N_1!N_2!\dots N_i!\dots }\right)\ }+\alpha \left(N-\sum^{\infty }_{i=1}{N_i}\right)+\beta \left(E-\sum^{\infty }_{i=1}{N_i{\epsilon }_i}\right) \\[4pt] &\approx N{ \ln N-N-\sum^{\infty }_{i=1}{N_i{ \ln N_i\ }}\ }+\sum^{\infty }_{i=1}{N_i}+\sum^{\infty }_{i=1}{N_i}{ \ln g_i\ }+\alpha \left(N-\sum^{\infty }_{i=1}{N_i}\right)+\beta \left(E-\sum^{\infty }_{i=1}{N_i{\epsilon }_i}\right) \end{align*} \]Taking the partial derivative with respect to \(N_i\) gives\[\frac{\partial F_{mn}}{\partial N_i}=-N_i\left(\frac{1}{N_1}\right)-{ \ln N_i\ }+1+{ \ln g_i\ }-\alpha -\beta {\epsilon }_i=-{ \ln N_i\ }+{ \ln g_i\ }-\alpha -\beta {\epsilon }_i \nonumber \]from which we have, for the population set with the largest possible thermodynamic probability,\[-{ \ln N^{\textrm{⦁}}_i\ }+{ \ln g_i\ }-\alpha -\beta {\epsilon }_i=0 \nonumber \] or \[N^{\textrm{⦁}}_i=g_i{\mathrm{exp} \left(-\alpha \right)\ }{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ } \nonumber \]We can again make use of the constraint on the total number of molecules to find \({\mathrm{exp} \left(-\alpha \right)\ }\):\[N=\sum^{\infty }_{i=1}{N^{\textrm{⦁}}_i}={\mathrm{exp} \left(-\alpha \right)\ }\sum^{\infty }_{i=1}{g_i{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ }} \nonumber \]so that \({\mathrm{exp} \left(-\alpha \right)\ }=Nz^{-1}\), where \(z\) is the partition function, \(z=\sum^{\infty }_{i=1}{g_i{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ }}\). Therefore, in the most probable population set, the number of molecules having energy \({\epsilon }_i\) is\[N^{\textrm{⦁}}_i=Nz^{-1}g_i{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ } \nonumber \]The fraction with this energy is\[\dfrac{N^{\textrm{⦁}}_i}{N}=z^{-1}g_i{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ } \nonumber \]This fraction is also the probability of finding an arbitrary molecule in one of the quantum states whose energy is \({\epsilon }_i\). When the isolated system and the corresponding constant-temperature system are functionally equivalent, this probability is \(P_i\). As in the two previous analyses, we have\[\begin{align*} P_i &=g_i\rho \left({\epsilon }_i\right) \\[4pt] &=z^{-1}g_i\ \mathrm{exp}\left(-\beta {\epsilon }_i\right). \end{align*} \]This derivation of Boltzmann’s equation from \(W_{max}\) is the most common introductory treatment. It relies on the assumption that all of the \(N_i\) are large enough to justify treating them as continuous variables. This assumption proves to be invalid for many important systems. (For ideal gases, we find that \(N_i=0\) or \(N_i=1\) for nearly all of the very large number of energy levels that are available to a given molecule.) Nevertheless, the result obtained is clearly correct; not only is it the same as the result of our two previous arguments, but also it leads to satisfactory agreement between microscopic models and the macroscopic properties of a wide variety of systems.This page titled 21.4: Deriving the Boltzmann Equation II is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.5: Partition Functions and Equilibrium - Isomeric Molecules

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In Section 20.11, we discuss chemical equilibrium between isomers from the perspective afforded by Boltzmann’s definition of entropy. Now, let us consider equilibrium in this system from the perspective afforded by the energy-level probabilities. Let us assign even-integer labels to energy levels of isomer \(A\) and odd-integer labels to energy levels of isomer \(B\). A group of atoms that can arrange itself into either a molecule of \(A\) or a molecule of \(B\) can occupy any of these energy levels. The partition function for this group of molecules to which all energy levels are available is\[z_{A+B}=\sum^{\infty }_{i=1}{g_i{\mathrm{exp} \left(-\beta {\epsilon }_i\right)\ }} \nonumber \]The fraction of molecules in the first (odd) energy level associated with molecules of isomer \(B\) is\[\dfrac{N^{\textrm{⦁}}_1}{N_{A+B}}=g_1{\left(z_{A+B}\right)}^{-1}{\mathrm{exp} \left(-\beta {\epsilon }_1\right)\ } \nonumber \]and the fraction in the next is\[\dfrac{N^{\textrm{⦁}}_3}{N_{A+B}}=g_3{\left(z_{A+B}\right)}^{-1}{\mathrm{exp} \left(-\beta {\epsilon }_3\right)\ } \nonumber \]The total number of \(B\) molecules is\[N^{\textrm{⦁}}_B=\sum_{i\ odd}{N_i} \nonumber \]so that the fraction of all of the molecules that are \(B\) molecules is\[\dfrac{N^{\textrm{⦁}}_B}{N_{A+B}}={\left(z_{A+B}\right)}^{-1}\sum_{i\ odd}{g_i{exp \left(-\beta {\epsilon }_i\right)\ }}={z_B}/{z_{A+B}} \nonumber \]Likewise, the fraction that is \(A\) molecules is\[\dfrac{N^{\textrm{⦁}}_A}{N_{A+B}}={\left(z_{A+B}\right)}^{-1}\sum_{i\ even}{g_i{exp \left(-\beta {\epsilon }_i\right)\ }}={z_A}/{z_{A+B}} \nonumber \]The equilibrium constant for the equilibrium between \(A\) and \(B\) is\[K_{eq}=\frac{N^{\textrm{⦁}}_B}{N^{\textrm{⦁}}_A}=\frac{z_B}{z_A} \nonumber \]We see that the equilibrium constant for the isomerization reaction is simply equal to the ratio of the partition functions of the isomers.It is always true that the equilibrium constant is a product of partition functions for reaction-product molecules divided by a product of partition functions for reactant molecules. However, the partition functions for the various molecules must be expressed with a common zero of energy. Choosing the infinitely separated component atoms as the zero-energy state for every molecule assures that this is the case. However, it is often convenient to express the partition function for a molecule by measuring each molecular energy level, \({\epsilon }_i\), relative to the lowest energy state of that isolated molecule. When we do this, the zero of energy is different for each molecule.To adjust the energies in a molecule’s partition function so that they are expressed relative to the energy of the molecule’s infinitely separated atoms, we must add to each molecular energy the energy required to take the molecule from its lowest energy state to its isolated component atoms. If \(z\) is the partition function when the \({\epsilon }_i\) are measured relative to the lowest energy state of the isolated molecule, \(\mathrm{\Delta }\epsilon\) is the energy released when the isolated molecule is formed from its component atoms, and \(z^{\mathrm{*}}\) is the partition function when the \({\epsilon }_i\) are measured relative to the molecule’s separated atoms, we have \(z^{\mathrm{*}} = z\mathrm{\ }\mathrm{exp}\left({ + \mathrm{\Delta }\epsilon }/{kT}\right)\).This page titled 21.5: Partition Functions and Equilibrium - Isomeric Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.6: Finding ß and the Thermodynamic Functions for Distinguishable Molecules

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All of a substance’s thermodynamic functions can be derived from the molecular partition function. We begin with the entropy. We consider closed (constant N) systems of independent, distinguishable molecules in which only pressure–volume work is possible. In Sections 20.10 and 20.14, we find that two different approaches give the entropy of this system,\[S=-Nk\sum^{\infty }_{i=1}{P_i}{ \ln \rho \left(\epsilon_i\right)\ }. \nonumber \]In Sections 20.1, 20.3, and 20.4, we find that three different approaches give the Boltzmann equation,\[P_i=g_i\rho \left(\epsilon_i\right)=z^{-1}g_i\ \mathrm{exp}\left(-\beta \epsilon_i\right). \nonumber \]We have\[ \ln \rho \left(\epsilon_i\right)=-\ln z -\beta \epsilon_i \nonumber \]Substituting, and recognizing that the energy of the N-molecule system is \(E=N\left\langle \epsilon \right\rangle\), we find that the entropy of the system is\[S=kN\sum^{\infty }_{i=1}{P_i}\left[{ \ln z\ }+\beta \epsilon_i\right]=kN{ \ln z\ }\sum^{\infty }_{i=1}{P_i}+k\beta N\sum^{\infty }_{i=1}{P_i}\epsilon_i=kN{ \ln z\ }+k\beta E \nonumber \]In Section 10.1, we find that the fundamental equation implies that\[{\left(\frac{\partial E}{\partial S}\right)}_V=T \nonumber \]Since the \(\epsilon_i\) are fixed when the volume and temperature of the system are fixed, \({ \ln z\ }\) is constant when the volume and temperature of the system are constant. Differentiating \(S=kN{ \ln z\ }+k\beta E\) with respect to \(S\) at constant \(V\), we find\[1=k\beta {\left(\frac{\partial E}{\partial S}\right)}_V=k\beta T \nonumber \] so that \[\beta =\frac{1}{kT} \nonumber \]This is an important result: Because we have now identified all of the parameters in our microscopic model, we can write the results we have found in forms that are more useful:To express the system energy in terms of the molecular partition function, we first observe that\[E=N\left\langle \epsilon \right\rangle =N\sum^{\infty }_{i=1}{P_i\epsilon_i}=Nz^{-1}\sum^{\infty }_{i=1}{g_i\epsilon_i}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]Then we observe that\[\begin{align*} \left(\frac{\partial \ln z}{\partial T}\right)_V &= z^{-1} \sum^{\infty}_{i=1} g_i \left(\frac{\epsilon_i}{kT^2}\right) \mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \\[4pt] &= \left(\frac{1}{NkT^2}\right) N z^{-1} \sum^{\infty}_{i = 1} g_i \epsilon_i \mathrm{exp} \left(\frac{-\epsilon_i}{kT}\right) \\[4pt] &=\frac{E}{NkT^2} \end{align*} \nonumber \]The system energy becomes\[\underbrace{E=NkT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V}_{\text{energy of an N-molecule system}} \nonumber \]By definition, \(A=E-TS\). Rearranging our entropy result, \(S=kN{ \ln z\ }+{E}/{T}\), we have \(E-TS=-NkT{ \ln z\ }\). Thus, \[A=-NkT{ \ln z\ } \nonumber \] (Helmholtz free energy of an N-molecule system)From \(dA=-SdT-PdV\), we have\[{\left(\frac{\partial A}{\partial V}\right)}_T=-P \nonumber \](Here, of course, \(P\) is the pressure of the system, not a probability.) Differentiating \(A=-NkT{ \ln z\ }\) with respect to \(V\) at constant \(T\), we find\[P=NkT{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T \nonumber \] (pressure of an N-molecule system)The pressure–volume product becomes\[PV=NkTV{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T \nonumber \]Substituting into \(H=E+PV\), the enthalpy becomes\[H=NkT\left[T{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V+V{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\right] \nonumber \] (enthalpy of an N-molecule system)The Gibbs free energy is given by \(G=A+PV\). Substituting, we find\[G=-NkT{ \ln z\ }+NkTV{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T \nonumber \] (Gibbs free energy of an N-molecule system)The chemical potential can be found from\[\mu ={\left(\frac{\partial A}{\partial n}\right)}_{V,T} \nonumber \]At constant volume and temperature, \(kT{ \ln z\ }\) is constant.Substituting \(N=n\overline{N}\) into \(A=-NkT{ \ln z\ }\) and taking the partial derivative, we find\[\underbrace{\mu =-\overline{N}kT{ \ln z\ }=-RT \ln z}_{\text{chemical potential of distinguishable molecules}} \nonumber \]In statistical thermodynamics we frequently express the chemical potential per molecule, rather than per mole; then, \[\mu ={\left(\frac{\partial A}{\partial N}\right)}_{V,T} \nonumber \] and\[\mu =-kT{ \ln z\ } \nonumber \](chemical potential per molecule)This page titled 21.6: Finding ß and the Thermodynamic Functions for Distinguishable Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.7: The Microscopic Model for Reversible Change

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Now let us return to the closed (constant-\(N\)) system to develop another perspective on the dependence of its macroscopic thermodynamic properties on the molecular energy levels and their probabilities. We undertake to describe the system using volume and temperature as the independent variables. In thinking about the energy-level probabilities, we stipulate that any parameters that affect the state of the system remain constant. Specifically, we mean that any parameters that appear in the Schrödinger equation remain constant. For example, the energy levels of a particle in a box depend on the mass of the particle and the length of the box. Any such parameter is called an exogenous variable. If we change an exogenous variable (say the length of the box) by a small amount, all of the energy levels change by a small amount, and all of the probabilities change by a small amount. The energy levels and their probabilities are smooth functions of the exogenous variable. If \(\xi\) is the exogenous variable, we have\[P_i=P\left(\epsilon_i\right)=g_i\rho \left(\epsilon_i\left(\xi \right)\right) \nonumber \]A change in the exogenous variable corresponds to a reversible macroscopic process.For a particle in a box, the successive \({\psi }_i\) are functions that depend on the quantum number, \(i\), and the length of the box, \(\ell\). When we change the length of the box, the wavefunction and its associated energy both change. Both are continuous functions of the length of the box. The energy is\[\epsilon_i=\frac{i^2h^2}{8m{\ell }^2} \nonumber \]Changing the length of the box is analogous to changing the volume of a system. A reversible volume change entails work. We see that changing the length of the box does work on the particle-in-a-box, just as changing the volume of a three-dimensional system does work on the system.Temperature plays a central role in the description of equilibrium from the macroscopic perspective. We can see that temperature enters the description of equilibrium from the microscopic perspective through its effect on the probability factors. When we increase the temperature of a system, its energy increases. The average energy of its molecules increases. The probability of an energy level must depend on temperature. Evidently, the probabilities of energy levels that are higher than the original average energy increase when the temperature increases. The probabilities of energy levels that are lower than the original average energy decrease when the temperature increases. The effects of heat and work on the energy levels and their equilibrium populations are diagrammed in .If our theory is to be useful, the energy we measure for a macroscopic system must be indistinguishably close to the expected value of the system energy as calculated from our microscopic model:\[E_{\mathrm{experiment}}\approx \left\langle E\right\rangle =N\left\langle \epsilon \right\rangle =N\sum^{\infty }_{i=1}{P_i\epsilon_i} \nonumber \]We can use this equation to relate the probabilities, \(P_i\), to other thermodynamic functions. Dropping the distinction between the experimental and expected energies, and assuming that the \(\epsilon_i\) and the \(P_i\) are continuous variables, we find the total differential\[dE=N\sum^{\infty }_{i=1}{\epsilon_idP_i}+N\sum^{\infty }_{i=1}{{P_id\epsilon }_i} \nonumber \]This equation is important because it describes a reversible macroscopic process in terms of the microscopic variables \(\epsilon_i\) and \(P_i\).Let us consider the first term. Since \(N\) is a constant, we have from \(N^{\textrm{⦁}}_i=P_iN\) that \(dN^{\textrm{⦁}}_i=NdP_i\). Substituting, we have\[\left(dE\right)_{\epsilon_i}=N\sum^{\infty }_{i=1}{\epsilon_idP_i}=\sum^{\infty }_{i=1}{\epsilon_i}dN^{\textrm{⦁}}_i \nonumber \]This asserts that the energy of the system changes if we redistribute the molecules among the various energy levels. If the redistribution takes molecules out of lower energy levels and puts them into higher energy levels, the energy of the system increases. This is our statistical-mechanical picture of the shift in the equilibrium position that occurs when we heat a system of independent molecules; the allocation of molecules among the available energy levels shifts to put more molecules in higher energy levels and fewer in lower ones. This corresponds to an increase in the temperature of the macroscopic system.In terms of the macroscopic system, the first term represents an increment of heat added to the system in a reversible process; that is,\[dq^{rev}=N\sum^{\infty }_{i=1}{\epsilon_idP_i} \nonumber \]The second term, \(N\sum^{\infty }_{i=1}{{P_id\epsilon }_i}\), is a contribution to the change in the energy of the system from reversible changes in the energy of the various quantum states, while the number of molecules in each quantum state remains constant. This term corresponds to a process in which the quantum states (and their energies) evolve in a continuous way as the state of the system changes. The second term represents an increment of work done on the system in a reversible process; that is\[dw^{rev}=N\sum^{\infty }_{i=1}{{P_id\epsilon }_i} \nonumber \]Evidently, the total differential expression for \(dE\) is the fundamental equation of thermodynamics expressed in terms of the variables we use to characterize the molecular system. It enables us to relate the variables that characterize our microscopic model of the molecular system to the variables that characterize the macroscopic system.For a system in which the reversible work is pressure–volume work, the energy levels depend on the volume. At constant temperature we have\[dw^{rev}=-PdV=N\sum^{\infty }_{i=1}{{P_id\epsilon }_i}=N\sum^{\infty }_{i=1}{P_i{\left(\frac{\partial \epsilon_i}{\partial V}\right)}_TdV} \nonumber \]so that the system pressure, \(P\), is related to the energy-level probabilities, \(P_i\), as\[P=-N\sum^{\infty }_{i=1}{P_i{\left(\frac{\partial \epsilon_i}{\partial V}\right)}_T} \nonumber \]To evaluate the pressure, we must know how the energy levels depend on the volume of the system.The first term relates the entropy to the energy-level probabilities. Since \(dq^{rev}=TdS=N\sum^{\infty }_{i=1}{\epsilon_idP_i}\), we have \[dS=\frac{N}{T}\sum^{\infty }_{i=1}{\epsilon_idP_i} \nonumber \]From the Boltzmann distribution function we have\(P_i=z^{-1}g_i\mathrm{exp}\left({-\epsilon_i}/{kT}\right)\), or\[\epsilon_i=-kT\ln P_i +kT\ln g_i -kT\ln z \nonumber \]Substituting into our expression for \(dS\), we find\[dS=-Nk\sum^{\infty }_{i=1}{\left(\ln P_i \right)}dP_i+Nk\sum^{\infty }_{i=1}{\left(\ln g_i \right)}dP_i-Nk\left(\ln z \right)\sum^{\infty }_{i=1}{dP_i} \nonumber \]Since \(\sum^{\infty }_{i=1}{P_i}=1\), we have \(\sum^{\infty }_{i=1}{dP_i}=0\), and the last term vanishes. Also,\[\sum^{\infty }_{i=1}{d\left(P_i\ln P_i \right)}=\sum^{\infty }_{i=1}{\left(\ln P_i \right){dP}_i}+\sum^{\infty }_{i=1}{dP_i}=\sum^{\infty }_{i=1}{\left(\ln P_i \right){dP}_i} \nonumber \]so that\[dS=-Nk\sum^{\infty }_{i=1}{d\left(P_i\ln P_i \right)}+Nk\sum^{\infty }_{i=1}{\left(\ln g_i \right)}dP_i \nonumber \]At any temperature, the probability ratio for any two successive energy levels is\[\frac{P_{i+1}\left(T\right)}{P_i\left(T\right)}=\frac{P_{i+1}}{P_i}=\frac{g_{i+1}}{g_i}\mathrm{exp}\left(\frac{-\left(\epsilon_{i+1}-\epsilon_i\right)}{kT}\right) \nonumber \]In the limit as the temperature goes to zero,\[\frac{P_{i+1}}{P_i}\to 0 \nonumber \]It follows that \(P_1\left(0\right)=1\) and \(P_i\left(0\right)=0\) for \(i>1\). Integrating from \(T=0\) to \(T\), the entropy of the system goes from \(S\left(0\right)=S_0\) to \(S\left(T\right)\), and the energy-level probabilities go from \(P_i\left(0\right)\) to \(P_i\left(T\right)\). We have\[\int^{S\left(T\right)}_{S_0}{dS}=-Nk\sum^{\infty }_{i=1}{\int^{P_i\left(T\right)}_{P_i\left(0\right)}{d\left(P_i\ln P_i \right)}}+Nk\sum^{\infty }_{i=1}{\int^{P_i\left(T\right)}_{P_i\left(0\right)}{\left(\ln g_i \right)dP_i}} \nonumber \]so that\[S\left(T\right)-S_0=-Nk\sum^{\infty }_{i=1}{P_i\left(T\right)}\ln P_i\left(T\right) +NkP_1\left(0\right)\ln P_1\left(0\right) +Nk\sum^{\infty }_{i=1}{\left(\ln g_i \right)P_i\left(T\right)}-Nk\left(\ln g_1 \right)P_1\left(0\right) \nonumber \]Since \(P_1\left(0\right)=1\), \(\ln P_1\left(0\right) \) vanishes. The entropy change becomes\[S\left(T\right)-S_0=-Nk\sum^{\infty }_{i=1}{P_i}\left[\ln P_i -\ln g_i \right]-Nk\ln g_1 =-Nk\sum^{\infty }_{i=1}{P_i}\ln \rho \left(\epsilon_i\right) -Nk\ln g_1 \nonumber \]We have \(S_0=Nk\ln g_1 \). If \(g_1=1\), the lowest energy level is non-degenerate, and \(S_0=0\); then we have\[S=-Nk\sum^{\infty }_{i=1}{P_i}\ln \rho \left(\epsilon_i\right) \nonumber \]This is the entropy of an \(N\)-molecule, constant-volume, constant-temperature system that is in thermal contact with its surroundings at the same temperature. We obtain this same result in Sections 20.10 and 20.14 by arguments in which we assume that the system is isolated. In all of these arguments, we assume that the constant-temperature system and its isolated counterpart are functionally equivalent; that is, a group of population sets that accounts for nearly all of the probability in one system also accounts for nearly all of the probability in the other.Because we obtain this result by assuming that the system is composed of \(N\), independent, non-interacting, distinguishable molecules, the entropy of this is system is \(N\) times the entropy contribution of an individual molecule. We can write\[S_{\mathrm{molecule}}=-k\sum^{\infty }_{i=1}{P_i}\ln \rho \left(\epsilon_i\right) \nonumber \]This page titled 21.7: The Microscopic Model for Reversible Change is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.8: The Third Law of Thermodynamics

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In Section 21.7, we obtain the entropy by a definite integration. We take the lower limits of integration, at \(T=0\), as \(P_1\left(0\right)=1\) and \(P_i\left(0\right)=0\), for \(i>1\). In doing so, we apply the third law of thermodynamics, which states that the entropy of a perfect crystal can be chosen to be zero when the temperature is at absolute zero. The idea behind the third law is that, at absolute zero, the molecules of a crystalline substance all are in the lowest energy level that is available to them. The probability that a molecule is in the lowest energy state is, therefore, \(P_1=1\), and the probability that it is any higher energy level, \(i>1\), is \(P_i=0\).While the fact is not relevant to the present development, we note in passing that the energy of a perfect crystal is not zero at absolute zero. While all of the constituent particles will be in their lowest vibrational energy levels at absolute zero, the energies of these lowest vibrational levels are not zero. In the harmonic oscillator approximation, the lowest energy possible for each oscillator is \({h\nu }/{2}\). (See Section 18.5).By a perfect crystalline substance we mean one in which the lowest energy level is non-degenerate; that is, for which \(g_1=1\). We see that our entropy equation conforms to the third law when we let\[S_0=Nk \ln g_1 \nonumber \]so that \(S_0=0\) when \(g_1=1\).Let us consider a crystalline substance in which the lowest energy level is degenerate; that is, one for which \(g_1>1\). This substance is not a perfect crystal. In this case, the temperature-zero entropy is\[S_0=Nk \ln g_1 >0 \nonumber \]The question arises: How can we determine whether a crystalline substance is a perfect crystal? In Chapter 11, we discuss the use of the third law to determine the absolute entropy of substances at ordinary temperatures. If we assume that the substance is a perfect crystal at zero degrees when it is not, our theory predicts a value for the absolute entropy at higher temperatures that is too small, because it does not include the term \(S_0=Nk\ln g_1\). When we use this too-small absolute entropy value to calculate entropy changes for processes involving the substance, the results do not agree with experiment.Absolute entropies based on the third law have been experimentally determined for many substances. As a rule, the resulting entropies are consistent with other experimentally observed entropy changes. In some cases, however, the assumption that the entropy is zero at absolute zero leads to absolute entropy values that are not consistent with other experiments. In these cases, the absolute entropies can be brought into agreement with other entropy measurements by assuming that, indeed, \(g_1>1\) for such substances. In any particular case, the value of \(g_1\) that must be used is readily reconciled with other information about the substance.For example, the third law entropy for carbon monoxide must be calculated taking \(g_1=2\) in order to obtain a value that is consistent with other entropy measurements. This observation is readily rationalized. In perfectly crystalline carbon monoxide, all of the carbon monoxide molecules point in the same direction, as sketched in equally energetic states for a carbon monoxide molecule in a carbon monoxide crystal at absolute zero, and we can take \(g_1=2\). (We are over-simplifying here. We explore this issue further in Section 22-7.)This page titled 21.8: The Third Law of Thermodynamics is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.9: The Partition Function for a System of N Molecules

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At a given temperature, the Boltzmann equation gives the probability of finding a molecule in any of the energy levels that the molecule can occupy. Throughout our development, we assume that there are no energies of interaction among the molecules of the system. The molecular partition function contains information about the energy levels of only one molecule. We obtain equations for the thermodynamic functions of an \(N\)-molecule system in terms of this molecular partition function. However, since these results are based on assigning the same isolated-molecule energy levels to each of the molecules, they do not address the real-system situation in which intermolecular interactions make important contributions to the total energy of the system.As we mention in Sections 20.1 and 20.3, the ensemble theory of statistical thermodynamics extends our arguments to express the thermodynamic properties of a macroscopic system in terms of all of the total energies that are available to the macroscopic system. The molecular origins of the energies of the system enter the ensemble treatment only indirectly. The theory deals with the relationships between the possible values of the energy of the system and its thermodynamic state. How molecular energy levels and intermolecular interactions give rise to these values of the system energy becomes a separate issue. Fortunately, ensemble theory just reuses—from a different perspective—all of the ideas we have just studied. The result is just the Boltzmann equation, again, but now the energies that appear in the partition function are the possible energies for the collection of \(N\) molecules, not the energies available to a single molecule.This page titled 21.9: The Partition Function for a System of N Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

21.10: Problems

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1. Consider a system with three non-degenerate quantum states having energies \({\epsilon }_1=0.9\ kT\), \({\epsilon }_2=1.0\ kT\), and \({\epsilon }_3=1.1\ kT\). The system contains \(N=3\times {10}^{10}\) molecules. Calculate the partition function and the number of molecules in each quantum state when the system is at equilibrium. This is the equilibrium population set \(\{N^{\textrm{⦁}}_1,N^{\textrm{⦁}}_2,N^{\textrm{⦁}}_3\}\). Let \(W_{mp}\) be the number of microstates associated with the equilibrium population set. Consider the population set when \({10}^{-5}\) of the molecules in \({\epsilon }_2\) are moved to each of \({\epsilon }_1\) and \({\epsilon }_3\). This is the population set \(\{N^{\textrm{⦁}}_1+{10}^{-5}N^{\textrm{⦁}}_2,\ \ \ N^{\textrm{⦁}}_2-2\times {10}^{-5},\ \ \ N^{\textrm{⦁}}_3+{10}^{-5}N^{\textrm{⦁}}_2\}\). Let \(W\) be the number of microstates associated with this non-equilibrium population set.(a) What percentage of the molecules are moved in converting the first population set into the second?(b) How do the energies of these two populations sets differ from one another?(c) Find \({W_{mp}}/{W}\). Use Stirling’s approximation and carry as many significant figures as your calculator will allow. You need at least six.(d) What does this calculation demonstrate?2. Find the approximate number of energy levels for which \(\epsilon for a molecule of molecular weight \(40\) in a box of volume \({10}^{-6}\ {\mathrm{m}}^3\) at \(300\) K.3. The partition function plays a central role in relating the probability of finding a molecule in a particular quantum state to the energy of that state. The energy levels available to a particle in a one-dimensional box are\[{\epsilon }_n=\frac{n^2h^2}{8m{\ell }^2} \nonumber \]where \(m\) is the mass of the particle and \(\ell\) is the length of the box. For molecular masses and boxes of macroscopic lengths, the factor \({h^2}/{8m{\ell }^2}\) is a very small number. Consequently, the energy levels available to a molecule in such a box can be considered to be effectively continuous in the quantum number, \(n\). That is, the partition function sum can be closely approximated by an integral in which the variable of integration, \(n\), runs from \(0\) to \(\infty\).(a) Obtain a formula for the partition function of a particle in a one-dimensional box. Integral tables give \[\int^{\infty }_0 \mathrm{exp} \left(-an^2\right) dn=\sqrt{\pi /4a} \nonumber \](b) The expected value of the energy of a molecule is given by \[\left\langle \epsilon \right\rangle =kT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V \nonumber \]What is \(\left\langle \epsilon \right\rangle\) for a particle in a box?(c) The relationship between the partition function and the per-molecule Helmholtz free energy is \(A=-kT{ \ln z\ }\). For a molecule in a one-dimensional box, we have \(dA=-SdT-\rho \ell\), where \(\rho\) is the per-molecule “pressure” on the ends of the box and \(\ell\) is the length of the box. (The increment of work associated with changing the length of the box is \(dw=-\rho \ d\ell\). In this relationship, \(d\ell\) is the incremental change in the length of the box and \(\rho\) is the one-dimensional “pressure” contribution from each molecule. \(\rho\) is, of course, just the force required to push the end of the box outward by a distance \(d\ell\). \(\rho d\ell\) is the one-dimensional analog of \(PdV\).) For the one-dimensional system, it follows that \[\rho =-{\left(\frac{\partial A}{\partial \ell }\right)}_T \nonumber \]Use this information to find \(\rho\) for a molecule in a one-dimensional box.(d) We can find \(\rho\) for a molecule in a one-dimensional box in another way. The per-molecule contribution to the pressure of a three-dimensional system is related to the energy-level probabilities, \(P_i\), by\[P^{\mathrm{system}}_{\mathrm{molecule}}=-\sum^{\infty }_{n=1}{P_n}{\left(\frac{\partial {\epsilon }_n}{\partial V}\right)}_T \nonumber \]By the same argument we use for the three-dimensional case, we find that the per-molecule contribution to the “pressure” inside a one-dimensional box is\[\rho =-\sum^{\infty }_{n=1}{P_n}{\left(\frac{\partial {\epsilon }_n}{\partial \ell }\right)}_T \nonumber \]From the equation for the energy levels of a particle in a one dimensional box, find an equation for\[{\left(\frac{\partial {\epsilon }_n}{\partial \ell }\right)}_T \nonumber \](Hint: We can express this derivative as a simple multiple of \({\epsilon }_n\).)(e) Using your result from part (d), show that the per molecule contribution, \(\rho\), to the “one-dimensional pressure” of \(N\) molecules in a one-dimensional box is \[\rho ={2\left\langle \epsilon \right\rangle }/{\ell } \nonumber \](f) Use your results from parts (b) and (e) to express \(\rho\) as a function of \(k\), \(T\), and \(\ell\).(g) Let \(\mathrm{\Pi }\) be the pressure of a system of \(N\) molecules in a one-dimensional box. From your result in part (c) or part (f), give an equation for \(\mathrm{\Pi }\). Show how this equation is analogous to the ideal gas equation.This page titled 21.10: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.1: Interpreting the Partition Function

