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1.9.4: Entropy- Dependence on Temperature

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

    This page titled 1.9.4: Entropy- Dependence on Temperature was last modified on Mon, 20 May 2024 07:03:08 GMT and is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer and Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform.