1.19.2: Perfect Gas: The Gas Constant
Throughout these Topics, the Gas Constant, symbol \(\mathrm{R}\), plays an important role. Here we examine how such an important quantity emerges [1].
An important concept in chemical thermodynamics is the perfect gas. In practice the properties of real gases differ from those of the perfect gas but the concept provides a useful basis for understanding the properties of real gases and by extension the properties of liquid mixtures and solutions. After all, nothing is perfect.
The starting point for the analysis is the following equation (see Topic 2500) for the change in thermodynamic energy of a closed system \(\mathrm{dU}\) at temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and affinity for spontaneous change \(\mathrm{A}\) [1].
\[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
Then for processes at equilibrium where \(\mathrm{A}\) is zero,
\[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV} \nonumber \]
For one mole of chemical substance \(\mathrm{j}\), equation (b) can be written in the following form.
\[\mathrm{dU}_{\mathrm{j}}=\mathrm{T} \, \mathrm{dS}_{\mathrm{j}}-\mathrm{p} \, \mathrm{dV_{ \textrm {j } }} \nonumber \]
Then,
\[\mathrm{dS}_{\mathrm{j}}=\frac{\mathrm{dU}_{\mathrm{j}}+\mathrm{p} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}}{\mathrm{T}} \nonumber \]
The molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) describes the differential dependence of molar thermodynamic energy \(\mathrm{U}_{j}\) on temperature at fixed volume. Thus
\[\mathrm{C}_{\mathrm{Vj}}=\left(\partial \mathrm{U}_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{V}(\mathrm{j})} \nonumber \]
Using equation (d),
\[\mathrm{dS}_{\mathrm{j}}=\frac{\mathrm{C}_{\mathrm{v}_{\mathrm{j}}}}{\mathrm{T}} \, \mathrm{dT}+\frac{\mathrm{p}}{\mathrm{T}} \, \mathrm{dV}_{\mathrm{j}} \nonumber \]
The latter equation emerges from an equation expressing the molar entropy of an ideal gas \(j\) as a function of the independent variables \(\mathrm{T}\) and \(\mathrm{V}_{j}\). Thus,
\[\mathrm{S}_{\mathrm{j}}=\mathrm{S}_{\mathrm{j}}\left[\mathrm{T}, \mathrm{V}_{\mathrm{j}}\right] \nonumber \]
According to Joules Law [2]. The molar thermodynamic energy of a perfect gas depends only on temperature. Hence from equation (e) the molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) is solely a function of temperature. Therefore equation (f) yields the following two important equations [3].
\[\left(\frac{\partial S_{j}}{\partial T}\right)_{v}=\frac{C_{v_{j}}}{T} \nonumber \]
\[\left(\frac{\partial S_{j}}{\partial V}\right)_{T}=\frac{p}{T} \nonumber \]
According to Boyles Law, the molar volume of gas \(j\) is inversely proportional to the pressure at fixed temperature. Thus
\[V_{j}=f(T) / p \nonumber \]
Alternatively
\[\mathrm{p}=\mathrm{f}(\mathrm{T}) / \mathrm{V}_{\mathrm{j}} \nonumber \]
Hence using equation (i),
\[\left(\frac{\partial S_{j}}{\partial V}\right)_{T}=\frac{f(T)}{T \, V_{j}} \nonumber \]
A calculus condition requires that
\[\frac{\partial}{\partial V}\left(\frac{\partial S}{\partial T}\right)=\frac{\partial}{\partial T}\left(\frac{\partial S}{\partial V}\right) \nonumber \]
In other words,
\[\frac{\partial\left(\mathrm{C}_{\mathrm{vj}} / \mathrm{T}\right)}{\partial \mathrm{V}}=\frac{\partial(\mathrm{p} / \mathrm{T})}{\partial \mathrm{T}} \nonumber \]
Or, using equation (k),
\[\frac{\partial\left(\mathrm{C}_{\mathrm{v}_{\mathrm{j}}} / \mathrm{T}\right)}{\partial \mathrm{V}}=\frac{\partial\left[\mathrm{f}(\mathrm{T}) / \mathrm{T} \, \mathrm{V}_{\mathrm{j}}\right]}{\partial \mathrm{T}} \nonumber \]
But the isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) is independent of volume. Hence
\[\frac{\partial\left(\mathrm{C}_{\mathrm{v}_{\mathrm{j}}} / \mathrm{T}\right)}{\partial \mathrm{V}}=0 \nonumber \]
Then,
\[\frac{\partial\left[\mathrm{f}(\mathrm{T}) / \mathrm{T} \, \mathrm{V}_{\mathrm{j}}\right]}{\partial \mathrm{T}}=0 \nonumber \]
In other words \([\mathrm{f}(\mathrm{T}) / \mathrm{T}]\) must be a constant, conventionally called the Gas Constant with symbol \(\mathrm{R}\). As the name implies \(\mathrm{R}\) is a constant used to describe the properties of all gases [3]. We can therefore rewrite equation (k) as follows (recalling that \(\mathrm{V}_{\mathrm{j}}\) is the molar volume of a perfect gas) [4,5].
\[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T} \nonumber \]
The perfect gas is an artificial chemical substance having defined properties. The link with reality stems from the idea that the properties of real gases approach those of an ideal gas as the pressure is reduced.
In addition to the definition given by equation (q), the ideal gas is defined by the following equation which requires that the thermodynamic energy of an ideal gas is independent of volume, being nevertheless a function of temperature.
\[\left(\partial \mathrm{U}_{\mathrm{j}} / \partial \mathrm{V}_{\mathrm{j}}\right)_{\mathrm{T}}=0 \nonumber \]
Footnotes
[1] I. Prigogine and R Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1954, chapter X.
[2] Reference 1, page 116.
[3]
\[\begin{aligned}
&\mathrm{R}=8.31450 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \\
&\mathrm{R}=\mathrm{N}_{\mathrm{A}} \, \mathrm{k}
\end{aligned} \nonumber \]
where \(\mathrm{N}_{\mathrm{A}} =\) Avogadro’s constant and k = Boltzmann’s constant
\[\begin{aligned}
&\mathrm{k}=1.380658 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1} \\
&\mathrm{~N}_{\mathrm{A}}=6.0221367 \times 10^{23} \mathrm{~mol}^{-1}
\end{aligned} \nonumber \]
[4] G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, 1923, page 63.
[5] P. W. Atkins, Concepts in Physical Chemistry, Oxford University Press, Oxford,1995.