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 \(\mathrm{p}-\mathrm{V}-\mathrm{T}\).

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