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When it is a good approximation to say that the energy of a molecule is the sum of translational, rotational, vibrational, and electronic components, we have\[{\epsilon }_{i,j,k,m}={\epsilon }_{t,i}+{\epsilon }_{r,j}+{\epsilon }_{v,k}+{\epsilon }_{e,m} \nonumber \]where the indices \(i\), \(j\), \(k\), and \(m\) run over all possible translational, rotational, vibrational, and electronic quantum states, respectively. Then the partition function for the molecule can be expressed as a product of the individual partition functions \(z_t\), \(z_r\), \(z_v\), and \(z_e\); that is,\[\begin{align*} z_{\mathrm{molecule}} &=\sum_t{\sum_r{\sum_v{\sum_e{g_{t,i}}}}}g_{r,j}g_{v,k}g_{e,m}\mathrm{exp}\left(\frac{-{\epsilon }_{i,j,k,m}}{kT}\right) \\[4pt] &=\sum_t{g_{t,i}}exp\left(\frac{-{\epsilon }_{t,i}}{kT}\right)\sum_r{g_{r,j}}exp\left(\frac{-{\epsilon }_{r,j}}{kT}\right) \sum_v{g_{v,k}}exp\left(\frac{-{\epsilon }_{v,k}}{kT}\right)\sum_e{g_{e,m}}exp\left(\frac{-{\epsilon }_{e,m}}{kT}\right) \\[4pt] &=z_tz_rz_vz_e \end{align*} \]The magnitude of an individual partition function depends on the magnitudes of the energy levels associated with that kind of motion. Table 1 gives the contributions made to their partition functions by levels that have various energy values.We see that only quantum states whose energy is less than \(kT\) can make substantial contributions to the magnitude of a partition function. Very approximately, we can say that the partition function is equal to the number of quantum states for which the energy is less than \(kT\). Each such quantum state will contribute approximately one to the sum that comprises the partition function; the contribution of the corresponding energy level will be approximately equal to its degeneracy. If the energy of a quantum state is large compared to \(kT\), the fraction of molecules occupying that quantum state will be small. This idea is often expressed by saying that such states are “unavailable” to the molecule. It is then said that the value of the partition function is approximately equal to the number of available quantum states. When most energy levels are non-degenerate, we can also say that the value of the partition function is approximately equal to the number of available energy levels.This page titled 22.1: Interpreting the Partition Function is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.2: Conditions under which Integrals Approximate Partition Functions

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The Boltzmann equation gives the equilibrium fraction of particles in the \(i^{th}\) energy level, \(\epsilon_i\), as\[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]so the fraction of particles in energy levels less than \(\epsilon_n\) is\[f\left(\epsilon_n\right)=z^{-1}\sum^{n-1}_{i=1}{g_i}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]where \(z=\sum^{\infty }_{i=1}{g_i}\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). We can represent either of these sums as the area under a bar graph, where the height and width of each bar are \(g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\) and unity, respectively. If \(g_i\) and \(\epsilon_i\) can be approximated as continuous functions, this area can be approximated as the area under the continuous function \(y\left(i\right)=g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). That is,\[\sum^{n-1}_{i=1}g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\approx \int^n_{i=0}{g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)}di \nonumber \]To evaluate this integral, we must know how both \(g_i\) and \(\epsilon_i\) depend on the quantum number, \(i\).Let us consider the case in which \(g_i=1\) and look at the constraints that the \(\epsilon_i\) must satisfy in order to make the integral a good approximation to the sum. The graphical description of this case is sketched in Since \(\epsilon_i>\epsilon_{i-1}>0\), we have\[e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0 \nonumber \]For the integral to be a good approximation, we must have\[e^{-\epsilon_{i-1}/kT}\gg e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0, \nonumber \]which means that\[1\gg 1-e^{-\Delta \epsilon /kT}>0 \nonumber \]where \(\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\). Now,\[e^x\approx 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots \nonumber \]so that the approximation will be good if\[1\gg 1-\left(1-\frac{\Delta \epsilon }{kT}+\dots \right) \nonumber \] or \[1\gg \frac{\Delta \epsilon }{kT} \nonumber \] or \[kT\gg \Delta \epsilon \nonumber \]We can be confident that the integral is a good approximation to the exact sum whenever there are many pairs of energy levels, \(\epsilon_i\) and \(\epsilon_{i-1}\), that satisfy the condition\[\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\ll kT. \nonumber \]If there are many energy levels that satisfy \(\epsilon_i\ll kT\), there are necessarily many intervals, \(\Delta \epsilon\), that satisfy \(\Delta \epsilon \ll kT\). In short, if a large number of the energy levels of a system satisfy the criterion \(\epsilon \ll kT\), we can use integration to approximate the sums that appear in the Boltzmann equation. In Section 24.3, we use this approach and the energy levels for a particle in a box to find the partition function for an ideal gas.This page titled 22.2: Conditions under which Integrals Approximate Partition Functions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.3: Probability Density Functions from the Energies of Classical-mechanical Models

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Guided by our development of the Maxwell-Boltzmann probability density function for molecular velocities, we could postulate that similar probability density functions apply to other energies derived from classical-mechanical models for molecular motion. We will see that this can indeed be done. The results correspond to the results that we get from the Boltzmann equation, where we assume for both derivations that many energy levels satisfy \(\epsilon \ll kT\). The essential point is that, at a sufficiently high temperature, the behavior predicted by the quantum mechanical model and that predicted from classical mechanics converge. This high-temperature approximation is a good one for translational motions but a very poor one for vibrational motions. These results further illuminate the differences between the classical-mechanical and the quantum-mechanical models for the behavior of molecules.Let us look at how we can generate probability density functions based on the energies of classical-mechanical models for molecular motions. In the classical mechanical model, a particle moving in one dimension with velocity \(v\) has kinetic energy \({mv^2}/{2}\). From the discussion above, if many velocities satisfy \(kT\gg {mv^2}/{2}\), we can postulate a probability density function of the form\[\frac{df}{dv}=B_{\mathrm{trans}}\mathrm{\ exp}\left(\frac{-mv^2}{2kT}\right) \nonumber \]where \(B_{\mathrm{trans}}\) is fixed by the condition\[\int^{\infty }_{-\infty }{\left(\frac{df}{dv}\right)dv}=B_{\mathrm{trans}}\int^{\infty }_{-\infty }{\mathrm{exp}\left(\frac{-mv^2}{2kT}\right)}dv=1 \nonumber \]Evidently, this postulate assumes that each velocity constitutes a quantum state and that the degeneracy is the same for all velocities. This assumption is successful for one-dimensional translation, but not for translational motion in two or three dimensions. The definite integral is given in Appendix D. We find\[B_{\mathrm{trans}}=\left({m}/{2\pi k}T\right)^{1/2} \nonumber \] and\[\frac{df}{dv}={\left(\frac{m}{2\pi kT}\right)}^{1/2}\mathrm{exp}\left(\frac{-mv^2}{2kT}\right) \nonumber \]With \(m/kT=\lambda\), this is the same as the result that we obtain in Section 4.4. With \(B_{\mathrm{trans}}\) in hand, we can calculate the average energy associated with the motion of a gas molecule in one dimension\[\left\langle \epsilon \right\rangle =\int^{\infty}_{-\infty}{\left(\frac{mv^2}{2}\right)\left(\frac{df}{dv}\right)dv}={\left(\frac{m^3}{8\pi kT}\right)}^{1/2}\int^{\infty }_{-\infty }{v^2\mathrm{exp}\left(\frac{-mv^2}{2kT}\right)}dv \nonumber \]This definite integral is also given in Appendix D. We find \[\left\langle \epsilon_{\mathrm{trans}}\right\rangle =\frac{kT}{2} \nonumber \]We see that we can obtain the average kinetic energy for one degree of translational motion by a simple argument that uses classical-mechanical energies in the Boltzmann equation. We can make the same argument for each of the other two degrees of translational motion. We conclude that each degree of translational freedom contributes \({kT}/{2}\) to the average energy of a gas molecule. For three degrees of translational freedom, the total contribution is \({3kT}/{2}\), which is the result that we first obtained in Section 2.10.Now let us consider a classical-mechanical model for a rigid molecule rotating in a plane. The classical kinetic energy is \(\epsilon_{\mathrm{rot}}={I{\omega }^2}/{2}\), where \(I\) is the molecule’s moment of inertia about the axis of rotation, and \(\omega\) is the angular rotation rate. This has the same form as the translational kinetic energy, so if we assume \(kT\gg {I{\omega }^2}/{2}\) and a probability density function of the form\[\frac{df}{d\omega }=B_{\mathrm{rot\ }}\mathrm{exp}\left(\frac{-I{\omega }^2}{2kT}\right) \nonumber \]finding \(B_{\mathrm{rot\ }}\) and \(\left\langle \epsilon_{\mathrm{rot\ }}\right\rangle\) follows exactly as before, and the average rotational kinetic energy is\[\left\langle \epsilon_{\mathrm{rot}}\right\rangle ={kT}/{2} \nonumber \]for a molecule with one degree of rotational freedom.For a classical harmonic oscillator, the vibrational energy has both kinetic and potential energy components. They are \({mv^2}/{2}\) and \({kx^2}/{2}\) where \(v\) is the oscillator’s instantaneous velocity, \(x\) is its instantaneous location, and \(k\) is the force constant. Both of these have the same form as the translational kinetic energy equation. If we can assume that \(kT\gg {mv^2}/{2}\), that \(kT\gg {kx^2}/{2}\), and that the probability density functions are\[\frac{df}{dv}=B^{\mathrm{kinetic}}_{\mathrm{vib}}\ \mathrm{exp}\left(\frac{-mv^2}{2kT}\right) \nonumber \] and \[\frac{df}{dx}=B^{\mathrm{potential}}_{\mathrm{vib}}\mathrm{exp}\left(\frac{-kx^2}{2kT}\right) \nonumber \]the same arguments show that the average kinetic energy and the average potential energy are both \({kT}/{2}\):\[\left\langle \epsilon^{\mathrm{kinetic}}_{\mathrm{vib}}\right\rangle ={kT}/{2} \nonumber \] and \[\left\langle \epsilon^{\mathrm{potential}}_{\mathrm{vib}}\right\rangle ={kT}/{2} \nonumber \]so that the average total vibrational energy is\[\left\langle \epsilon^{\mathrm{total}}_{\mathrm{vib}}\right\rangle =kT \nonumber \]In summary, because the energy for translational motion in one dimension, the energy for rotational motion about one axis, the energy for vibrational kinetic energy in one dimension, and the energy for vibrational potential energy in one dimension all have the same form (\(\epsilon =Xu^2\)) each of these modes can contribute \({kT}/{2}\) to the average energy of a molecule. For translation and rotation, the total is \({kT}/{2}\) for each degree of translational or rotational freedom. For vibration, because there is both a kinetic and a potential energy contribution, the total is \(kT\) per degree of vibrational freedom.Let us illustrate this for the particular case of a non-linear, triatomic molecule. From our discussion in Section 18.4, we see that there are three degrees of translational freedom, three degrees of rotational freedom, and three degrees of vibrational freedom. The contributions to the average molecular energy areSince the heat capacity is\[C_V=\left(\frac{\partial \epsilon }{\partial T}\right)_v \nonumber \]each translational degree of freedom can contribute \({k}/{2}\) to the heat capacity. Each rotational degree of freedom can also contribute \({k}/{2}\) to the heat capacity. Each vibrational degree of freedom can contribute \(k\) to the heat capacity. It is important to remember that these results represent upper limits for real molecules. These limits are realized at high temperatures, or more precisely, at temperatures where many energy levels, \(\epsilon_i\), satisfy \(\epsilon_i\ll kT\)This page titled 22.3: Probability Density Functions from the Energies of Classical-mechanical Models is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.4: Partition Functions and Average Energies at High Temperatures

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It is enlightening to find the integral approximations to the partition functions and average energies for our simple quantum-mechanical models of translational, rotational, and vibrational motions. In doing so, however, it is important to remember that the use of integrals to approximate Boltzmann-equation sums assumes that there are a large number of energy levels, \(\epsilon_i\), for which \(\epsilon_i\ll kT\). If we select a high enough temperature, the energy levels for any motion will always satisfy this condition. The energy levels for translational motion satisfy this condition even at sub-ambient temperatures. This is the reason that Maxwell’s derivation of the probability density function for translational motion is successful.Rotational motion is an intermediate case. At sub-ambient temperatures, the classical-mechanical derivation can be inadequate; at ordinary temperatures, it is a good approximation. This can be seen by comparing the classical-theory prediction to experimental values for diatomic molecules. For diatomic molecules, the classical model predicts a constant-volume heat capacity of \({5k}/{2}\) from \(3\) degrees of translational and \(2\) degrees of rotational freedom. Since this does not include the contributions from vibrational motions, constant-volume heat capacities for diatomic molecules must be greater than \({5k}/{2}\) if both the translational and rotational contributions are accounted for by the classical model. For diatomic molecules at \(298\) K, the experimental values are indeed somewhat larger than \({5k}/{2}\). (Hydrogen is an exception; its value is \(2.47\ k\).)Vibrational energies are usually so big that only a minor fraction of the molecules can be in higher vibrational levels at reasonable temperatures. If we try to increase the temperature enough to make the high-temperature approximation describe vibrational motions, most molecules decompose. Likewise, electronic partition functions must be evaluated from the defining equation.The high-temperature limiting average energies can also be calculated from the Boltzmann equation and the appropriate quantum-mechanical energies. Recall that we find the following quantum-mechanical energies for simple models of translational, rotational, and vibrational motions:Translation\[\epsilon^{\left(n\right)}_{\mathrm{trans}}=\frac{n^2h^2}{8m{\ell }^2} \nonumber \](\(\mathrm{n\ =\ 1,\ 2,\ 3,\dots .}\) Derived for a particle in a box)Rotation\[\epsilon^{\left(m\right)}_{\mathrm{rot}}=\frac{m^2h^2}{8{\pi }^2I} \nonumber \] (\(\mathrm{m\ =\ 1,\ 2,\ 3,\ \dots .}\) Derived for rotation about one axis—each energy level is doubly degenerate)Vibration\[\epsilon^{\left(n\right)}_{\mathrm{vibration}}=h\nu \left(n+\frac{1}{2}\right) \nonumber \] (\(\mathrm{n\ =\ 0,\ 1,\ 2,\ 3,\dots .}\) Derived for simple harmonic motion in one dimension)When we assume that the temperature is so high that many \(\epsilon_i\) are small compared to \(kT\), we find the following high-temperature limiting partition functions for these motions:\[z_{\mathrm{translation}}=\sum^{\infty }_{n=1}{\mathrm{exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)\approx \int^{\infty }_0{\mathrm{exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)dn={\left(\frac{2\pi mkT{\ell }^2}{h^2}\right)}^{1/2} \nonumber \]\[z_{\mathrm{rotation}}=\sum^{\infty }_{m=1}{\mathrm{2\ exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)\approx 2\int^{\infty }_0{\mathrm{exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)dn={\left(\frac{8{\pi }^3IkT}{h^2}\right)}^{1/2} \nonumber \] \[z_{\mathrm{vibration}}=\sum^{\infty }_{n=0}{\mathrm{exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)\approx \int^{\infty }_0{\mathrm{exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)dn=\frac{kT}{h\nu }\mathrm{exp}\ \left(\frac{-h\nu }{2kT}\right) \nonumber \]We can then calculate the average energy for each mode as\[\left\langle \epsilon \right\rangle =z^{-1}\int^{\infty }_0{\epsilon_n}{\mathrm{exp} \left(\frac{-\epsilon_n}{kT}\right)\ }dn \nonumber \]and find\[\begin{align*} \left\langle \epsilon_{\mathrm{translation}}\right\rangle &=z^{-1}_{\mathrm{translation}}\int^{\infty }_0{\left(\frac{n^2h^2}{8m{\ell }^2}\right)\mathrm{\ exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)dn \\[4pt] &=\frac{kT}{2} \\[4pt] \left\langle \epsilon_{\mathrm{rotation}}\right\rangle &=z^{-1}_{\mathrm{rotation}}\int^{\infty }_0{2\left(\frac{m^2h^2}{8{\pi }^2I}\right)\mathrm{\ exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)dm \\[4pt] &=\frac{kT}{2} \\[4pt] \left\langle \epsilon_{\mathrm{vibration}}\right\rangle &=z^{-1}_{\mathrm{vibration}} \times \int^{\infty }_0{h\nu \left(n+\frac{1}{2}\right) \mathrm{\ exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)dn \\[4pt] &=kT+\frac{h\nu }{2} \\[4pt] &\approx kT \end{align*} \]where the last approximation assumes that \({h\nu }/{2}\ll kT\). In the limit as \(T\to 0\), the average energy of the vibrational mode becomes just \({h\nu }/{2}\). This is just the energy of the lowest vibrational state, implying that all of the molecules are in the lowest vibrational energy level at absolute zero.This page titled 22.4: Partition Functions and Average Energies at High Temperatures is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.5: Energy Levels for a Three-dimensional Harmonic Oscillator

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One of the earliest applications of quantum mechanics was Einstein’s demonstration that the union of statistical mechanics and quantum mechanics explains the temperature variation of the heat capacities of solid materials. In Section 7.14, we note that the heat capacities of solid materials approach zero as the temperature approaches absolute zero. We also review the law of Dulong and Petit, which describes the limiting heat capacity of many solid elements at high (ambient) temperatures. The Einstein model accounts for both of these observations.The physical model underlying Einstein’s development is that a monatomic solid consists of atoms vibrating about fixed points in a lattice. The particles of this solid are distinguishable from one another, because the location of each lattice point is uniquely specified. We suppose that the vibration of any one atom is independent of the vibrations of the other atoms in the lattice. We assume that the vibration results from a Hooke’s Law restoring force\[\mathop{F}\limits^{\rightharpoonup}=-\lambda \mathop{r}\limits^{\rightharpoonup}=-\lambda \left(x\mathop{\ i}\limits^{\rightharpoonup}+y\ \mathop{j}\limits^{\rightharpoonup}+z\ \mathop{k}\limits^{\rightharpoonup}\right) \nonumber \]that is zero when the atom is at its lattice point, for which \(\mathop{r}\limits^{\rightharpoonup}=\left(0,0,0\right)\). The potential energy change when the atom, of mass m, is driven from its lattice point to the point \(\mathop{r}\limits^{\rightharpoonup}=\left(x,y,x\right)\) is\[V=\int^{\mathop{r}\limits^{\rightharpoonup}}_{\mathop{r}\limits^{\rightharpoonup}=\mathop{0}\limits^{\rightharpoonup}}{-\mathop{F}\limits^{\rightharpoonup}\bullet d\mathop{r}\limits^{\rightharpoonup}}=\lambda \int^x_{x=0}{xdx}+\lambda \int^y_{y=0}{ydy}+\lambda \int^z_{z=0}{zdz}=\lambda \frac{x^2}{2}+\lambda \frac{y^2}{2}+\lambda \frac{z^2}{2} \nonumber \]The Schrödinger equation for this motion is\[-\frac{h^2}{8{\pi }^2m}\left[\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}+\frac{{\partial }^2\psi }{\partial z^2}\right]+\lambda \left[\frac{x^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}\right]\psi =\epsilon \psi \nonumber \]where \(\psi\) is a function of the three displacement coordinates; that is \(\psi =\psi \left(x,y,z\right)\). We assume that motions in the \(x\)-, \(y\)-, and \(z\)-directions are completely independent of one another. When we do so, it turns out that we can express the three-dimensional Schrödinger equation as the sum of three one-dimensional Schrödinger equations\[\ \ \ \left[-\frac{h^2}{8{\pi }^2m}\frac{{\partial }^2{\psi }_x}{\partial x^2}+\lambda \frac{x^2{\psi }_x}{2}\right] \nonumber \] \[+\left[-\frac{h^2}{8{\pi }^2m}\frac{{\partial }^2{\psi }_y}{\partial y^2}+\lambda \frac{y^2{\psi }_y}{2}\right] \nonumber \] \[+\left[-\frac{h^2}{8{\pi }^2m}\frac{{\partial }^2{\psi }_z}{\partial z^2}+\lambda \frac{z^2{\psi }_z}{2}\right] \nonumber \] \[=\epsilon \ {\psi }_x+\epsilon {\ \psi }_y+\epsilon \ {\psi }_z \nonumber \]where any wavefunction \({\psi }^{\left(n\right)}_x\) is the same function as \({\psi }^{\left(n\right)}_y\) and \({\psi }^{\left(n\right)}_z\), and the corresponding energies \({\epsilon }^{\left(n\right)}_x\), \({\epsilon }^{\left(n\right)}_y\), and \({\epsilon }^{\left(n\right)}_z\) have the same values. The energy of the three-dimensional atomic motion is simply the sum of the energies for the three one-dimensional motions. That is,\[{\epsilon }_{n,m,p}={\epsilon }^{\left(n\right)}_x+{\epsilon }^{\left(m\right)}_y+{\epsilon }^{\left(p\right)}_z, \nonumber \]which, for simplicity, we also write as\[{\epsilon }_{n,m,p}={\epsilon }_n+{\epsilon }_m+{\epsilon }_p. \nonumber \]This page titled 22.5: Energy Levels for a Three-dimensional Harmonic Oscillator is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.6: Energy and Heat Capacity of the "Einstein Crystal"

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In Section 22.4, we find an approximate partition function for the harmonic oscillator at high temperatures. Because it is a geometric series, the partition function for the harmonic oscillator can also be obtained exactly at any temperature. By definition, the partition function for the harmonic oscillator is\[z=\sum^{\infty }_{n=0}{\mathrm{exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)=\mathrm{exp}\left(\frac{-h\nu }{2kT}\right)\sum^{\infty }_{n=0}{\mathrm{exp}}\left(\frac{-nh\nu }{kT}\right)=\mathrm{exp}\left(\frac{-h\nu }{2kT}\right)\sum^{\infty }_{n=0}{{\left[{\mathrm{exp} \left(\frac{-h\nu }{kT}\right)\ }\right]}^n} \nonumber \]This is just the infinite sum\[z=a\sum^{\infty }_{n=0}{r^n}=\frac{a}{1-r} \nonumber \] with \[a=\mathrm{exp}\left(\frac{-h\nu }{2kT}\right) \nonumber \] and \[r={\mathrm{exp} \left(\frac{-h\nu }{kT}\right)\ } \nonumber \]Hence, the exact partition function for the one-dimensional harmonic oscillator is\[z=\frac{\mathrm{exp} \left(-h\nu /2kT\right)}{1- \mathrm{exp} \left(-h\nu /kT\right)} \nonumber \]The partition function for vibration in each of the other two dimensions is the same. To get the partition function for oscillation in all three dimensions, we must sum over all possible combinations of the three energies. Distinguishing the energies associated with motion in the \(x\)-, \(y\)-, and \(z\)-directions by the subscripts \(n\), \(m\), and \(p\), respectively, we have for the three-dimensional harmonic oscillator:\[\begin{aligned} z_{3D} & =\sum^{\infty }_{p=0} \sum^{\infty }_{m=0} \sum^{\infty }_{n=0} \mathrm{exp} \left[\frac{-\left({\epsilon }_n+{\epsilon }_m+{\epsilon }_p\right)}{kT}\right] \\ ~ & =\sum^{\infty }_{p=0} \mathrm{exp} \frac{-{\epsilon }_p}{kT} \sum^{\infty }_{m=0} \mathrm{exp} \frac{-{\epsilon }_m}{kT} \sum^{\infty }_{n=0} \mathrm{exp} \frac{-{\epsilon }_n}{kT} \\ ~ & =z^3 \end{aligned} \nonumber \]Hence,\[z_{3D}=\left[\frac{\mathrm{exp} \left(-h\nu /2kT\right)}{1- \mathrm{exp} \left(-h\nu /kT\right)} \right]^3 \nonumber \]and the energy of a crystal of \(N\), independent, distinguishable atoms is\[\begin{aligned} E & =N\left\langle \epsilon \right\rangle \\ ~ & =NkT^2 \left(\frac{\partial \ln z_{3D}}{ \partial T} \right)_V \\ ~ & =\frac{3Nh\nu }{2}+\frac{3Nh\nu \mathrm{exp} \left(-h\nu /kT\right)}{1-\mathrm{exp} \left(-h\nu /kT\right)} \end{aligned} \nonumber \]Taking the partial derivative with respect to temperature gives the heat capacity of this crystal. The molar heat capacity can be expressed in two ways that are useful for our purposes:\[\begin{aligned} C_V & =\left(\frac{\partial \overline{E}}{\partial T}\right)_V \\ ~ & =3\overline{N}k \left(\frac{h\nu }{kT}\right)^2\left[\frac{\mathrm{exp} \left(-h\nu /kT\right)}{\left(1-\mathrm{exp} \left(-h\nu /kT\right) \right)^2}\right] \\ ~ & =3\overline{N}k \left(\frac{h\nu }{kT}\right)^2\left[\frac{\mathrm{exp} \left(h\nu /kT\right)}{\left(\mathrm{exp} \left(h\nu /kT\right)-1\right)^2}\right] \end{aligned} \nonumber \]Consider the heat capacity at high temperatures. As the temperature becomes large, \(h\nu /kT\) approaches zero. Then\[\mathrm{exp} \left(\frac{h\nu }{kT}\right) \approx 1+\frac{h\nu }{kT} \nonumber \]Using this approximation in the second representation of \(C_V\) gives for the high temperature limit\[\begin{aligned} C_V & \approx 3\overline{N}k \left(\frac{h\nu }{kT}\right)^2\left[\frac{1+h\nu /kT}{\left(1+{h\nu }/{kT}-1\right)^2}\right] \\ ~ & \approx 3\overline{N}k\left(1+\frac{h\nu }{kT}\right) \\ ~ & \approx 3\overline{N}k=3R \end{aligned} \nonumber \]Since \(C_V\) and \(C_P\) are about the same for solids at ordinary temperatures, this result is essentially equivalent to the law stated by Dulong and Petit. Indeed, it suggests that the law would be more accurate if stated as a condition on \(C_V\) rather than \(C_P\), and this proves to be the case.At low temperatures, \(h\nu /kT\) becomes arbitrarily large and \(\mathrm{exp} \left(-h\nu /kT\right)\) approaches zero. From the first representation of \(C_V,\) we see that\[\mathop{\mathrm{lim}}_{T\to 0} \left(\frac{\partial \overline{E}}{\partial T}\right)_V =C_V=0 \nonumber \]In Section 10.9, we see that \(C_P-C_V\to 0\) as \(T\to 0\). Hence, the theory also predicts that \(C_P\to 0\) as \(T\to 0\), in agreement with experimental results.The Einstein model assumes that energy variations in a solid near absolute zero are entirely due to variations in the vibrational energy. From the assumption that all of these vibrational motions are characterized by a single frequency, it predicts the limiting values for the heat capacity of a solid at high and low temperatures. At intermediate temperatures, the quantitative predictions of the Einstein model leave room for improvement. An important refinement developed by Peter Debye assumes a spectrum of vibrational frequencies and results in excellent quantitative agreement with experimental values at all temperatures.We can give a simple qualitative interpretation for the result that heat capacities decrease to zero as the temperature goes to absolute zero. The basic idea is that, at a sufficiently low temperature, essentially all of the molecules in the system are in the lowest available energy level. Once essentially all of the molecules are in the lowest energy level, the energy of the system can no longer decrease in response to a further temperature decrease. Therefore, in this temperature range, the heat capacity is essentially zero. Alternatively, we can say that as the temperature approaches zero, the fraction of the molecules that are in the lowest energy level approaches one, and the energy of the system of \(N\) molecules approaches the smallest value it can have.The weakness in this qualitative view is that there is always a non-zero probability of finding molecules in a higher energy level, and this probability changes as the temperature changes. To firm up the simple picture, we need a way to show that the energy decreases more rapidly than the temperature near absolute zero. More precisely, we need a way to show that\[{\mathop{\mathrm{lim}}_{T\to 0} {\left(\frac{\partial \overline{E}}{\partial T}\right)}_V\ }=C_V=0 \nonumber \]Since the Einstein model produces this result, it constitutes a quantitative validation of our qualitative model.This page titled 22.6: Energy and Heat Capacity of the "Einstein Crystal" is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.7: Applications of Other Entropy Relationships

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In most cases, calculation of the entropy from information about the energy levels of a system is best accomplished using the partition function. Occasionally other entropy relationships are useful. We illustrate this by using the entropy relationship\[S=-Nk\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}{ \ln \rho \left({\epsilon }_i\right)\ }+Nk{ \ln g_1\ } \nonumber \]to find the entropy of an \(N\)-molecule disordered crystal at absolute zero. To be specific, let us consider a crystal of carbon monoxide.We can calculate the entropy of carbon monoxide at absolute zero from either of two perspectives. Let us first assume that the energy of a molecule is almost completely independent of the orientations of its neighbors in the crystal. Then the energy of any molecule in the crystal is essentially the same in either of the two orientations available to it. In this model for the system, we consider that there are two, non-degenerate, low-energy quantum states available to the molecule. We suppose that all other quantum states lie at energy levels whose probabilities are very small when the temperature is near absolute zero. We have \(g_1=g_2=1\), \({\epsilon }_2\approx {\epsilon }_1\). Near absolute zero, we have \(\rho \left({\epsilon }_2\right)\approx \rho \left({\epsilon }_1\right)\approx {1}/{2}\); for \(i>2\), \(\rho \left({\epsilon }_i\right)\approx 0\). The entropy becomes\[\begin{aligned} S & =-Nk\sum^{\infty }_{i=1}{g_i\rho \left({\epsilon }_i\right)}{ \ln \rho \left({\epsilon }_i\right)\ }+Nk{ \ln g_1\ } \\ ~ & =-Nk\left(\frac{1}{2}\right){ \ln \left(\frac{1}{2}\right)\ }-Nk\left(\frac{1}{2}\right){ \ln \left(\frac{1}{2}\right)\ } \\ ~ & =-Nk{ \ln \left(\frac{1}{2}\right)\ } \\ ~ & =Nk{ \ln 2\ } \end{aligned} \nonumber \]Alternatively, we can consider that there is just one low-energy quantum state available to the molecule but that this quantum state is doubly degenerate. In this model, the energy of the molecule is the same in either of the two orientations available to it. We have \(g_1=2\). Near absolute zero, we have \(\rho \left({\epsilon }_1\right)\approx 1\); for \(i>1\), \(\rho \left({\epsilon }_i\right)\approx 0\). The summation term vanishes, and the entropy becomes\[S=Nk{ \ln g_1\ }=Nk{ \ln 2\ } \nonumber \]Either perspective implies the same value for the zero-temperature entropy of the \(N\)-molecule crystal.Either of these treatments involves a subtle oversimplification. In our first model, we recognize that the carbon monoxide molecule must have a different energy in each of its two possible orientations in an otherwise perfect crystal. The energy of the orientation that makes the crystal perfect is slightly less than the energy of the other orientation. We introduce an approximation when we say that \(\rho \left({\epsilon }_2\right)\approx \rho \left({\epsilon }_1\right)\approx {1}/{2}\). However, if \({\epsilon }_2\) is not exactly equal to \({\epsilon }_1\), this approximation cannot be valid at an arbitrarily low temperature. To see this, we let the energy difference between these orientations be \({\epsilon }_2-{\epsilon }_1=\Delta \epsilon >0\). At relatively high temperatures, at which \(\Delta \epsilon \ll kT\), we have\[\frac{\rho \left({\epsilon }_2\right)}{\rho \left({\epsilon }_1\right)}={\mathrm{exp} \left(\frac{-\Delta \epsilon }{kT}\right)\ }\approx 1 \nonumber \]and \(\rho \left({\epsilon }_2\right)\approx \rho \left({\epsilon }_1\right)\approx {1}/{2}\). At such temperatures, the system behaves as if the lowest energy level were doubly degenerate, with \({\epsilon }_2={\epsilon }_1\). However, since \(T\) can be arbitrarily close to zero, this condition cannot always apply. No matter how small \(\Delta \epsilon\) may be, there are always temperatures at which \(\Delta \epsilon \gg kT\) and at which we have\[\frac{\rho \left({\epsilon }_2\right)}{\rho \left({\epsilon }_1\right)}\approx 0 \nonumber \]This implies that the molecule should always adopt the orientation that makes the crystal perfectly ordered when the temperature becomes sufficiently close to zero. This conclusion disagrees with the experimental observations.Our second model assumes that the energy of a carbon monoxide molecule is the same in either of its two possible orientations. However, its interactions with the surrounding molecules cannot be exactly the same in each orientation; consequently, its energy cannot be exactly the same. From first principles, therefore, our second model cannot be strictly correct.To resolve these apparent contradictions, we assume that the rate at which a carbon monoxide molecule can change its orientation within the lattice depends on temperature. For some temperature at which \(\Delta \epsilon \ll kT\), the reorientation process occurs rapidly, and the two orientations are equally probable. As the temperature decreases, the rate of reorientation becomes very slow. If the reorientation process effectively ceases to occur while the condition \(\Delta \epsilon \ll kT\) applies, the orientations of the component molecules remain those that occur at higher temperatures no matter how much the temperature decreases thereafter. This is often described by saying that molecular orientations become “frozen.” The zero-temperature entropy of the system is determined by the energy-level probabilities that describe the system at the temperature at which reorientation effectively ceases to occur.This page titled 22.7: Applications of Other Entropy Relationships is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

