1.5.2: Chemical Potentials- Gases
- Page ID
- 352533
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.