Cp - CV for perfect gases

2.10 \(C_p-C_V\) for perfect gases

Here we will prove a relation for \(C_p - C_V\) which holds for nearly all solids and liquids and for ideal gases; so we discuss a very general feature of heat capacities. Including the expansion coefficient \(\alpha\) of Eq. (2.6) in Eq. (2.11) we get

\begin{equation*} dU = \pi_T dV + C_V dT \quad \Rightarrow \quad \left(\frac{\partial U}{\partial T}\right)_p = \pi_T \left(\frac{\partial V}{\partial T}\right)_p + C_V = \pi_T\, \alpha V + C_V \label{eq:dU_TV_3} \end{equation*}

(2.27)

We learned already:

\(\alpha\): Expansion coefficient is small for solids and liquids.
\(\pi_T\): Is small for gases and zero for perfect gases.

\begin{equation*} \Rightarrow \quad \left(\frac{\partial U}{\partial T}\right)_p \approx C_V \label{eq:dUdTapprox} \end{equation*}

(2.28)

\begin{equation*} \mbox{Thus using} \quad \left(\frac{\partial U}{\partial T}\right)_p \approx \left(\frac{\partial U}{\partial T}\right)_V = C_V \quad\mbox{we get}\quad C_p - C_V = \left(\frac{\partial H}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_V \approx \left(\frac{\partial H}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_p \label{eq:Cp-CV_1} \end{equation*}

(2.29)

\begin{equation*} \mbox{i.e.} \quad C_p - C_V = \left(\frac{\partial (U+pV)}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_p = \left(\frac{\partial (U+n\,R\,T)}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_p = n\,R \label{eq:Cp-CV_2} \end{equation*}

(2.30)

So for many materials the specific heat capacities \(C_p/n\) and \(C_V/n\) just differ by a constant \(R\).