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10.10: The Dependence of Cv on Volume and of Cp on Pressure

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    The heat capacities of a substance increase with temperature. The rate of increase decreases as the temperature increases. To achieve adequate accuracy in calculations, we often need to know how heat capacities depend on temperature. In contrast, the dependence of heat capacities on pressure and volume is usually negligible; that is, the dependence of \(C_V\) on \(V\) and the dependence of \(C_P\) on \(P\) can usually be ignored. Nevertheless, we need to know how to find them.

    An exact equation for the dependence of \(C_V\) on \(V\) follows readily from \(dS\) expressed as a function of \(dT\) and \(dV\)

    \[dS=\frac{C_V}{T}dT+{\left(\frac{\partial P}{\partial T}\right)}_VdV \nonumber \]

    Since the mixed second-partial derivatives must be equal, we have

    \[{\left[\frac{\partial }{\partial V}\left(\frac{C_V}{T}\right)\right]}_T = {\left[\frac{\partial }{\partial T}{\left(\frac{\partial P}{\partial T}\right)}_V\right]}_V \nonumber \]

    and thus

    \[\left(\frac{\partial C_V}{\partial V}\right)_T = T \left(\frac{\partial^{\mathrm{2}}P}{\partial T^{\mathrm{2}}}\right)_V \nonumber \]

    Similarly, the dependence of \(C_P\) on \(P\) follows from \(dS\) expressed as a function of \(dT\) and \(dP\),

    \[dS = \frac{C_P}{T}dT + {\left(\frac{\partial V}{\partial T}\right)}_PdP \nonumber \] Equating the mixed second-partial derivatives, we have

    \[\left[\frac{\partial }{\partial P}\left(\frac{C_P}{T}\right)\right]_T = \left[ + \frac{\partial }{\partial T}{\left(\frac{\partial V}{\partial T}\right)}_P\right]_P \nonumber \]

    and thus

    \[\left(\frac{\partial C_P}{\partial P}\right)_T = -T\left(\frac{\partial ^{\mathrm{2}}V}{\partial T^{\mathrm{2}}}\right)_P \nonumber \]

    For an ideal gas, it follows that \(C_V\) is independent of \(V\), and \(C_P\) is independent of \(P\).

    When we use the coefficient of thermal expansion to describe the variation of volume with temperature, we have \[{\left(\frac{\partial V}{\partial T}\right)}_P=\alpha V \nonumber \]

    When it is adequate to approximate \(\alpha\) as a constant, another partial differentiation with respect to temperature gives

    \[{\left(\frac{\partial C_P}{\partial P}\right)}_T = -T{\left(\frac{\partial \left(\alpha V\right)}{\partial T}\right)}_P = -{\alpha }^{\mathrm{2}}TV \nonumber \]

    Since \(\mathrm{\alpha }\) is normally small, this result predicts weak dependence of \(C_P\) on \(P\). If \(\mathrm{\alpha }\) and \(\mathrm{\beta }\) are both adequately approximated as constants, we have from

    \[{\left(\frac{\partial P}{\partial T}\right)}_V=\frac{\alpha }{\beta } \nonumber \] that \[{\left(\frac{\partial C_V}{\partial V}\right)}_T = T{\left(\frac{\partial \left({\alpha }/{\beta }\right)}{\partial T}\right)}_V\mathrm{=0} \nonumber \]

    This page titled 10.10: The Dependence of Cv on Volume and of Cp on Pressure is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.