22.8: Problems

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1. The gravitational potential energies available to a molecule near the surface of the earth are \(\epsilon \left(h\right)=mgh\). Each height, \(h\), corresponds to a unique energy, so we can infer that the degeneracy of \(\epsilon \left(h\right)\) is unity. Derive the probability density function for the distribution of molecules in the earth’s atmosphere. (See Problem 19 in Chapter 3.)2. The value of the molecular partition function approximates the number of quantum states that are available to the molecule and whose energy is less than \(kT\). How many such quantum states are available to a molecule of molecular weight \(40\) that is confined in a volume of \({10}^{-6}\ {\mathrm{m}}^3\) at \(300\) K?This page titled 22.8: Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

23.1: Ensembles of N-molecule Systems

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When we begin our discussion of Boltzmann statistics in Chapter 20, we note that there exists, in principle, a Schrödinger equation for an \(N\)-molecule system. For any particular set of boundary conditions, the solutions of this equation are a set of infinitely many wavefunctions, \(\Psi_{i,j}\), for the \(N\)-molecule system. For every such wavefunction, there is a corresponding system energy, \(E_i\). The wavefunctions reflect all of the attractive and repulsive interactions among the molecules of the system. Likewise, the energy levels of the system reflect all of these interactions.In Section 20.12, we introduce the symbol \(\Omega_E\) to denote the degeneracy of the energy, \(E\), of an \(N\)-molecule system. Because the constituent molecules are assumed to be distinguishable and non-interacting, we have\[\Omega_E=\sum_{\left\{N_i\right\},E}{W\left(N_i,g_i\right)}\nonumber \]In the solution of the Schrödinger equation for a system of \(N\) interacting molecules, each system-energy level, \(E_i\), can be degenerate. We again let \(\Omega\) denote the degeneracy of an energy level of the system. We use \(\Omega_i\) (rather than \(\Omega_{E_i}\)) to represent the degeneracy of \(E_i\). It is important to recognize that the symbol “\(\Omega_i\)” now denotes an intrinsic quantum-mechanical property of the N-particle system.In Chapters 21 and 22, we denote the parallel properties of an individual molecule by \({\psi }_{i,j}\) for the molecular wavefunctions, \({\epsilon }_i\) for the corresponding energy levels, and \(g_i\) for the degeneracy of the \(i^{th}\) energy level. We imagine creating an \(N\)-molecule system by collecting \(N\) non-interacting molecules in a fixed volume and at a fixed temperature.In exactly the same way, we now imagine collecting \(\hat{N}\) of these \(N\)-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-interacting molecules we consider earlier, we assume that no forces act among the systems of the ensemble. However, as we emphasize above, our model for the systems of an ensemble recognizes that intermolecular forces among the molecules of an individual system can be important. We can imagine specifying the properties of the individual systems in a variety of ways. A collection is called a canonical ensemble if each of the systems in the ensemble has the same values of \(N\), \(V\), and \(T\). (The sense of this name is that by specifying constant \(N\), \(V\), and \(T\), we create the ensemble that can be described most simply.)The canonical ensemble is a collection of \(\hat{N}\) identical systems, just as the \(N\)-molecule system is a collection of \(N\) identical molecules. We imagine piling the systems that comprise the ensemble into a gigantic three-dimensional stack. We then immerse the entire stack—the ensemble—in a constant temperature bath. The ensemble and its constituent systems are at the constant temperature \(T\). The volume of the ensemble is \(\hat{N}V\). Because we can specify the location of any system in the ensemble by specifying its \(x\)-, \(y\)-, and \(z\)-coordinates in the stack, the individual systems that comprise the ensemble are distinguishable from one another. Thus the ensemble is analogous to a crystalline \(N\)-molecule system, in which the individual molecules are distinguishable from one another because each occupies a particular location in the crystal lattice, the entire crystal is at the constant temperature, \(T\), and the crystal volume is \(NV_{\mathrm{molecule}}\).Since the ensemble is a conceptual construct, we can make the number of systems in the ensemble, \(\hat{N}\), as large as we please. Each system in the ensemble will have one of the quantum-mechanically allowed energies, \(E_i\). We let the number of systems that have energy \(E_1\) be \(\hat{N}_1\). Similarly, we let the number with energy \(E_2\) be \(\hat{N}_2\), and the number with energy \(E_i\) be \(\hat{N}_i\). Thus at any given instant, the ensemble is characterized by a population set, \(\{\hat{N}_1,\ \hat{N}_2,\ \dots ,\ \hat{N}_i,\dots \}\), in exactly the same way that an \(N\)-molecule system is characterized by a population set, \(\{N_1,\ N_2,\dots ,\ N_i,\dots \}\). We have\[\hat{N}=\sum^{\infty }_{i=1}{\hat{N}_i}\nonumber \]While all of the systems in the ensemble are immersed in the same constant-temperature bath, the energy of any one system in the ensemble is completely independent of the energy of any other system. This means that the total energy of the ensemble, \(\hat{E}\), is given by\[\hat{E}=\sum^{\infty }_{i=1}{\hat{N}_i}E_i\nonumber \]The population set, \(\{\hat{N}_1,\ \hat{N}_2,\ \dots ,\ \hat{N}_i,\dots \}\), that characterizes the ensemble is not constant in time. However, by the same arguments that we apply to the N-molecule system, there is a population set\[\{\hat{N}^{\textrm{⦁}}_1,\ \hat{N}^{\textrm{⦁}}_2,\dots ,\ \hat{N}^{\textrm{⦁}}_i,\dots \}\nonumber \]which characterizes the ensemble when it is at equilibrium in the constant-temperature bath.We define the probability, \(\hat{P}_i\), that a system of the ensemble has energy \(E_i\) to be the fraction of the systems in the ensemble with this energy, when the ensemble is at equilibrium at the specified temperature. Thus, by definition,\[\hat{P}_i=\dfrac{\hat{N}^{\textrm{⦁}}_i}{\hat{N}}.\nonumber \]We define the probability that a system is in one of the states, \(\Psi_{i,j}\), with energy \(E_i\), as\[\widehat{\rho }\left(E_i\right)=\frac{\hat{P}_i}{\Omega_i}\nonumber \]The method we have used to construct the canonical ensemble insures that the entire ensemble is always at the specified temperature. If the component systems are at equilibrium, the ensemble is at equilibrium. The expected value of the ensemble energy is\[\left\langle \hat{E}\right\rangle =\hat{N}\sum^{\infty }_{i=1}{\hat{P}_iE_i=}\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i}\nonumber \]Because the number of systems in the ensemble, \(\hat{N}\), is very large, we know from the central limit theorem that any observed value for the ensemble energy will be indistinguishable from the expected value. To an excellent approximation, we have at any time,\[\hat{E}=\left\langle \hat{E}\right\rangle\nonumber \]and\[\hat{N}^{\textrm{⦁}}_i=\hat{N}_i\nonumber \]The table above summarizes the terminology that we have developed to characterize molecules, \(N\)-molecule systems, and \(\hat{N}\)-system ensembles of \(N\)-molecule systems.We can now apply to an ensemble of \(\hat{N}\), distinguishable, non-interacting systems the same logic that we applied to a system of \(N\), distinguishable, non-interacting molecules. The probability that a system is in one of the energy levels is\[1=\hat{P}_1+\hat{P}_2+\dots +\hat{P}_i+\dots\nonumber \]The total probability sum for the constant-temperature ensemble is\[1={\left(\hat{P}_1+\hat{P}_2+\dots +\hat{P}_i+\dots \right)}^\hat{N}=\sum_{\{\hat{N}_i\}}{\hat{W}\left(\hat{N}_i,\Omega_i\right)}{\widehat{\rho }\left(E_1\right)}^{\hat{N}_1}{\widehat{\rho }\left(E_2\right)}^{\hat{N}_2}\dots {\widehat{\rho }\left(E_i\right)}^{\hat{N}_i}\dots\nonumber \]where\[\hat{W}\left(\hat{N}_i,\Omega_i\right)=\hat{N}!\prod^{\infty }_{i=1}{\frac{\Omega^{\hat{N}_i}_i}{\hat{N}_i!}}\nonumber \]Moreover, we can imagine instantaneously isolating the ensemble from the temperature bath in which it is immersed. This is a wholly conceptual change, which we effect by replacing the fluid of the constant-temperature bath with a solid blanket of insulation. The ensemble is then an isolated system whose energy, \(\hat{E}\), is constant. Every system of the isolated ensemble is immersed in a constant-temperature bath, where the constant-temperature bath consists of the \(\hat{N}-1\) systems that make up the rest of the ensemble. This is an important feature of the ensemble treatment. It means that any conclusion we reach about the systems of the constant-energy ensemble is also a conclusion about each of the \(\hat{N}\) identical, constant-temperature systems that comprise the isolated, constant-energy ensemble.Only certain population sets, \(\{\hat{N}_1,\ \hat{N}_2,\ \dots ,\ \hat{N}_i,\dots \}\), are consistent with the fixed value, \(\hat{E}\), of the isolated ensemble. For each of these population sets, there are \(\hat{W}\left(\hat{N}_i,\Omega_i\right)\) system states. The probability of each of these system states is proportional to \({\widehat{\rho }\left(E_1\right)}^{\hat{N}_1}{\widehat{\rho }\left(E_2\right)}^{\hat{N}_2}\dots {\widehat{\rho }\left(E_i\right)}^{\hat{N}_i}\dots\). By the principle of equal a priori probability, every system state of the fixed-energy ensemble occurs with equal probability. We again conclude that the population set that characterizes the equilibrium state of the constant-energy ensemble, \(\{\hat{N}^{\textrm{⦁}}_1,\ \hat{N}^{\textrm{⦁}}_2,\dots ,\ \hat{N}^{\textrm{⦁}}_i,\dots \}\), is the one for which \(\hat{W}\) or \({ \ln \hat{W}\ }\) is a maximum, subject to the constraints\[\hat{N}=\sum^{\infty }_{i=1}{\hat{N}_i}\nonumber \]and\[\hat{E}=\sum^{\infty }_{i=1}{\hat{N}_i}E_i\nonumber \]The fact that we can make \(\hat{N}\) arbitrarily large ensures that any term, \(\hat{N}^{\textrm{⦁}}_i\), in the equilibrium-characterizing population set can be very large, so that \(\hat{N}^{\textrm{⦁}}_i\) can be found using Stirling’s approximation and Lagrange’s method of undetermined multipliers. We have the mnemonic function \[F_{mn}=\hat{N}{ \ln \hat{N}-\hat{N}+\sum^{\infty }_{i=1}{\left(\hat{N}_i{ \ln \Omega_i\ }-\hat{N}_i{ \ln \hat{N}_i\ }+\hat{N}_i\right)}\ }+\alpha \left(\hat{N}-\sum^{\infty }_{i=1}{\hat{N}_i}\right)+\beta \left(\hat{E}-\sum^{\infty }_{i=1}{\hat{N}_i}E_i\right)\nonumber \] so that\[{\left(\frac{\partial F_{mn}}{\partial \hat{N}^{\textrm{⦁}}_i}\right)}_{j\neq i}={ \ln \Omega_i\ }-\frac{\hat{N}^{\textrm{⦁}}_i}{\hat{N}^{\textrm{⦁}}_i}-{ \ln \hat{N}^{\textrm{⦁}}_i\ }+1-\alpha -\beta E_i=0\nonumber \]and\[{ \ln \hat{N}^{\textrm{⦁}}_i\ }={ \ln \Omega_i\ }-\alpha -\beta E_i\nonumber \]or\[\hat{N}^{\textrm{⦁}}_i=\Omega_i\exp \left(-\alpha \right) \exp -\beta E_i\nonumber \]When we make use of the constraint on the total number of systems in the ensemble, we have\[\hat{N}=\sum^{\infty}_{i=1} \hat{N}^{\textrm{⦁}}_i =\exp \left(-\alpha \right)\sum^{\infty }_{i=1} \Omega_i \exp \left(-\beta E_i\right)\nonumber \]so that\[\exp \left(-\alpha \right)=\hat{N}Z^{-1}\nonumber \]where the partition function for a system of \(N\) possibly-interacting molecules is\[Z=\sum^{\infty}_{i=1} \Omega_i \exp \left(-\beta E_i\right)\nonumber \]The probability that a system has energy \(E_i\) is equal to the equilibrium fraction of systems in the ensemble that have energy \(E_i\), so that\[\hat{P}_i=\frac{\hat{N}^{\textrm{⦁}}_i}{\hat{N}}=\frac{\Omega_i\exp \left(-\beta E_i\right)}{Z}\nonumber \]This page titled 23.1: Ensembles of N-molecule Systems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

23.2: The Ensemble Entropy and the Value of ß

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At equilibrium, the entropy of the \(\hat{N}\)-system ensemble, \(S_{\text{ensemble}}\), must be a maximum. By arguments that parallel those in Chapter 20, \(\hat{W}\) is a maximum for the ensemble population set that characterizes this equilibrium state. Applying the Boltzmann definition to the ensemble, the ensemble entropy is \(S_{\text{ensemble}}=k{ \ln {\hat{W}}_{\text{max}}\ }\). Since all \(\hat{N}\) systems in the ensemble have effectively the same entropy, \(S\), we have \(S_{\text{ensemble}}=\hat{N}S\). When we assume that \({\hat{W}}_{\text{max}}\) occurs for the equilibrium population set, \(\left\{\hat{N}^{\textrm{⦁}}_1,\ {\hat{N}}^{\textrm{⦁}}_2,\dots ,\ {\hat{N}}^{\textrm{⦁}}_i,\dots \right\}\), we have\[{\hat{W}}_{\text{max}}=\hat{N}!\prod^{\infty }_{i=1}{\frac{\Omega^{\hat{N}^{\textrm{⦁}}_i}_i}{\hat{N}^{\textrm{⦁}}_i!}} \nonumber \]so that\[S_{\text{ensemble}}=\hat{N}S=k \ln \hat{N}! +k \sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_i} {\ln \Omega_i} - k \sum^{\infty }_{i=1} { \ln \left(\hat{N}^{\textrm{⦁}}_i!\right) } \nonumber \]From the Boltzmann distribution function, \({\hat{N}^{\textrm{⦁}}_i}/{\hat{N}}=Z^{-1}\Omega_i{\mathrm{exp} \left(-\beta E_i\right)\ }\), we have\[{ \ln \Omega_i\ }={ \ln Z\ }+{ \ln {\hat{N}}^{\textrm{⦁}}_i\ }+\beta E_i-{ \ln \hat{N}\ } \nonumber \]Substituting, and introducing Stirling’s approximation, we find\[\begin{align*} \hat{N}S &=k\hat{N}{ \ln \hat{N}-k\hat{N}\ } + k\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_i\left({ \ln Z+{ \ln {\hat{N}}^{\textrm{⦁}}_i\ }\ }+\beta E_i-{ \ln \hat{N}\ }\right)}-k\sum^{\infty }_{i=1}{\left({\hat{N}}^{\textrm{⦁}}_i{ \ln {\hat{N}}^{\textrm{⦁}}_i-{\hat{N}}^{\textrm{⦁}}_i\ }\right)} \\[4pt] &=\hat{N}k{ \ln Z\ }+k\beta \sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i} \end{align*} \]Since \(\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i}\) is the energy of the \(\hat{N}\)-system ensemble and the energy of each system is the same, we have\[\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i}=E_{\text{ensemble}}=\hat{N}E \nonumber \]Substituting, we find\[S=k\beta E+k{ \ln Z\ } \nonumber \]where \(S\), \(E\), and \(Z\) are the entropy, energy, and partition function for the \(N\)-molecule system. From the fundamental equation, we have\[{\left(\frac{\partial E}{\partial S}\right)}_V=T \nonumber \]Differentiating \(S=k\beta E+k{ \ln Z\ }\) with respect to entropy at constant volume, we find\[1=k\beta {\left(\frac{\partial E}{\partial S}\right)}_V \nonumber \] and it follows that \[\beta =\frac{1}{kT} \nonumber \]We have, for the \(N\)-molecule system\[Z=\sum^{\infty }_{i=1}{\Omega_i}{\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ } \nonumber \] (System partition function)\[{\hat{P}}_i=Z^{-1}\Omega_i{\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ } \nonumber \] (Boltzmann’s equation)\[S=\frac{E}{T}+k{ \ln Z\ } \nonumber \] (Entropy of the N-molecule system)This page titled 23.2: The Ensemble Entropy and the Value of ß is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

