\[P_r = \frac{P}{P_c}\]

where \(P_r\) is the reduced pressure, \(P\) is pressure, and \(P_c\) is the critical pressure.

\[T_r = \frac{T}{T_c}\]

where \(T_r\) is the reduced temperature, \(T\) is temperature, and \(T_c\) is the critical temperature (must be absolute temperatures).

\[V_r = \frac{V}{V_c}\]

where \(V_r\) is the reduced molar volume, \(V\) is molar volume, and \(V_c\) is the critical molar volume.

\[Z = \frac{PV}{RT}\]

where \(Z\) is the compressibility factor, and R is the ideal gas constant.

van der Waals Equation of State (EOS):

\[P = \frac{RT}{V – b} – \frac{a}{V^2} = \frac{\rho RT}{1 – b\rho} – a\rho ^2\]

where V is molar volume, \(\rho\) is molar density and \(a\) and \(b\) are constants. The \(a\) term accounts for attractive interactions between molecules and the \(b\) term accounts for the volume occupied by molecules (replusive term). The \(a\) and \(b\) constants are determined from critical pressures and temperatures:

\[a = \frac{27R^2T^2}{64PC} \hspace{1cm}   b = \frac{RT_c}{8P_c}\]

\[P = \frac{\rho RT}{1 – b\rho} – \frac{a\rho ^2}{(1 + 2b\rho – b^2 \rho ^2)}\]

and \(a = a_c \alpha\) , \(b = 0.0779607\frac{RT_c}{P_c}\)

where \(a_c = 0.45723553R^2T_c^2/P_c\) and \(a = [1 + \kappa(1 – T_r^{0.5})]^2\), and \(\kappa = 0.37464 + 1.5422\omega – 0.26992\omega ^2\)

At saturation conditions (vapor-liquid equilibrium):

\[f_{liquid} = f_{vapor}\]

where \(f\) is the fugacity, which can be calculated from the equation of state.