1.19.1: Perfect and Real Gases
- Page ID
- 394371
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In a description of the properties of gases, the term ‘perfect’ means that there are no intermolecular forces, either attractive or repulsive. The equation of state for \(\mathrm{n}_{\mathrm{j}}\) moles of perfect gas \(\mathrm{j}\) takes the following form where \(\mathrm{R}\) is the Gas Constant, \(8.314 \mathrm{J K}^{-1} \mathrm{~mol}^{-1}\) [1].
\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \nonumber \]
The chemical potential of a perfect gas \(\mu_{j}^{\text {id }}\) at temperature \(\mathrm{T}\) is related to pressure \(\mathrm{p}_{j}\) using equation (b).
\[\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{\mathrm{j}} / \mathrm{p}^{0}\right) \nonumber \]
Thus \(\mu_{j}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)\) is the chemical potential of gas \(j\) at pressure \(\mathrm{p}_{j}\) whereas \(\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)\) is the corresponding chemical potential at the standard pressure \(\mathrm{p}^{0}\) [2].
The ratio \(\left(\mathrm{V}_{\mathrm{j}} / \mathrm{n}_{\mathrm{j}}\right)\) is the molar volume of gas \(j\), \(\mathrm{V}_{\mathrm{mj}}\). Equation (a) describing a perfect gas can be written as follows.
\[\mathrm{p}_{\mathrm{j}}^{\mathrm{id}} \, \mathrm{V}_{\mathrm{mj}}=\mathrm{R} \, \mathrm{T} \nonumber \]
No real gas is perfect at all temperatures and pressures although at high temperatures and low pressures the product \(\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{mj}}\) is arithmetically almost equal to the product, \(\mathrm{R} \, \mathrm{T}\). Generally however equation (c) does not describe real gases. The properties of real gases are described in several ways.
In one approach \(\mu_{j}\left(T, p_{j}\right)\) is related to \(\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)\) using equation (d) where \(\mathrm{f}_{j}\) is the fugacity.
\[\mu_{\mathrm{j}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\mu_{\mathrm{j}}\left(\mathrm{T}, \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{j}} / \mathrm{p}^{0}\right) \nonumber \]
Thus
\[\operatorname{limit}\left(\mathrm{p}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}=\mathrm{p}_{\mathrm{j}} \nonumber \]
Another approach uses virial coefficients [3]. Thus pressure \(\mathrm{p}_{j}\) is related to molar volume \(\mathrm{V}_{\mathrm{mj}\) using a power series in the term \(\mathrm{V}_{\mathrm{mj}}\). Thus,
\[\mathrm{p}_{\mathrm{j}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{mj}}}\left[1+\frac{\mathrm{B}}{\mathrm{V}_{\mathrm{mj}}}+\frac{\mathrm{C}}{\mathrm{V}_{\mathrm{mj}}^{2}}+\ldots \ldots\right] \nonumber \]
In the event that a given gas is only slightly imperfect the terms C, D,…. are negligibly small. Then,
\[\mathrm{p}_{\mathrm{j}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{mj}}}\left[1+\frac{\mathrm{B}}{\mathrm{V}_{\mathrm{mj}}}\right] \nonumber \]
At low temperatures \(\mathrm{B}\) tends to be negative but at high temperatures \(\mathrm{B}\) is positive.
Footnotes
[1] For equation (a),
\[\left[\mathrm{N} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \nonumber \]
where \([\mathrm{J}]=[\mathrm{Nm}]\)
[2]
\[\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \quad \mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \nonumber \]
[3] I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1953, chapter 11.