Maxwell relations, calculation of residual functions

3.28 Maxwell relations, calculation of residual functions

The residual function \(\Gamma^{res}(T,p)\) of a caloric state function \(\Gamma(T,p)\) is defined as the difference between the real and the ideal state function

\begin{equation*} \Gamma(T,p) = \Gamma^{real}(T,p) = \Gamma^{res}(T,p) + \Gamma^{ideal}(T,p) \label{eq:residual_function} \end{equation*}

(3.67)

The calculation of the residual function and other thermodynamic properties starts from the volume-explicit thermal equation of state \(V(T,p,n)\) of a real fluid. As an example we calculate the residual entropy. Taking into account that for zero pressure all gases/fluids behave ideally and using a Maxwell relation we start with

\begin{equation*} S(T,0) - S^{ideal}(T,0) = 0 \qquad \left(\frac{\partial S}{\partial p} \right)_T = - \left(\frac{\partial V}{\partial T} \right)_p \qquad \left(\frac{\partial S^{ideal}}{\partial p} \right)_T = - \left(\frac{\partial V^{ideal}}{\partial T} \right)_p = - \frac{n\,R}{p} \label{eq:S_Sideal} \end{equation*}

(3.68)

Using

\begin{equation*} S(T,p) = S(T,0) + \int_0^p \left(\frac{\partial S}{\partial p} \right)_T dp \label{eq:S_T_p} \end{equation*}

(3.69)

we find for the residual entropy

\begin{equation*} \begin{split} S^{res}(T,p) = &\;\; S(T,p) - S^{ideal}(T,p)\\ = &\;\; S(T,0) - S^{ideal}(T,0) + \int_0^p \left(\left(\frac{\partial S}{\partial p} \right)_T -\left(\frac{\partial S^{ideal}}{\partial p} \right)_T \right) dp \\ \Rightarrow S^{res}(T,p) = &\;\; \int_0^p \left(- \left(\frac{\partial V}{\partial T} \right)_p + \frac{n\,R}{p} \right) dp \\ \end{split} \label{eq:S_res} \end{equation*}

(3.70)

So no explicit measurements of entropies are necessary to calculate the residual entropy. The experimentally much more easily accessible \(V(T,p,n)\) relation allows for the complete calculation.