Departure function

General expressions for the enthalpy H, entropy S and Gibbs free energy G are given by \begin{align} \frac{H^\mathrm{ig}-H}{RT} &= \int_V^\infty \left[ T \left(\frac{\partial Z}{\partial T}\right)_V \right] \frac{dV}{V} + 1 - Z \\[2ex] \frac{S^\mathrm{ig}-S}{R} &= \int_V^\infty \left[ T \left(\frac{\partial Z}{\partial T}\right)_V - 1 + Z\right] \frac{dV}{V} - \ln Z \\[2ex] \frac{G^\mathrm{ig}-G}{RT} &= \int_V^\infty (1-Z) \frac{dV}{V} + \ln Z +1 - Z \end{align} == Departure functions for Peng–Robinson equation of state ==

The Peng–Robinson equation of state relates the three interdependent state properties pressure P, temperature T, and molar volume Vm. From the state properties (P, Vm, T), one may compute the departure function for enthalpy per mole (denoted h) and entropy per mole (s): :\begin{align} h_{T,P}-h_{T,P}^{\mathrm{ideal}} &= RT_C\left[T_r(Z-1)-2.078(1+\kappa)\sqrt{\alpha}\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)\right] \\[1.5ex] s_{T,P}-s_{T,P}^{\mathrm{ideal}} &= R\left[\ln(Z-B)-2.078\kappa\left(\frac{1+\kappa}{\sqrt{T_r}}-\kappa\right)\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)\right] \end{align} where \alpha is defined in the Peng-Robinson equation of state, Tr is the reduced temperature, Pr is the reduced pressure, Z is the compressibility factor, and :\kappa = 0.37464 + 1.54226\;\omega - 0.26992\;\omega^2 :B = 0.07780\frac{P_r}{T_r} Typically, one knows two of the three state properties (P, Vm, T), and must compute the third directly from the equation of state under consideration. To calculate the third state property, it is necessary to know three constants for the species at hand: the critical temperature Tc, critical pressure Pc, and the acentric factor ω. But once these constants are known, it is possible to evaluate all of the above expressions and hence determine the enthalpy and entropy departures. == References ==