# 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 $\frac{C_P}{T} = \frac{C_V}{T} + {\left(\frac{\partial P}{\partial T}\right)}_V{\left(\frac{\partial V}{\partial T}\right)}_P \nonumber$

and

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

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

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

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