1.19.1: Perfect and Real Gases

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.