1.17.4: Isochoric Properties
A given closed system is characterised by a given intensive variable \(\mathrm{X}\). In this section we have in mind an intensive property such as the relative permittivity of a liquid. The variable \(\mathrm{X}\) may also refer to an equilibrium constant and related parameters such as the enthalpy of reaction, \(\Delta_{\mathrm{r}}\mathrm{H}(\mathrm{T},\mathrm{p})\). In all cases we assert that the closed system is at thermodynamic equilibrium where the affinity for spontaneous change is zero. Thus we may define \(\mathrm{X}\) for a given system in terms of the temperature and pressure.
\[\mathrm{X}=\mathrm{X}[\mathrm{T}, \mathrm{p}] \nonumber \]
The molar volume of the system is defined in analogous fashion.
\[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}] \nonumber \]
Then
\[\mathrm{dV}_{\mathrm{m}}=\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp} \nonumber \]
In other words the dependence of molar volume on \(\mathrm{T}\) and \(\mathrm{p}\) is characterised by the partial derivatives \(\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) and \(\left(\frac{\partial V_{m}}{\partial p}\right)_{T}\).
With equation (b) and (c) in mind we return the intensive property \(\mathrm{X}\) described in equation (a). The dependence of \(\mathrm{X}\) on \(\mathrm{T}\) and \(\mathrm{p}\) is similarly characterized by the two partial derivatives, \(\left(\frac{\partial X}{\partial T}\right)_{p}\) and \(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\). A calculus operation yields an equation for the partial derivative \(\left(\frac{\partial \mathrm{X}}{\partial T}\right)_{\mathrm{V}(\mathrm{m})}\). Thus
\[\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})}=\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})} \nonumber \]
The property \(\left(\frac{\partial \mathrm{X}}{\partial T}\right)_{\mathrm{V}(\mathrm{m})}\) is the isochoric differential dependence of \(\mathrm{X}\) on \(\mathrm{T}\). Now (cf. equation (c)) volume \(\mathrm{V}_{\mathrm{m}\) depends on \(\mathrm{T}\). Hence to hold \(\mathrm{V}_{\mathrm{m}}\) constant, the pressure has to change. In fact equation (c) is used to find the required change in pressure for a given change in \(\mathrm{T}\); equation (e).
\[\mathrm{dp}=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}_{\mathrm{m}}}\right)_{\mathrm{T}} \, \mathrm{dT} \nonumber \]
In other words the required change in pressure is determined by the equation of state for the system and is characteristic of the system, \(\mathrm{T}\) and \(\mathrm{p}\). For a given change in temperature, \(\delta \mathrm{T}(\exp )\) there is a defined change in pressure, \(\delta \mathrm{p}(\operatorname{def})\). The isochoric condition takes the following form granted that in the experiment we decide to change the temperature by an amount \(\delta \mathrm{T}\).
\[\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}]=\mathrm{V}_{\mathrm{m}}[\mathrm{T}+\delta \mathrm{T}(\exp ) ; \mathrm{p}+\delta \mathrm{p}(\operatorname{def})] \nonumber \]
We now return to the property \(\mathrm{X}\) defined in equation (a). We consider the property \(\mathrm{X}\) at the two conditions highlighted in equation (f);
\[\mathrm{X}[\mathrm{T}, \mathrm{p}] ; \quad \mathrm{X}[\mathrm{T}+\delta \mathrm{T}(\exp ) ; \mathrm{p}+\delta \mathrm{p}(\operatorname{def})] \nonumber \]
The term \(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{T}}\right)_{\mathrm{V}(\mathrm{m})[\mathrm{T}, \mathrm{p}]}\) defines an isochoric dependence of \(\mathrm{X}\) on \(\mathrm{T}\) at pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\). At each temperature the isochoric dependence of \(\mathrm{X}\) on \(\mathrm{T}\) reflects the dependence of \(\mathrm{V}_{\mathrm{m}}\) on \(\mathrm{T}\).
The analysis outlined above is repeated but in terms of the isochoric dependence of \(\mathrm{X}\) on pressure. In order that the volume of a system does not change when the pressure is changed by \(\delta \mathrm{p}(\exp )\), the temperature must be changed by an amount \(\delta \mathrm{T}(\operatorname{def})\) determined by the equation of state for the system.
\[\mathrm{V}_{\mathrm{m}}[\mathrm{T}, \mathrm{p}]=\mathrm{V}_{\mathrm{m}}[\mathrm{T}+\delta \mathrm{T}(\operatorname{def}) ; \mathrm{p}+\delta \mathrm{p}(\exp )] \nonumber \]
We compare property \(\mathrm{X}\) under the isochoric condition given in equation (h);
\[\mathrm{X}[\mathrm{T}, \mathrm{p}] ; \quad \mathrm{X}[\mathrm{T}+\delta \mathrm{T}(\operatorname{def}) ; \mathrm{p}+\delta \mathrm{p}(\exp )] \nonumber \]
\(\left(\frac{\partial \mathrm{X}}{\partial \mathrm{p}}\right)_{\mathrm{v}_{(\mathrm{m})[\mathrm{T}, \mathrm{p}]}}\) describes the isochoric dependence of \(\mathrm{X}\) on pressure.
We have carefully examined the concept of an isochoric dependence of a given variable on either \(\mathrm{T}\) or \(\mathrm{p}. The reason for this care emerges from the observation that the literature describes a number of isochoric parameters. In some cases the analysis is recognized as extrathermodynamic. In other cases a patina of thermodynamics is introduced into an analysis leading to further debate.