Law of corresponding states

Remarks

The critical point is $V_c = 3b$, $T_c = 8a/(27Rb)$, $p_c = a/(27b^2)$. Substituting $p = p_c p_r$, $V_m = V_c V_r$ and $T = T_c T_r$ and dividing by $p_c$ gives the displayed equation. The same argument works for other two-constant cubic equations of state with a critical point, such as the Redlich–Kwong equation. There the constants only set the scales, and the reduced equation is universal. As an empirical principle, the law of corresponding states says that real fluids at the same reduced temperature and pressure have approximately the same compressibility factor $pV_m/(RT)$. It holds well for simple fluids such as argon, krypton and methane, and less well for polar or strongly non-spherical molecules, which led to corrections with a third parameter.

Going deeper

For a student

The law shows that, in the van der Waals model, the two constants only set the scales of pressure and volume. Consequences hold for all fluids at once. For example, the critical compressibility factor is $p_c V_c/(R T_c) = 3/8$ for every van der Waals fluid. The reduced coexistence curve given by the Maxwell construction is also universal. In statistical mechanics the law follows for molecules whose pair potentials have the same shape and differ only by an energy scale and a length scale.