Lee-Kesler correlation for the volumetric Equation of State (EoS)

It is blessing that the volumetric properties of many fluid at typical temperature and pressure conform to the description when expressed using the reduced properties $P_{\rm r}$ and $T_{\rm r}$. In the simplest form, the compressibility factor may be written $$ Z \equiv \frac{PV}{RT} = Z_0(P_{\rm r}, T_{\rm r}),\quad \text{with } P_{\rm r} \equiv \frac{P}{P_{\rm c}},\quad T_{\rm r} \equiv \frac{T}{T_{\rm c}}, $$ in which the functional form $Z_0$ is assumed universal. The applicability of universality is closely linked to the fact that the inter-molecular interactions are dominated by the two-body potential containing a long-ranged Van der Waals attraction and a short-ranged repulsion. Shown below are the isotherms, including the vapor-liquid equilibrium dome. The initial decay of $Z$ with $P_{\rm r}$ at modest temperature, $T_{\rm r} \lesssim 2$, is due to attrative interaction, whereas the increase at higher density and higher temperature is caused by repulsion. The slope at the dilute limit, $P_{\rm r} \to 0$, is either negative at low $T_{\rm r}$ or positive at high $T_{\rm r}$, depending on if attraction or repulsion dominates. The crossover is at the Boyle temperature, where the two effects cancel, resulting in a wide ideal gas behavior in the dilute limit. In the high-density limit, the compressibility is high, the molar volume may be approximated as constant, and $Z$ increases linearly with $P_{\rm r}$ at fixed $T_{\rm r}$.

Not all molecules can be sufficiently described by the 2-parameter form. The 3-parameter form introduces the acentric factor $\omega$ to account the departure due to non-spherical shape of molecules, giving rise to the Lee-Kesler correlation $$ Z = Z_0(P_{\rm r}, T_{\rm r}) + \omega\, Z_1(P_{\rm r}, T_{\rm r}) .$$ The values of acentric factor and the critical properties have been tabulated in the standard table, which allow for the estimate to the realistic volumetric EoS of many pure fluids.