1.5.1: Chemical Potentials, Composition and the Gas Constant
In many Topics describing the thermodynamic properties of liquid mixtures and solutions, key equations relate the chemical potentials of components to the composition of a given system. For example in the case of a binary aqueous mixture the chemical potential of water \(\mu_{1}(\mathrm{~T}, \mathrm{p}, \mathrm{mix})\) is related to the mole fraction of water \(x_{1}\) at temperature \(\mathrm{T}\) and pressure \(p\) using equation (a).
\[\mu_{1}(\mathrm{~T}, \mathrm{p}, \operatorname{mix})=\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
\[\text { By definition, limit }\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{1}=1.0 \nonumber \]
Here \(\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)\) is the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(p\); \(\mathrm{f}_{1}\) is the rational activity coefficient of water in the mixture.
Similarly for solute \(j\) in an aqueous solution at temperature \(\mathrm{T}\) and pressure \(p\), the chemical potential of solute \(j\), \(\mu_{j}(T, p, a q)\) is related to the molality mj using equation (c) where \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\).
\[\mu_{\mathrm{j}}(\mathrm{aq}, \mathrm{T}, \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
\[\text { By definition, at all T and } p \text { limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \nonumber \]
Here \(\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})\) is the chemical potentials of solute \(j\) in an aqueous solution at the same \(\mathrm{T}\) and \(p\) where \(\mathrm{m}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{~kg}\) and \(\gamma_{\mathrm{j}}=1.0\).
In equations (a) and (c) the parameter \(\mathrm{R}\) is the Gas Constant, \(8.314 \mathrm{~J mol}^{-1} \mathrm{~K}^{-1}\). The word ‘Gas’ in the latter sentence is interesting bearing in mind that equations (a) and (c) describe the properties of liquids, mixtures and solutions. Here we examine how this parameter emerges in these equations.
The starting point is a description of a closed system containing \(i\)–chemical substances, the amount of chemical substance \(j\) being \(n_{j}\).
\[\text { Then, } \mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
The chemical potential \(\mu_{j}(T, p)\) of chemical substance \(j\) is given by equation (f).
\[\mu_{\mathrm{j}}(\mathrm{T}, \mathrm{p})=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
Moreover the partial molar volume \(\mathrm{V}_{j}\) of chemical substance \(j\) is given by equation (g).
\[V_{j}=\left(\frac{\partial \mu_{j}}{\partial p}\right)_{T} \nonumber \]
We simplify the argument by considering a system comprising pure chemical substance 1.
\[\text { Then } \quad \mathrm{V}_{1}^{*}=\left(\frac{\partial \mu_{1}^{*}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Thus \(\mathrm{V}_{1}^{*}(\mathrm{~T}, \mathrm{p})\) is the molar volume of pure substance 1 at temperature \(\mathrm{T}\) and pressure \(p\). In the event that chemical substance 1 is a perfect (ideal) gas, the following equation describes the \(p-\mathrm{V}-\mathrm{T}\) properties.
\[p_{1}^{*} \, V_{1}^{*}(g)=R \, T \nonumber \]
We write equation (h) in the following form describing an ideal gas at constant temperature \(\mathrm{T}\).
\[d \mu_{1}^{*}(g)=V_{1}^{*}(g) \, d p \nonumber \]
Equations (i) and (j) yield equation (k).
\[\mathrm{d} \mu_{1}^{*}(\mathrm{~g})=\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \mathrm{p}_{1}^{*} \nonumber \]
We integrate equation (k) between limits \(p_{1}^{*}\) and \(p^{0}\) where \(p^{0}\) is the standard pressure, \(101325 \mathrm{~N m}^{-2}\).
\[\text { Hence, at temperature } \mathrm{T}, \mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}_{1}^{*}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{*} / \mathrm{p}^{0}\right) \nonumber \]
In a more complicated system, the gas phase is a gaseous mixture, comprising two components, component 1 and component 2 with partial pressures \(\mathrm{p}_{1}\) and \(\mathrm{p}_{2}\). We assume the thermodynamic properties of the gas phase in equilibrium with a liquid phase are ideal. Hence equation (l) takes the following form where \(\mu_{1}\left(g ; \text { mix; } p_{1}\right)\) is the chemical potential of gas-1 at partial pressure \(\mathrm{p}_{1}\).
\[\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}_{1}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right) \nonumber \]
Liquid Mixtures
A given closed system contains chemical substances 1 and 2, present in two phases, gas and a liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). Thus \(\mathrm{p}_{1}^{\text {eq }}\) is the equilibrium partial pressure of chemical substance 1 in the gas phase. At equilibrium the chemical potentials of chemical substance 1 in the vapour and liquid mixture phases are equal.
\[\mu_{1}^{\mathrm{eq}}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{1}^{\mathrm{eq}}\right) \nonumber \]
Thus \(\mathrm{p}_{1}^{\text {eq }}\) is the partial pressure of chemical substance 1 in the gas phase, the superscript ‘eq’ indicating an equilibrium with the liquid phase at pressure \(\mathrm{p}\); the complete system is at temperature \(\mathrm{T}\).
