1.7.4: Compressibilities- Isentropic- Related Properties
A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains chemical substances 1 and \(j\). The system at specified \(\mathrm{T}\) and \(\mathrm{p}\) is at equilibrium where the affinity for spontaneous change is zero. We describe the volume and the entropy of the system using the following two equations.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right] \nonumber \]
\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right] \nonumber \]
The system is perturbed by a change in pressure. We envisage two possible paths tracked by the system accompanying a change in volume. In the first case the temperature is constant along the path for which ‘\(\mathrm{A}=0\)’. The isothermal equilibrium dependence of volume on pressure, namely the equilibrium isothermal compression \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\), is defined by equation (c).
\[K_{T}(A=0)=-\left(\frac{\partial V}{\partial p}\right)_{T, A=0} \nonumber \]
In the second case the entropy remains constant along the path travelled by the system where ‘\(\mathrm{A}=0\)’. The differential equilibrium isentropic compression is given by equation (d); isentropic = adiabatic + equilibrium
\[\mathrm{K}_{\mathrm{s}}(\mathrm{A}=0)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0} \nonumber \]
For all stable phases the volume of a given system decreases with increase in pressure at fixed temperature. The minus signs in equations (c) and (d) mean that compressions are positive variables. Neither \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) or \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) are strong functions of state because both variables describe pathways between states. The partial differentials in equations (c) and (d) differ in an important respect. The isothermal condition refers to an intensive variable whereas the isentropic condition refers to an extensive variable. The two properties \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) and \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) are related using a calculus operation.
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{S}}\right)_{\mathrm{p}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{P}, \mathrm{A}=0} \nonumber \]
Hence, [1]
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}\right]^{2} \, \frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)} \nonumber \]
But the (equilibrium) isobaric expansibility,
\[\alpha_{p}(A=0)=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, A=0} \nonumber \]
\[\operatorname{Then}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V}^{2} \, \mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)} \nonumber \]
By definition, the equilibrium isobaric heat capacity per unit volume [2] {also called heat capacitance [3]},
\[\sigma(A=0)=C_{p}(A=0) / V \nonumber \]
In terms of compressions,
\[\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V} \, \mathrm{T}}{\sigma(\mathrm{A}=0)} \nonumber \]
Three terms in equation (j), \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\), \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) and \(\mathrm{V}\), are volume extensive variables. However it is convenient to rewrite these equations using volume intensive variables. Two equations define the isentropic equilibrium compressibility \(\kappa_{\mathrm{S}}(\mathrm{A}=0)\) and isothermal equilibrium compressibility \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) of a given system.
\[\kappa_{\mathrm{T}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)}{\mathrm{V}} \nonumber \]
\[\kappa_{\mathrm{S}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)}{\mathrm{V}} \nonumber \]
\[\text { Therefore } \kappa_{\mathrm{S}}(\mathrm{A}=0)=\kappa_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)} \nonumber \]
\[\text { By definition, } \delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}} \nonumber \]
\[\text { Then } \delta(A=0)=\left[\alpha_{p}(A=0)\right]^{2} \, \frac{T}{\sigma(A=0)} \nonumber \]
Footnotes
[1] From a Maxwell relationship for the condition at ‘\(\mathrm{A}=0\)’; i.e. at equilibrium, \(\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T}\). Then, \(\mathrm{E}_{\mathrm{p}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}\) From the Gibbs - Helmholtz equation, we combine the equations, \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\) and \(\mathrm{S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\). Hence, \(\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\) Then, \((\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\)
[2] \(\kappa_{\mathrm{S}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1} \quad \kappa_{\mathrm{T}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1}\)
\(\begin{aligned}
&{\left[\alpha_{p}(\mathrm{~A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)}=\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}{ }^{-1} \mathrm{~m}^{-3}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\mathrm{Pa}^{-1}} \\
&\sigma(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0) / \mathrm{V}=\left[\mathrm{J} \mathrm{K}{ }^{-1}\right] \,[\mathrm{m}]^{-3}
\end{aligned}\)
[3] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc Rev., 2001, 30 ,8.