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10.9: The Relationship Between Cv and Cp for Any Substance

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    In Chapter 7, we derive the relationship between \(C_P\) and \(C_V\) for an ideal gas. It is useful to have a relationship between these quantities that is valid for any substance. We can derive this relationship from the equations for \(dS\) that we develop in Sections 10.4 and 10.5. If we apply the divide-through rule to \(dS\) expressed as a function of \(dT\) and dV, at constant pressure, we have

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

    From \(dS\) expressed as a function of \(T\) and \(P\),

    \[dS = \frac{C_P}{T}dT + {\left(\frac{\partial V}{\partial T}\right)}_PdP \nonumber \]

    we have

    \[{\left(\frac{\partial S}{\partial T}\right)}_P = \frac{C_P}{T} \nonumber \] so that \[\frac{C_P}{T} = \frac{C_V}{T} + {\left(\frac{\partial P}{\partial T}\right)}_V{\left(\frac{\partial V}{\partial T}\right)}_P \nonumber \]


    \[C_P + C_V = T{\left(\frac{\partial P}{\partial T}\right)}_V{\left(\frac{\partial V}{\partial T}\right)}_P \label{eq10} \]

    For an ideal gas, the right side of Equation \ref{eq10} reduces to \(R\), in agreement with our previous result. Note also that, for any substance, \(C_P\) and \(C_V\) become equal when the temperature goes to zero.

    The partial derivatives on the right hand side can be related to the coefficients of thermal expansion, \(\alpha\), and isothermal compressibility, \(\beta\). Using

    \[\frac{\alpha }{\beta } = {\left(\frac{\partial P}{\partial T}\right)}_V \nonumber \]

    we can write the relationship between \(C_P\) and \(C_V\) as

    \[C_P + C_V = \frac{VT{\alpha }^{\mathrm{2}}}{\beta } \nonumber \]

    This page titled 10.9: The Relationship Between Cv and Cp for Any Substance 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.