23.3: The Thermodynamic Functions of the N-molecule System

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With the results of Section 23.2 in hand, we can find the other thermodynamic functions for the \(N\)-molecule system from the equations for \(Z\) and \(\hat{P}_i\) by the arguments we use in Chapters 20 and 21. Let us summarize these arguments. From\[E=\sum^{\infty }_{i=1}{\hat{P}_i}E_i\nonumber \]we have\[dE=\sum^{\infty }_{i=1}{E_id\hat{P}_i}+\sum^{\infty }_{i=1}{\hat{P}_i}{dE}_i\nonumber \]We associate the first term with \({dq}^{rev}\) and the second term with \(dw=-PdV\); that is,\[dq^{rev}=TdS=\sum^{\infty }_{i=1} E_id\hat{P}_i = -kT\sum^{\infty}_{i=1} \ln \left(\frac{\hat{P}_i}{\Omega_i}\right) d\hat{P}_i-kT \ln Z \sum^{\infty }_{i=1} d\hat{P}_i\nonumber \]Where we substitute\[E_i=-kT{ \ln \left(\frac{\hat{P}_i}{\Omega_i}\right)\ }-kT{ \ln Z\ }\nonumber \]which we obtain by taking the natural logarithm of the partition function. Since \(\sum^{\infty }_{i=1}{d\hat{P}_i}=0\), we have for each system,\[dS=-k\sum^{\infty }_{i=1} \ln \left(\frac{\hat{P}_i}{\Omega_i}\right) d\hat{P}_i=-k\sum^{\infty }_{i=1}{\left\{\Omega_id\left(\frac{\hat{P}_i}{\Omega_i}{ \ln \frac{\hat{P}_i}{\Omega_i}\ }\right)-d\hat{P}_i\right\}}=-k\sum^{\infty }_{i=1}{d\left(\hat{P}_i{ \ln \frac{\hat{P}_i}{\Omega_i}\ }\right)}\nonumber \]The system entropy, \(S\), and the system-energy-level probabilities, \(\hat{P}_i\), are functions of temperature. Integrating from \(T=0\) to \(T\) and choosing the lower limits for the integrations on the right to be \(\hat{P}_1\left(0\right)=1\) and \(\hat{P}_i\left(0\right)=0\) for \(i>1\), we have\[\int^S_{S_0}{dS}=-k\sum^{\infty }_{i=1}{\int^{\hat{P}_i\left(T\right)}_{\hat{P}_i\left(0\right)}{d\left(\hat{P}_i{ \ln \frac{\hat{P}_i}{\Omega_i}\ }\right)}}\nonumber \]Letting \(\hat{P}_i\left(T\right)=\hat{P}_i\), the result is\[ \begin{align*} S-S_0 &= -k\hat{P}_1{ \ln \frac{\hat{P}_1}{\Omega_1}\ }+k \ln \frac{1}{\Omega_1} -k\sum^{\infty }_{i=2}{\hat{P}_i{ \ln \frac{\hat{P}_i}{\Omega_i}\ }} \\[4pt] &=-k\sum^{\infty }_{i=1}{\hat{P}_i{ \ln \frac{\hat{P}_i}{\Omega_i}\ }}-k \ln \Omega_1 \end{align*}\nonumber \]From the partition function, we have\[{ \ln \left(\frac{\hat{P}_i}{\Omega_i}\right)\ }=-\frac{E_i}{kT}+{ \ln Z\ }\nonumber \]so that\[\begin{align*} S-S_0 &= -k\sum^{\infty }_{i=1}{\hat{P}_i}\left(-\frac{E_i}{kT}+{ \ln Z\ }\right)-k{ \ln \Omega_1\ } \\[4pt] &= \frac{1}{T}\sum^{\infty }_{i=1}{\hat{P}_i}E_i+k{ \ln Z\ }\sum^{\infty }_{i=1}{\hat{P}_i}-k{ \ln \Omega_1\ } \\[4pt]&= \frac{E}{T}+k{ \ln Z\ }-k{ \ln \Omega_1\ } \end{align*}\nonumber \]We take the system entropy at absolute zero, \(S_0\), to be\[S_0=k{ \ln \Omega_1\ }\nonumber \]If the lowest energy state is non-degenerate, \(\Omega_1=1\), and \(S_0=0\), so that\[S\left(T\right)=\frac{E}{T}+k{ \ln Z\ }\nonumber \]As in Section 21.6, we observe that\[E=\sum^{\infty }_{i=1}{\hat{P}_i}E_i=Z^{-1}\sum^{\infty }_{i=1}{\Omega_i}E_i{\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ }\nonumber \] and that\[{\left(\frac{\partial { \ln Z\ }}{\partial T}\right)}_V=Z^{-1}\sum^{\infty }_{i=1}{\Omega_i}\left(\frac{E_i}{kT^2}\right){\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ }=\frac{E}{kT^2}\nonumber \]so that\[E=kT^2{\left(\frac{\partial { \ln Z\ }}{\partial T}\right)}_V\nonumber \]From \(A=E-TS\) and the entropy equation, \(S={E}/{T}+k{ \ln Z\ }\), the Helmholtz free energy of the system is\[A=-kT{ \ln Z\ }\nonumber \]For the system pressure, we find from\[P=-{\left(\frac{\partial A}{\partial V}\right)}_T\nonumber \] that \[P=kT{\left(\frac{\partial { \ln Z\ }}{\partial V}\right)}_T\nonumber \]From \(H=E+PV\), we find\[H=kT^2{\left(\frac{\partial { \ln Z\ }}{\partial T}\right)}_V+VkT{\left(\frac{\partial { \ln Z\ }}{\partial V}\right)}_T\nonumber \]and from \(G=E+PV-TS\), we find\[G=VkT{\left(\frac{\partial { \ln Z\ }}{\partial V}\right)}_T-kT{ \ln Z\ }\nonumber \]For the chemical potential per molecule in the \(N\)-molecule system, we obtain\[\mu ={\left(\frac{\partial A}{\partial N}\right)}_{VT}=-kT{\left(\frac{\partial { \ln Z\ }}{\partial N}\right)}_{VT}\nonumber \]Thus, we have found the principle thermodynamic functions for the \(N\)-molecule system expressed in terms of \({ \ln Z\ }\) and its derivatives. The system partition function, \(Z\), depends on the energy levels available to the \(N\)-molecule system. The thermodynamic functions we have obtained are valid for any system, including systems in which intermolecular forces make large contributions to the system energy. Of course, the system partition function, \(Z\), must accurately reflect the effects of these forces.In Chapter 24 we find that the partition function, \(Z\), for a system of \(N\), distinguishable, non-interacting molecules is related in a simple way to the molecular partition function, \(z\). We find \(Z=z^N\). When we substitute this result for \(Z\) into the system partition functions developed above, we recover the same results that we developed in Chapters 20 and 21 for the thermodynamic properties of a system of \(N\), distinguishable, non-interacting molecules.This page titled 23.3: The Thermodynamic Functions of the N-molecule System is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.1: The Partition Function for N Distinguishable, Non-interacting Molecules

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In Chapter 21, our analysis of a system of \(N\) distinguishable and non-interacting molecules finds that the system entropy is given by\[S=\frac{E}{T}+Nk{ \ln z\ }=\frac{E}{T}+k{ \ln z^N\ } \nonumber \]where \(E\) is the system energy and \(z\) is the molecular partition function. From ensemble theory, we found\[S=\frac{E}{T}+k{ \ln Z\ } \nonumber \]where \(Z\) is the partition function for the \(N\)-molecule system. Comparison implies that, for a system of \(N\), distinguishable, non-interacting molecules, we have\[Z=z^N \nonumber \]We can obtain this same result by writing out the energy levels for the system in terms of the energy levels of the distinguishable molecules that make up the system. First we develop the obvious notation for the energy levels of the individual molecules. We let the energy levels of the first molecule be the set \(\{{\epsilon }_{1,i}\}\), the energy levels of the second molecule be the set \(\{{\epsilon }_{2,i}\}\), and so forth to the last molecule for which the energy levels are the set \(\{{\epsilon }_{N,i}\}\). Thus, the \(i^{th}\) energy level of the \(r^{th}\) molecule is \({\epsilon }_{r,i}\). We let the corresponding energy-level degeneracy be \(g_{r,i}\) and the partition function for the \(r^{th}\) molecule be \(z_r\). Since all of the molecules are identical, each has the same set of energy levels; that is, we have \({\epsilon }_{p,i}={\epsilon }_{r,i}\) and \(g_{p,i}=g_{r,i}\) for any two molecules, \(p\) and \(r\), and any energy level, \(i\). It follows that the partition function is the same for every molecule\[z_1=z_2=\dots =z_j=\dots =z_N=z=\sum^{\infty }_{i=1}{g_{r,i}}{\mathrm{exp} \left(\frac{-{\epsilon }_{r,i}}{kT}\right)\ } \nonumber \] so that \[z_1z_2\dots z_r\dots z_N=z^N \nonumber \]We can write down the energy levels available to the system of \(N\) distinguishable, non-interacting molecules. The energy of the system is just the sum of the energies of the constituent molecules, so the possible system energies consist of all of the possible sums of the distinguishable-molecule energies. Since there are infinitely many molecular energies, there are infinitely many system energies.\[E_1={\epsilon }_{1,1}+{\epsilon }_{2,1}+\dots +{\epsilon }_{r,1}+\dots +{\epsilon }_{N,1} \nonumber \] \[E_2={\epsilon }_{1,2}+{\epsilon }_{2,1}+\dots +{\epsilon }_{r,1}+\dots +{\epsilon }_{N,1} \nonumber \] \[E_3={\epsilon }_{1,3}+{\epsilon }_{2,1}+\dots +{\epsilon }_{r,1}+\dots +{\epsilon }_{N,1} \nonumber \] \[\dots \nonumber \] \[E_m={\epsilon }_{1,i}+{\epsilon }_{2,j}+\dots +{\epsilon }_{r,k}+\dots +{\epsilon }_{N,p} \nonumber \] \[\dots \nonumber \]The product of the \(N\) molecular partition functions is\[z_1z_2\dots z_r\dots z_N=\sum^{\infty }_{i=1}{g_{1,i}}{\mathrm{exp} \left(\frac{-{\epsilon }_{1,i}}{kT}\right)\ } \nonumber \] \[\times \sum^{\infty }_{j=1}{g_{2,j}}{\mathrm{exp} \left(\frac{-{\epsilon }_{2,j}}{kT}\right)\ }\times \dots \times \sum^{\infty }_{k=1}{g_{r,k}}{\mathrm{exp} \left(\frac{-{\epsilon }_{r,k}}{kT}\right)\ }\times \nonumber \] \[\dots \times \sum^{\infty }_{p=1}{g_{N,p}}{\mathrm{exp} \left(\frac{-{\epsilon }_{N,p}}{kT}\right)\ } \nonumber \]\[=\sum^{\infty }_{i=1}{\sum^{\infty }_{j=1}{\dots \sum^{\infty }_{k=1}{\dots \sum^{\infty }_{p=1}{g_{1,i}g_{2,j}\dots g_{r,k}\dots g_{N,p}}}}} \nonumber \] \[\times {\mathrm{exp} \left[\frac{-\left({\epsilon }_{1,i}+{\epsilon }_{2,j}+\dots +{\epsilon }_{r,k}+\dots +{\epsilon }_{N,p}\right)}{kT}\right]\ } \nonumber \]The sum in each exponential term is just the sum of \(N\) single-molecule energies. Moreover, every possible combination of \(N\) single-molecule energies occurs in one of the exponential terms. Each of these possible combinations is a separate energy level available to the system of \(N\) distinguishable molecules.The system partition function is\[Z=\sum^{\infty }_{i=1}{{\mathit{\Omega}}_i}{\mathrm{exp} \left(\frac{{-E}_i}{kT}\right)\ } \nonumber \]The \(i^{th}\) energy level of the system is the sum\[E_i={\epsilon }_{1,v}+{\epsilon }_{2,w}+\dots +{\epsilon }_{r,k}+\dots +{\epsilon }_{N,y} \nonumber \]The degeneracy of the \(i^{th}\) energy level of the system is the product of the degeneracies of the molecular energy levels that belong to it. We have\[{\mathit{\Omega}}_i=g_{1,v}g_{2,w}\dots g_{r,k}\dots g_{N,y} \nonumber \]Thus, by a second, independent argument, we see that\[z_1z_2\dots z_r\dots z_N=z^N=Z \nonumber \] (\(\mathrm{N}\) distinguishable, non-interacting molecules)This page titled 24.1: The Partition Function for N Distinguishable, Non-interacting Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.2: The Partition Function for N Indistinguishable, Non-interacting Molecules

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In all of our considerations to this point, we focus on systems in which the molecules are distinguishable. This effectively confines the practical applications to crystalline solids. Since there is no way to distinguish one molecule of a given substance from another in the gas phase, it is evident that the assumptions we have used so far do not apply to gaseous systems. The number and importance of practical applications increases dramatically if we can extend the theory to describe the behavior of ideal gases.We might suppose that distinguishability is immaterial—that there is no difference between the behavior of a system of distinguishable particles and an otherwise-identical system of indistinguishable particles. Indeed, this is an idea well worth testing. We know the partition function for a particle in box, and we have every reason to believe that this should be a good model for the partition function describing the translational motion of a gas particle. If an ideal gas behaves as a collection of \(N\) distinguishable particles-in-a-box, the translational partition of the gas is just \(z^N\). Thermodynamic properties calculated on this basis for, say, argon should agree with those observed experimentally. Indeed, when the comparison is made, this theory gives some properties correctly. The energy is correct; however, the entropy is not.Thus, experiment demonstrates that the partition function for a system of indistinguishable molecules is different from that of an otherwise-identical system of distinguishable molecules. The reason for this becomes evident when we compare the microstates available to a system of distinguishable molecules to those available to a system of otherwise-identical indistinguishable molecules. Consider the distinguishable-molecule microstate whose energy is\[E_i={\epsilon }_{1,v}+{\epsilon }_{2,w}+\dots +{\epsilon }_{r,k}+\dots +{\epsilon }_{N,y} \nonumber \]As a starting point, we assume that every molecule is in a different energy level. That is, all of the \(N\) energy levels, \({\epsilon }_{i,j}\), that appear in this sum are different. For the case in which the molecules are distinguishable, we can write down additional microstates that have this same energy just by permuting the energy values among the \(N\) molecules. (A second microstate with this energy is \(E_i = {\epsilon }_{\mathrm{1,}w} + {\epsilon }_{\mathrm{2,}v}\mathrm{+\dots +}{\epsilon }_{r,k}\mathrm{+\dots +}{\epsilon }_{N,y}\).) Since there are \(N!\) such permutations, there are a total of \(N!\) quantum states that have this same energy, and each of them appears as an exponential term in the product \(z_1z_2\dots z_r\dots z_N=z^N\).If, however, the \(N\) molecules are indistinguishable, there is no way to tell one of these \(N!\) assignments from another. They all become the same thing. All we know is that some one of the \(N\) molecules has the energy \({\epsilon }_w\), another has the energy \({\epsilon }_v\), etc. This means that there is only one way that the indistinguishable molecules can have the energy \(E_i\). It means also that the difference between the distinguishable-molecules case and the indistinguishable-molecules case is that, while they contain the same system energy levels, each level appears \(N!\) more times in the distinguishable-molecules partition function than it does in the indistinguishable-molecules partition function. We have\[Z_{\mathrm{indistinguishable}}=\frac{1}{N!}Z_{\mathrm{distinguishable}}=\frac{1}{N!}z^N \nonumber \]In the next section, we see that nearly all of the molecules in a sample of gas must have different energies, so that this relationship correctly relates the partition function for a single gas molecule to the partition function for a system of \(N\) indistinguishable gas molecules.Before seeing that nearly all of the molecules in a macroscopic sample of gas actually do have different energies, however, let us see what happens if they do not. Suppose that just two of the indistinguishable molecules have the same energy. Then there are not \(N!\) permutations of the energies among the distinguishable molecules; rather there are only \({N!}/{2!}\) such permutations. In this case, the relationship between the system and the molecular partition functions is\[Z_{\mathrm{indistinguishable}}=\frac{2!}{N!}Z_{\mathrm{distinguishable}}=\frac{2!}{N!}z^N \nonumber \]For the population set \(\{N_1,\ N_2,\dots ,N_r,\dots ,N_{\omega }\}\) the relationship is\[Z_{\mathrm{indistinguishable}}=\frac{N_1!N_2!\dots N_r!\dots N_{\omega }!}{N!}z^N \nonumber \] which is much more complex than the case in which all molecules have different energies. Of course, if we extend the latter case, so that the population set consists of N energy levels, each occupied by at most one molecule, the relationship reverts to the one with which we began.\[Z_{indistinguishable}=\frac{1}{N!}\left(\prod^{\infty }_{i=1}{N_i!}\right)z^N=\frac{1}{N!}\left(\prod^{\infty }_{i=1}{1}\right)z^N=\frac{1}{N!}z^N \nonumber \]This page titled 24.2: The Partition Function for N Indistinguishable, Non-interacting Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.3: Occupancy Probabilities for Translational Energy Levels

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The particle in a box is a quantum mechanical model for the motion of a point mass in one dimension. In Section 18.3, we find that the energy levels are\[{\epsilon }_n=\frac{n^2h^2}{8m{\ell }^2} \nonumber \]so that the partition function for a particle in a one-dimensional box is\[z=\sum^{\infty }_{n=1} \mathrm{exp} \left(\frac{-n^2h^2}{8mkT{\ell }^2}\right) \nonumber \]When the mass approximates that of a molecule, the length of the box is macroscopic, and the temperature is not extremely low, there are a very large number of energy levels for which \({\epsilon }_n. When this is the case, we find in Section 22-4 that this sum can be approximated by an integral to obtain an expression for z in closed form:\[z\approx \int^{\infty }_0 \mathrm{exp} \left(\frac{-n^2h^2}{8mkT{\ell }^2}\right)\ dn= \left(\frac{2\pi mkT}{h^2}\right)^{1/2}\ell \nonumber \]A particle in a three-dimensional rectangular box is a quantum mechanical model for an ideal gas molecule. The molecule moves in three dimensions, but the component of its motion parallel to any one coordinate axis is independent of its motion parallel to the others. This being the case, the kinetic energy of a particle in a three-dimensional box can be modeled as the sum of the energies for motion along each of the three independent coordinate axes that describe the translational motion of the particle. Taking the coordinate axes parallel to the faces of the box and labeling the lengths of the sides \({\ell }_x\), \({\ell }_y\), and \({\ell }_z\), the energy of the particle in the three-dimensional box becomes\[\epsilon ={\epsilon }_x+{\epsilon }_y+{\epsilon }_z \nonumber \]and the three-dimensional partition function becomes\[\begin{aligned} z_t & =\sum^{\infty }_{n_{x=1}} \sum^{\infty }_{n_{y=1}} \sum^{\infty }_{n_{z=1}} \mathrm{exp} \left[\left(\frac{-h^2}{8mkT}\right)\left(\frac{n^2_x}{{\ell }^2_x}+\frac{n^2_y}{{\ell }^2_y}+\frac{n^2_z}{{\ell }^2_z}\right)\right] \\ ~ & =\sum^{\infty }_{n_x=1} \mathrm{exp} \left(\frac{-n^2_xh^2}{8mkT{\ell }^2_x}\right) \sum^{\infty }_{n_y=1} \mathrm{exp} \left(\frac{-n^2_yh^2}{8mkT{\ell }^2_y}\right) \sum^{\infty }_{n_z=1} \mathrm{exp} \left(\frac{-n^2_zh^2}{8mkT{\ell }^2_z}\right) \end{aligned} \nonumber \]or, recognizing this as the product of three one-dimensional partition functions,\[z_t=z_xz_yz_z. \nonumber \]Approximating each molecular partition function as integrals gives\[z_t= \left(\frac{2\pi mkT}{h^2}\right)^{3/2}{\ell }_x{\ell }_y{\ell }_z=\left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \nonumber \]where the volume of the container is \(V={\ell }_x{\ell }_y{\ell }_z\).Let us estimate a lower limit for the molecular partition function for the translational motion of a typical gas at ambient temperature. The partition function increases with volume, \(V\), so we want to select a volume that is near the smallest volume a gas can have. We can estimate this as the volume of the corresponding liquid at the same temperature. Let us calculate the molecular translational partition function for a gas whose molar mass is \(0.040\ \mathrm{kg}\) in a volume of \(0.020\ \mathrm{L}\) at \(300\) K. We find \(z_t=5\times {10}^{27}\).Given \(z_t\), we can estimate the probability that any one of the energy levels available to this molecule is occupied. For any energy level, the upper limit to the term \(\mathrm{exp} \left({-\epsilon }_i/kT\right)\) is one. If the quantum numbers \(n_x\), \(n_y\), and \(n_z\) are different from one another, the corresponding molecular energy is non-degenerate. To a good approximation, we have \(g_i=1\). We find\[\frac{N_i}{\overline{N}}=\frac{g_i \mathrm{exp} \left(-{\epsilon }_i/kT\right)}{z_t}<\frac{1}{z_t}=2\times 10^{-28} \nonumber \]We calculate \(N_i\approx 1\times 10^{-4}\). When a mole of this gas occupies \(0.020\ \mathrm{L}\), the system density approximates that of a liquid. Therefore, even in circumstances selected to minimize the number of energy levels, there is less than one gas molecule per ten thousand energy levels.For translational energy levels of gas molecules, it is an excellent approximation to say that each molecule occupies a different translational energy level. This is a welcome result, because it assures us that the translational partition function for a system containing a gas of \(N\) indistinguishable non-interacting molecules is just\[Z_t=\frac{1}{N!} \left(\frac{2\pi mkT}{h^2}\right)^{3N/2}V^N \nonumber \]So that \(Z_t\) is the translational partition function for a system of \(N\) ideal gas molecules.We derive \(Z_t\) from the assumption that every equilibrium population number, \(N^{\textrm{⦁}}_i\), for the molecular energy levels satisfies \(N^{\textrm{⦁}}_i\le 1\). We use \(Z_t\) and the ensemble-treatment results that we develop in Chapter 23 to find thermodynamic functions for the \(N\)-molecule ideal-gas system. The ensemble development assumes that the number of systems, \(\hat{N}^{\textrm{⦁}}_i\), in the ensemble that have energy \(E_i\) is very large. Since the ensemble is a creature of our imaginations, we can imagine that \(\hat{N}\) is as big as it needs to be in order that \(\hat{N}^{\textrm{⦁}}_i\) be big enough. The population sets \(N^{\textrm{⦁}}_i\) and \(\hat{N}^{\textrm{⦁}}_i\) are independent; they characterize different distributions. The fact that \(N^{\textrm{⦁}}_i\le 1\) is irrelevant when we apply Lagrange’s method to find the distribution function for \(\hat{N}^{\textrm{⦁}}_i\), the partition function \(Z_t\), and the thermodynamic functions for the system. Consequently, the ensemble treatment enables us to find the partition function for an ideal gas, \(Z_{IG}\), by arguments that avoid the questions that arise when we apply Lagrange’s method to the distribution of molecular translational energies.This page titled 24.3: Occupancy Probabilities for Translational Energy Levels is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.4: The Separable-modes molecular Model

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At this point in our development, we have a theory that gives the thermodynamic properties of a polyatomic ideal gas molecule. To proceed, however, we must know the energy of every quantum state that is available to the molecule. There is more than one way to obtain this information. We will examine one important method—one that involves a further idealization of molecular behavior.We have made great progress by using the ideal gas model, and as we have noted repeatedly, the essential feature of the ideal gas model is that there are no attractive or repulsive forces between its molecules. Now we assume that the molecule’s translational, rotational, vibrational, and electronic motions are independent of one another. We could say that this idealization defines super-ideal gas molecules; not only does one molecule not interact with another molecule, an internal motion of one of these molecules does not interact with the other internal motions of the same molecule!The approximation that a molecule’s translational motion is independent of its rotational, vibrational, and electronic motions is usually excellent. The approximation that its intramolecular rotational, vibrational and electronic motions are also independent proves to be surprisingly good. Moreover, the very simple quantum mechanical systems that we describe in Chapter 18 prove to be surprisingly good models for the individual kinds of intramolecular motion. The remainder of this chapter illustrates these points.In Chapter 18, we note that a molecule’s wavefunction can be approximated as a product of a wavefunction for rotations, a wavefunction for vibrations, and a wavefunction for electronic motions. (As always, we are simply quoting quantum mechanical results that we make no effort to derive; we begin with the knowledge that the quantum mechanical problems have been solved and that the appropriate energy levels are available for our use.) Our goal is to see how we can apply the statistical mechanical results we have obtained to calculate the thermodynamic properties of ideal gases. To illustrate the essential features, we consider diatomic molecules. The same considerations apply to polyatomic molecules; there are additional complications, but none that introduce new principles.For diatomic molecules, we need to consider the energy levels for translational motion in three dimensions, the energy levels for rotation in three dimensions, the energy levels for vibration along the inter-nuclear axis, and the electronic energy states.This page titled 24.4: The Separable-modes molecular Model is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.5: The Partition Function for A Gas of Indistinguishable, Non-interacting, Separable-modes Molecules

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We represent the successive molecular energy levels as \({\epsilon }_i\) and the successive translational, rotational, vibrational, and electronic energy levels as \({\epsilon }_{t,a}\), \({\epsilon }_{r,b}\), \({\epsilon }_{v,c}\), and \({\epsilon }_{e,d}\). Now the first subscript specifies the energy mode; the second specifies the energy level. We approximate the successive energy levels of a diatomic molecule as\[{\epsilon }_1={\epsilon }_{t,1}+{\epsilon }_{r,1}+{\epsilon }_{v,1}+{\epsilon }_{e,1} \nonumber \] \[{\epsilon }_2={\epsilon }_{t,2}+{\epsilon }_{r,1}+{\epsilon }_{v,1}+{\epsilon }_{e,1} \nonumber \]\[\dots \nonumber \] \[{\epsilon }_i={\epsilon }_{t,a}+{\epsilon }_{r,b}+{\epsilon }_{v,c}+{\epsilon }_{e,d} \nonumber \]\[\dots \nonumber \]In Section 22.1, we find that the partition function for the molecule becomes\[\begin{align*} z&=\sum^{\infty }_{a=1}{\sum^{\infty }_{b=1}{\sum^{\infty }_{c=1}{\sum^{\infty }_{d=1}{g_{t,a}}}}}g_{r,b}g_{v,c}g_{e,d} \times {\mathrm{exp} \left[\frac{-\left({\epsilon }_{t,a}+{\epsilon }_{r,b}+{\epsilon }_{v,c}+{\epsilon }_{e,d}\right)}{kT}\right]\ } \\[4pt] &=z_tz_rz_vz_e \end{align*} \]where \(z_t\), \(z_r\), \(z_v\), and \(z_e\) are the partition functions for the individual kinds of motion that the molecule undergoes; they are sums over the corresponding energy levels for the molecule. This is essentially the same argument that we use in Section 22.1 to show that the partition function for an \(N\)-molecule system is a product of \(N\) molecular partition functions:\[Z=z^N. \nonumber \]We are now able to write the partition function for a gas containing \(N\) molecules of the same substance. Since the molecules of a gas are indistinguishable, we use the relationship\[Z_{\mathrm{indistinguishable}}=\frac{1}{N!}z^N=\frac{1}{N!}{\left(z_tz_rz_vz_e\right)}^N \nonumber \]To make the notation more compact and to emphasize that we have specialized the discussion to the case of an ideal gas, let us replace “\(Z_{\mathrm{indistinguishable}}\)” with “\(Z_{\mathrm{IG}}\)”. Also, recognizing that \(N!\) enters the relationship because of molecular indistinguishability, and molecular indistinguishability arises because of translational motion, we regroup the terms, writing\[Z_{\mathrm{IG}}=\left[\frac{{\left(z_t\right)}^N}{N!}\right]{\left(z_r\right)}^N{\left(z_v\right)}^N{\left(z_e\right)}^N \nonumber \]Our goal is to calculate the thermodynamic properties of the ideal gas. These properties depend on the natural logarithm of the ideal-gas partition function. This is a sum of terms:\[{ \ln Z_{IG}\ }={ \ln \left[\frac{{\left(z_t\right)}^N}{N!}\right]+N{ \ln z_r\ }+N{ \ln z_v\ }+N{ \ln z_e\ }\ } \nonumber \]In our development of classical thermodynamics, we find it convenient to express the properties of substance on a per-mole basis. For the same reasons, we focus on evaluating \({ \ln Z_{IG}\ }\) for one mole of gas; that is, for the case that \(N\) is Avogadro’s number, \(\overline{N}\). We now examine the relationships that enable us to evaluate each of these contributions to \({ \ln Z_{IG}\ }\).This page titled 24.5: The Partition Function for A Gas of Indistinguishable, Non-interacting, Separable-modes Molecules is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.6: The Translational Partition Function of An Ideal Gas

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We can make use of Stirling’s approximation to write the translational contribution to \({ \ln Z_{IG}\ }\) per mole of ideal gas. This is\[ \ln \left[\frac{\left(z_t\right)^{\overline{N}}}{\overline{N}!}\right] =\overline{N} \ln z_t -\overline{N} \ln \overline{N} +\overline{N}=\overline{N}+\overline{N} \ln \frac{z_t}{\overline{N}} \nonumber \](We omit the other factors in Stirling’s approximation. Their contribution to the thermodynamic values we calculate is less than the uncertainty introduced by the measurement errors in the molecular parameters we use.) In Section 24.3 we find the molecular partition function for translation:\[z_t= \left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \nonumber \]For one mole of an ideal gas, \(\overline{V}={\overline{N}kT}/{P}\). The translational contribution to the partition function for one mole of an ideal gas becomes\[ \ln \left[\frac{\left(z_t\right)^{\overline{N}}}{\overline{N}!}\right] =\overline{N}+\overline{N} \ln \left[ \left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{\overline{V}}{\overline{N}}\right] =\overline{N}+\overline{N} \ln \left[\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{kT}{P}\right] \nonumber \]This page titled 24.6: The Translational Partition Function of An Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.7: The Electronic Partition Function of an Ideal Gas

