1.14.25: Equation of State- Perfect Gas
A closed system contains \(\mathrm{n}_{j}\) moles of a gaseous substance \(j\). No chemical reaction takes place in the system. The system is at equilibrium where the affinity for spontaneous change is zero. The system is characterised by the thermodynamic energy \(\mathrm{U}\). The system is displaced to a neighbouring equilibrium state by a change in entropy \(\mathrm{dS}\) and a change in volume \(\mathrm{dV}\). The change in thermodynamic energy is given by the Master Equation.
\[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV} \nonumber \]
At equilibrium the isothermal dependence of thermodynamic energy on volume is given by equation (b).
\[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}-\mathrm{p} \nonumber \]
In attempting to understand from a chemical standpoint the properties of gases, liquid mixtures and solutions, a common approach formulates a set of properties which classify a given system as ideal. The definition of an ideal (or, perfect) system is made with practical chemistry in mind. When examining the properties of gases there is merit in identifying the properties of a perfect gas. [No real gas is perfect!] If the gaseous substance \(j\) is a perfect gas, the following conditions [1] are met at all temperatures and pressures.
- \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}=0\)
- \(\mathrm{p} \, \mathrm{V}=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\)
Here \(\mathrm{R}\) is the gas constant, \(8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\). Conditions (A) and (B) are equivalent [2]. In most cases condition (B) is quoted because the equation links three practical properties, \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\).
Footnotes
[1]
\[\begin{aligned}
&\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\frac{[\mathrm{J}]}{\left[\mathrm{m}^{3}\right]}=\frac{[\mathrm{N} \mathrm{m}]}{\left[\mathrm{m}^{3}\right]}=\left[\mathrm{N} \mathrm{m}{ }^{2}\right]=[\mathrm{Pa}] \\
&\mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}{ }^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\
&\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=[\mathrm{J}]
\end{aligned} \nonumber \]
[2] From definition (A) and equation (b),
\[p=T \,\left(\frac{\partial S}{\partial V}\right)_{T} \nonumber \]
We use a Maxwell equation;
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \nonumber \]
Hence from equations (i) and (ii),
\[p=T \,\left(\frac{\partial p}{\partial T}\right)_{v} \nonumber \]
From definition (B),
\[\left(\frac{\partial \mathrm{p}}{\partial T}\right)_{\mathrm{V}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} / \mathrm{V}=\mathrm{p} / \mathrm{T} \nonumber \]
Equations (iii) and (iv) are the same. Hence definition (B) is the integrated form of definition \(\mathrm{A}\). The gas constant \(\mathrm{R}\) is experimentally determined.