Residual functions of enthalpy and Gibbs potential

3.29 Residual functions of enthalpy and Gibbs potential

To calculate the residual function of the enthalpy \(H^{res}(T,p)\) we now start with

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

(3.71)

and get

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

(3.72)

Combining Eq. (3.70) and Eq. (3.72) we find according to the definition of the Gibbs potential

\begin{equation*} \begin{split} G^{res}(T,p) = &\;\; H^{res}(T,p) - T\; S^{res}(T,p)\\ = &\;\;\int_0^p \left(V - T \left(\frac{\partial V}{\partial T} \right)_p \right) dp - T\; \left[ \int_0^p \left(- \left(\frac{\partial V}{\partial T} \right)_p + \frac{n\,R}{p} \right) dp \right] \\ = &\;\; \int_0^p \left( V - \frac{n\, R\, T}{p} \right) dp \\ \end{split} \label{eq:G_res} \end{equation*}

(3.73)