Hence using equations (m) and (n) we obtain an equation for the equilibrium chemical potential of chemical substance 1 in an ideal liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)
\[\mu_{1}^{\text {eq }}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right) \nonumber \]
The thermodynamic analysis calls on the results of experiments in which the partial pressure \(\mathrm{p}_{i}\) of chemical substance-\(i\) in a liquid mixture at temperature \(\mathrm{T}\) is measured as a function of mole fraction \(\mathrm{x}_{i}\). It turns out that for nearly all liquid mixtures at fixed temperature, \(\mathrm{p}_{i}\) is approximately a linear function of the mole fraction \(\mathrm{x}_{1}\) at low \(\mathrm{x}_{1}\). We therefore define an ideal liquid mixture. By definition the (equilibrium) vapour pressure of chemical substance \(i\), one component of a liquid mixture, is related to the mole fraction composition at temperature \(\mathrm{T}\) using equation (p).
\[\text { Thus } \mathrm{p}_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{T} ; \text { mix } ; \mathrm{id})=\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T}) \nonumber \]
Here \(\mathrm{x}_{i}\) is the mole fraction of component-\(i\) in the liquid mixture; \(\mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T})\) is the vapour pressure of pure liquid substance 1 at temperature \(\mathrm{T}\).
For example if \(\mathrm{x}_{i}\) is \(0.5\), the contribution to the vapour pressure of the (ideal) mixture is one-half of the vapour pressure of the pure liquid-\(i\) at the same temperature. Equation (p) is Raoult’s law, describing the properties of an ideal liquid mixture having ideal thermodynamic properties. We note that the Gas Constant emerges in equation (o) because the r.h.s. of equation (o) describes the properties of chemical substance 1 in the vapour phase.
Combination of equations (o) and (p) yields equation (q).
\[\mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right] \nonumber \]
\[\text { Or, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right]+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right) \nonumber \]
For the pure liquid-\(i\) at pressure \(\mathrm{p}\),
\[\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}(\mathrm{T}) / \mathrm{p}^{0}\right] \nonumber \]
\[\text { Hence, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \text { mix } ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right) \nonumber \]
We notice that the Gas Constant in equation (t) emerged from equation (i) describing the properties of an ideal gas.
Solutions
A similar argument is used when we turn our attention to the thermodynamic properties of a solute, chemical substance \(j\). In this case we use Henry’s Law as the link between theory and the properties of solutions. This law relates the equilibrium partial pressure \(\mathrm{p}_{j}\) of solute \(j\) to the molality of solute \(j\), \(\mathrm{m}_{j}|) for a solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). Experiment shows that certainly for dilute solutions, the partial pressure \(\mathrm{p}_{j}\) is close to a linear function of molality \(\mathrm{m}_{j}\). Taking this experimental result as a lead we state that, by definition, in the event that the thermodynamic properties of the solution are ideal, equation (u) relates the partial pressure \(\mathrm{p}_{j}\) to the solute molality \(\mathrm{m}_{j}\); \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\).
\[\text { Thus, } \mathrm{p}_{\mathrm{j}}\left(\mathrm{s} \ln ; \mathrm{T} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right)=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Here \(\mathrm{H}_{j}\) is Henry’s Law constant characteristic of solute, solvent, \(\mathrm{T}\) and \(\mathrm{p}\). \(\mathrm{H}_{j}\) is a pressure being the partial pressure of solute \(j\) where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}\). In other words equation (u) is not thermodynamic in the sense of being derived from the Laws of Thermodynamics. Rather the basis is experiment. We return to equation (n) but written for the equilibrium for solute in solution and in the vapour phase, a mixture of solute \(j\) and solvent.
\[\mu_{j}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \operatorname{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right) \nonumber \]
For the vapour phase, \(\mu_{j}^{c q}\left(g ; \operatorname{mix} ; T ; \mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}\right)\) is related to the partial pressure \(\mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}\) using equation (w).
\[\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{p}^{0}\right) \nonumber \]
Hence using equations (u)-(w),
\[\mu_{\mathrm{j}}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}} \, \frac{\mathrm{m}_{\mathrm{j}}}{\mathrm{m}^{0}}\right] \nonumber \]
\[\text { Or, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\left\{\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}}\right]\right\}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
The term \(\left\{\mu_{j}^{0}\left(g ; T ; p^{0}\right)+R \, T \, \ln \left[\frac{H_{j}}{p^{0}}\right]\right\}\) characterises solute \(j\) in a solution at the same \(\mathrm{T}\) and \(\mathrm{p}\) when \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\). Thus we define a reference chemical potential for the solute-\(j\),
\[\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p}) \text { as given by }\left\{\mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}^{0}}\right]\right\} \nonumber \]
\[\text { Therefore, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Again we can trace the gas constant \(\mathrm{R}\) in equation (za) to a description of the vapour state although the term \(\mu_{j}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)\) describes the chemical potential of chemical substance \(j\), the solute, in solution.
Finally we should note that for real as opposed to ideal liquid mixtures and ideal solutions, activity coefficients express the extent to which the properties of these systems differ from those defined as ideal.