1.12.3: Heat Capacities- Isobaric and Isochoric
When heat \(\mathrm{q}\) passes smoothly (reversibly) into a closed system from the surroundings, the temperature of the system increases (if there are no phase changes, e.g. liquid to vapour). The increase in temperature \(\Delta \mathrm{T}\) is related to heat \(\mathrm{q}\) using equation (a).
\[\mathrm{q}=\mathrm{C} \, \Delta \mathrm{T} \nonumber \]
Heat capacity \(\mathrm{C}\) is an extensive property of a system whereas \(\Delta \mathrm{T}\) is the change in an intensive variable. For a given amount of heat, a more dramatic increase in temperature is produced the lower is the heat capacity \(\mathrm{C}\). Moreover as defined by equation (a) the heat capacity of a system is not a thermodynamic function of state because heat capacity describes a pathway accompanying a change in temperature. Hence, we define precisely the pathway taken by the system. Two important classes of heat capacities are
- isobaric, \(\mathrm{C}_{\mathrm{p}}\), and
- isochoric, \(\mathrm{CV}\).
Isochoric and isobaric heat capacities are related to the isobaric expansions \(\mathrm{E}_{\mathrm{p}}\) and isothermal compression \(\mathrm{KT}\) using equation (b) [1].
\[\mathrm{C}_{\mathrm{V}}=\mathrm{C}_{\mathrm{p}}-\mathrm{T} \,\left(\mathrm{E}_{\mathrm{p}}\right)^{2} / \mathrm{K}_{\mathrm{T}} \nonumber \]
Heat capacities and compressions are simply related [2].
\[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}} \nonumber \]
Footnotes
[1] According to a calculus operation, the dependences of entropy on temperature at constant volume and constant pressure are related. \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) A Maxwell equation requires that \(\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p}\) Hence,
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
But the isobaric expansion, \(E_{p}=\left(\frac{\partial V}{\partial T}\right)_{p}\) And the isothermal compression, \(\mathrm{K}_{\mathrm{T}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\) From the Gibbs –Helmholtz equation, \(\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\) And \(C_{p}=T \,\left(\frac{\partial S}{\partial T}\right)_{p}\) Then
\[C_{V}=C_{p}-T \,\left(E_{p}\right)^{2} / K_{T} \nonumber \]
The latter equation is correct under the condition of either ‘at constant affinity \(\mathrm{A}\)’ or ‘at constant composition’.
[2] The starting point is the following equation.
\[\begin{array}{r}
\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \\
\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{v}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}
\end{array} \nonumber \]
Then \((\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\) Hence,
\[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}} \nonumber \]