1.5.2: Chemical Potentials- Gases
A given closed system contains gas \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The chemical potential \(\mu_{\mathrm{j}}(\mathrm{g} ; \mathrm{T} ; \mathrm{p})\) is given by Equation \ref{a} where \(\mathrm{p}^{0}\) is the standard pressure and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T}, \mathrm{p})\) is the molar volume of the gas \(j\).
\[\mu_{j}(g ; T ; p)= \mu_{j}^{0}(p f g ; T)+ R T \ln \left(\dfrac{p}{p^{0}}\right)+\int_{0}^{p}\left[V_{j}^{*}(T ; p)-\left(\dfrac{R T}{p}\right)\right] d p \label{a} \]
\(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})\) is the molar volume at pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\). In the event that gas \(j\) has the properties of a perfect gas, the chemical potential is given by Equation \ref{b}.
\[\mu_{j}(p f g ; T ; p)=\mu_{j}^{0}(p f g ; T)+R T \ln \left( \dfrac{p}{p^{0}} \right) \label{b} \]
If gas \(j\) exists at mole fraction \(\mathrm{x}_{j}\) as one component of a mixture of \(\mathrm{k}\) gases the chemical potential of gas \(j\) is given by Equation \ref{c} where \(\mathrm{x}_{\mathrm{k}}\) is the set of mole fractions defining the composition of the mixture [1].
\[ \mu_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{k}}\right)= \mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T})+\mathrm{R} \mathrm{T} \ln \left(\mathrm{x}_{\mathrm{j}} \mathrm{p} / \mathrm{p}^{0}\right) +\int_{\mathrm{o}}^{\mathrm{p}}\left[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)-\left(\dfrac{RT}{p}\right)\right] \mathrm{dp} \label{c} \]
Here \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)\) is the molar volume of gas \(j\) in the gaseous mixture.
Footnote
[1] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 184.