1.9.4: Entropy- Dependence on Temperature
Using a calculus operation, the isochoric dependence of entropy of temperature is related to the corresponding isobaric dependence. Thus
\[\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\left(\frac{\partial S}{\partial p}\right)_{T} \,\left(\frac{\partial p}{\partial V}\right)_{T} \,\left(\frac{\partial V}{\partial T}\right)_{p} \nonumber \]
But
\[\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p} \nonumber \]
Hence,
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right]^{2} \nonumber \]
Or,
\[\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\frac{\left(E_{p}\right)^{2}}{K_{T}} \nonumber \]
The final term in equation (c) contains the variable \(\mathrm{p}-\mathrm{V}-\mathrm{T}\).