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Our quantum-mechanical model for a diatomic molecule takes the zero of energy to be the infinitely separated atoms at rest—that is, with no kinetic energy. The electrical interactions among the nuclei and electrons are such that, as the atoms approach one another, a bond forms and the energy of the two-atom system decreases. At some inter-nuclear distance, the energy reaches a minimum; at shorter inter-nuclear distances, the repulsive interactions between nuclei begin to dominate, and the energy increases. We can use quantum mechanics to find the wavefunction and energy of the molecule when the nuclei are separated to any fixed distance. By repeating the calculation at a series of inter-nuclear distances, we can find the distance at which the molecular energy is a minimum. We take this minimum energy as the electronic energy of the molecule, and the corresponding inter-nuclear distance as the bond length. This is the energy of the lowest electronic state of the molecule. The lowest electronic state is called the ground state.Excited electronic states exist, and their energies can be estimated from spectroscopic measurements or by quantum mechanical calculation. For most molecules, these excited electronic states are at much higher energy than the ground state. When we compare the terms in the electronic partition function, we see that\[{\mathrm{exp} \left({-{\epsilon }_{e,1}}/{kT}\right)\ }\gg {\mathrm{exp} \left({-{\epsilon }_{e,2}}/{kT}\right)\ } \nonumber \]The term for any higher energy level is insignificant compared to the term for the ground state. The electronic partition function becomes just\[z_e=g_1{\mathrm{exp} \left({-{\epsilon }_{e,1}}/{kT}\right)\ } \nonumber \]The ground-state degeneracy, \(g_1\), is one for most molecules. For unusual molecules the ground-state degeneracy can be greater; for molecules with one unpaired electron, it is two.The energy of the electronic ground state that we obtain by direct quantum mechanical calculation includes the energy effects of the motions of the electrons and the energy effects from the electrical interactions among the electrons and the stationary nuclei. Because we calculate it for stationary nuclei, the electronic energy does not include the energy of nuclear motions. The ground state electronic energy is the energy released when the atoms come together from infinite separation to a state in which they are at rest at the equilibrium inter-nuclear separation. This is just minus one times the work required to separate the atoms to an infinite distance, starting from the inter-nuclear separation with the smallest energy. On a graph of electronic (or potential) energy versus inter-nuclear distance, the ground state energy is just the depth of the energy well measured from the top down \(\left({\epsilon }_{e,1}<0\right)\). The work required to separate one mole of these molecules into their constituent atoms is called the equilibrium dissociation energy, and conventionally given the symbol \(D_e\). These definitions mean that \(D_e>0\) and \(D_e=-\overline{N}{\epsilon }_{e,1}\).In practice, the energy of the electronic ground state is often estimated from spectroscopic measurements. By careful study of its spectra, it is possible to find out how much energy must be added, as a photon, to cause a molecule to dissociate into atoms. Expressed per mole, this energy is called the spectroscopic dissociation energy, and it is conventionally given the symbol \(D_0\). These spectroscopic measurements involve the absorption of photons by real molecules. Before they absorb the photon, these molecules already have energy in the form of vibrational and rotational motions. So the real molecules that are involved in any spectroscopic measurement have energies that are greater than the energies of the hypothetical motionless-atom molecules at the bottom of the potential energy well. This means that less energy is required to separate the real molecule than is required to separate the hypothetical molecule at the bottom of the well. For any molecule, \(D_e\mathrm{>}D_0\).To have the lowest possible energy, a real molecule must be in its lowest rotational and lowest vibrational energy levels. As turns out, a molecule can have zero rotational energy, but its vibrational energy can never be zero. In Section 24.8 we review the harmonic oscillator approximation. In its lowest vibrational energy level \(\left(n=0\right)\), a diatomic molecule’s minimum vibrational energy is \({h\nu }/{\mathrm{2}}\). \(D_0\) and \(\nu\) can be estimated from spectroscopic experiments. We estimate\[{\epsilon }_{e,1}=-\frac{D_e}{\overline{N}}=-\left(\frac{D_0}{\overline{N}}+\frac{h\nu }{2}\right) \nonumber \]and the molecular electronic partition function becomes\[z_e=g_1{\mathrm{exp} \left(\frac{D_0}{\overline{N}kT}+\frac{h\nu }{2kT}\right)\ } \nonumber \]or\[z_e=g_1{\mathrm{exp} \left(\frac{D_0}{RT}+\frac{h\nu }{2kT}\right)\ } \nonumber \]This page titled 24.7: The Electronic Partition Function of an Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.8: The Vibrational Partition Function of A Diatomic Ideal Gas

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We base the electronic potential energy for a diatomic molecule on a model in which the nuclei are stationary at the bottom of the electronic potential energy well. We now want to expand this model to include vibrational motion of the atoms along the line connecting their nuclei. It is simple, logical, and effective to model this motion using the quantum mechanical treatment of the classical (Hooke’s law) harmonic oscillator.A Hooke’s law oscillator has a location, \(r_0\), at which the restoring force, \(F\left(r_0\right)\), and the potential energy, \(\epsilon \left(r_0\right)\), are zero. As it is displaced from \(r_0\), the oscillator experiences a restoring force that is proportional to the magnitude of the displacement, \(dF=-\lambda \ dr\). Then, we have\[\int^r_{r_0}{dF}=-\lambda \ \int^r_{r_0}{dr} \nonumber \]so that \(F\left(r\right)-F\left(r_o\right)=-\lambda \left(r-r_0\right)\). Since \(F\left(r_o\right)=0\), we have \(F\left(r\right)=-\lambda \left(r-r_0\right)\). The change in the oscillator’s potential energy is proportional to the square of the displacement,\[\epsilon \left(r\right)-\epsilon \left(r_o\right)=\int^r_{r_0}{-F\ dr}=\lambda \ \int^r_{r_0}{\left(r-r_0\right)dr\ }=\frac{\lambda }{2}{\left(r-r_0\right)}^2 \nonumber \]Since we take \(\epsilon \left(r_o\right)=0\), we have \(\epsilon \left(r\right)={\lambda {\left(r-r_0\right)}^2}/{2}\). Taking the second derivative, we find\[\frac{d^2\epsilon }{{dr}^2}=\lambda \nonumber \]Therefore, if we determine the electronic potential energy function accurately near \(r_0\), we can find \(\lambda\) from its curvature at \(r_0\).In Chapter 18, we note that the Schrödinger equation for such an oscillator can be solved and that the resulting energy levels are given by \({\epsilon }_n=h\nu \left(n+{1}/{2}\right)\) where \(\nu\) is the vibrational frequency. The relationship between frequency and force constant is\[\nu =\frac{1}{2\pi }\sqrt{\frac{\lambda }{m}} \nonumber \]where the oscillator consists of a single moving mass, \(m\). In the case where masses \(m_1\) and \(m_2\) oscillate along the line joining their centers, it turns out that the same equations describe the relative motion, if the mass, \(m\), is replaced by the reduced mass\[\mu =\frac{m_1m_2}{m_1+m_2} \nonumber \]Therefore, in principle, we can find the characteristic frequency, \(\nu\), of a diatomic molecule by accurately calculating the dependence of the electronic potential energy on \(r\) in the vicinity of \(r_0\). When we know \(\nu\), we know the vibrational energy levels available to the molecule. Alternatively, as discussed in Section 24.7, we can obtain information about the molecule’s vibrational energy levels from its infrared absorption spectrum and use these data to find \(\nu\). Either way, once we know \(\nu\), we can evaluate the vibrational partition function. We have\[z_v=\sum^{\infty }_{n=0} \mathrm{exp} \left[-\frac{h\nu }{kT}\left(n+\frac{1}{2}\right)\right] =\frac{\mathrm{exp} \left(-h\nu /2kT\right)}{1- \mathrm{exp} \left(-h\nu /kT\right)} \nonumber \]where we take advantage of the fact that the vibrational partition function is the sum of a geometric series, as we show in Section 22.6.This page titled 24.8: The Vibrational Partition Function of A Diatomic Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.9: The Rotational Partition Function of A Diatomic Ideal Gas

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For a diatomic molecule that is free to rotate in three dimensions, we can distinguish two rotational motions; however, their wave equations are intertwined, and the quantum mechanical result is that there is one set of degenerate rotational energy levels. The energy levels are\[{\epsilon }_{r,J}=\frac{J\left(J+1\right)h^2}{8{\pi }^2I} \nonumber \]with degeneracies \(g_J=2J+1\), where \(J=0,\ 1,\ 2,\ 3,\dots\).(Recall that \(I\) is the moment of inertia, defined as \(I=\sum{m_ir^2_i}\), where \(r_i\) is the distance of the \(i^{th}\) nucleus from the molecule’s center of mass. For a diatomic molecule, \(XY\), whose internuclear distance is \(r_{XY}\), the values of \(r_X\) and \(r_Y\) must satisfy the conditions \(r_X+r_Y=r_{XY}\) and \(m_Xr_X=m_Yr_Y\). From these relationships, it follows that the moment of inertia is \(I=\mu r^2_{XY}\), where \(\mu\) is the reduced mass.) For heteronuclear diatomic molecules, the rotational partition function is\[z_r=\sum^{\infty }_{J=0}{\left(2J+1\right)}{\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ } \nonumber \]For homonuclear diatomic molecules, there is a complication. This complication occurs in the quantum mechanical description of the rotation of any molecule for which there is more than one indistinguishable orientation in space. When we specify the locations of the atoms in a homonuclear diatomic molecule, like \(H_2\), we must specify the coordinates of each atom. If we rotate this molecule by \({360}^{\mathrm{o}}\) in a plane, the molecule and the coordinates are unaffected. If we rotate it by only \({180}^{\mathrm{o}}\) in a plane, the coordinates of the nuclei change, but the rotated molecule is indistinguishable from the original molecule. Our mathematical model distinguishes the \({180}^{\mathrm{o}}\)-rotated molecule from the original, unrotated molecule, but nature does not.This means that there are twice as many energy levels in the mathematical model as actually occur in nature. The rotational partition function for a homonuclear diatomic molecule is exactly one-half of the rotational partition function for an “otherwise identical” heteronuclear diatomic molecule. To cope with this complication in general, it proves to be useful to define a quantity that we call the symmetry number for any molecule. The symmetry number is usually given the symbol \(\sigma\); it is just the number of ways that the molecule can be rotated into indistinguishable orientations. For a homonuclear diatomic molecule, \(\sigma =2\); for a heteronuclear diatomic molecule, \(\sigma =1\).Making use of the symmetry number, the rotational partition function for any diatomic molecule becomes\[z_r=\left(\frac{1}{\sigma }\right)\sum^{\infty }_{J=0}{\left(2J+1\right)}{\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ } \label{exact} \]For most molecules at ordinary temperatures, the lowest rotational energy level is much less than \(kT\), and this infinite sum can be approximated to good accuracy as the corresponding integral. That is\[z_r \approx \left(\frac{1}{\sigma }\right)\int^{\infty }_{J=0}{\left(2J+1\right){\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ }}dJ \nonumber \]Initial impressions notwithstanding, this integral is easily evaluated. The substitutions \(a={h^2}/{8{\pi }^2IkT}\) and \(u=J\left(J+1\right)\) yield\[ \begin{align} z_r & \approx \left(\frac{1}{\sigma }\right)\int^{\infty }_{u=0} \mathrm{exp} \left(-au\right) du \\[4pt] & \approx \left(\frac{1}{\sigma }\right)\left(\frac{1}{a}\right)=\frac{8{\pi }^2IkT}{\sigma h^2} \label{approx}\end{align} \]To see that this is a good approximation for most molecules at ordinary temperatures, we calculate the successive terms in the partition function of the hydrogen molecule at \(25\ \mathrm{C}\). The results are shown in Table 1. We choose hydrogen because the energy difference between successive rotational energy levels becomes greater the smaller the values of \(I\) and \(T\). Since hydrogen has the smallest angular momentum of any molecule, the integral approximation will be less accurate for hydrogen than for any other molecule at the same temperature. For hydrogen, summing the first seven terms in the exact calculation (Equation \ref{exact}) gives \(z_{\mathrm{rotation}}=1.87989\), whereas the approximate calculation (Equation \ref{approx}) gives \(1.70284\). This difference corresponds to a difference of \(245\ \mathrm{J}\) in the rotational contribution to the standard Gibbs free energy of molecular hydrogen.This page titled 24.9: The Rotational Partition Function of A Diatomic Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.10: The Gibbs Free Energy for One Mole of An Ideal Gas

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In our discussion of ensembles, we find that the thermodynamic functions for a system can be expressed as functions of the system’s partition function. Now that we have found the molecular partition function for a diatomic ideal gas molecule, we can find the partition function, \(Z_{IG}\), for a gas of \(N\) such molecules. From this system partition function, we can find all of the thermodynamic functions for this \(N\)-molecule ideal-gas system. The system entropy, energy, and partition function are related to each other by the equation\[S=\frac{E}{T}+k{ \ln Z\ }_{IG} \nonumber \]Rearranging, and adding \(\left(PV\right)_{\mathrm{system}}\) to both sides, we find the Gibbs free energy\[G=E-TS+\left(PV\right)_{\mathrm{system}}= \left(PV\right)_{\mathrm{system}}-kT \ln Z_{IG} \nonumber \]For a system of one mole of an ideal gas, we have \(\left(PV\right)_{\mathrm{system}}=\overline{N}kT\). If the ideal gas is diatomic, we can substitute the molecular partition functions developed above to find\[ \begin{align*} G_{IG}&=\overline{N}kT-kT \ln Z_{IG} \\[4pt] &=\overline{N}kT-\mathrm{kT ln} \left[\frac{\left(z_t\right)^{\overline{N}}}{\overline{N}!}\right] -\overline{N}kT \ln z_r -\overline{N}kT \ln z_v -\overline{N}kT \ln z_e \\[4pt] &=\overline{N}kT-\overline{N}kT-\overline{N}kT \ln \left[\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{kT}{P}\right] -\overline{N}kT \ln \left(\frac{8{\pi }^2IkT}{\sigma h^2}\right) -\overline{N}kT \ln \left(\frac{\mathrm{exp} \left(-h\nu/2kT\right)}{1-\mathrm{exp} \left(-h\nu /kT\right)}\right) -\overline{N}kT \ln \left(\frac{D_0}{RT}+\frac{h\nu }{2kT}\right) \end{align*} \]For the standard Gibbs free energy of an ideal gas, we define the pressure to be one bar. Introduction of this condition \(\left(P=P^o=1\ \mathrm{bar}={10}^5\ \mathrm{Pa}\right)\) and further simplification gives\[G^o_{IG}=-RT \ln \left[\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{kT}{P^o}\right] -RT \ln \left(\frac{8{\pi }^2IkT}{\sigma h^2}\right) -RT \ln \left(\frac{\mathrm{exp} \left(-h\nu/2kT\right)}{1-\mathrm{exp} \left(-h\nu /kT\right)}\right)-RT\left(\frac{D_0}{RT}+\frac{h\nu }{2kT}\right) \nonumber \]In this form, the successive terms represent, respectively, the translational, rotational, vibrational, and electronic contributions to the Gibbs free energy. Further simplification results because vibrational and electronic contributions from terms involving \(h\nu /2kT\) cancel. This is a computational convenience. Factoring out \(RT\),\[G^o_{IG}=-RT\left\{ \ln \left[\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\frac{kT}{P^o}\right] + \ln \left(\frac{8{\pi }^2IkT}{\sigma h^2}\right) - \ln \left(1-\mathrm{exp} \left(-h\nu/kT \right) \right) +\frac{D_0}{RT}\right\} \nonumber \]This page titled 24.10: The Gibbs Free Energy for One Mole of An Ideal Gas is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

24.11: The Standard Gibbs Free Energy for H₂(g), I₂(g), and HI(g)

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/24%3A_Indistinguishable_Molecules_-_Statistical_Thermodynamics_of_Ideal_Gases/24.11%3A_The_Standard_Gibbs_Free_Energy_for_H(g)_I(g)_and_HI(g)

For many diatomic molecules, the data needed to calculate \(G^o_{IG}\) are readily available in various compilations. For illustration, we consider the molecules \(H_2\), \(I_2\), and \(HI\). The necessary experimental data are summarized in Table 2.The terms in the simplified equation for the standard Gibbs free energy at \(298.15\) K are given in Table 3.Finally, the standard molar Gibbs Free Energies at \(298.15\) K are summarized in Table 4.These results can be used to calculate the standard Gibbs free energy change, at \(298.15\) K, for the reaction\[H_2\left(g\right)+I_2\left(g\right)\to 2HI\left(g\right). \nonumber \]We find\[{\Delta }_rG^o_{298}=2G^o\left(HI,\ g,\ 298.15\ \mathrm{K}\right)-G^o\left(H_2,\ g,\ 298.15\ \mathrm{K}\right)-G^o\left(I_2,\ g,\ 298.15\ \mathrm{K}\right)=-16.20\ \mathrm{kJ} \nonumber \]This page titled 24.11: The Standard Gibbs Free Energy for H₂(g), I₂(g), and HI(g) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.