The van der Waals equation and cubic equations of state

To account for the finite size of molecules in the system. If we consider molecules to be of volume b, then the actual volume available to the system is approximately V — b. Therefore, we replace V in the ideal gas equation of state with V — b. In addition, at large distances, the molecules in a gas exert an attractive interaction between each other due to dispersion forces. The van der Waals equation of state, which accounts for both the short-ranged repulsions and long-ranged attractions between molecules, is given by 

A schematic plot of the variation of the pressure with volume, as predicted by the van der Waals equation of state, at various temperatures is given in Fig. 10.2. At temperatures above the critical temperature, the pressure-volume variation is monotonic and qualitatively similar to that of an ideal gas (see dotted-line). At temperatures below tire critical temperature, the pressure-volume curve begins to oscillate, exhibiting a van der Waals loop (see dashed-line). This behavior is unphysical, but represents tire vapor-liquid transition, and should be replaced by the solid line. The precise location of the solid line is given by the Maxwell construction. 

When the temperature is equal to the critical temperature, the pressure-volume curve exhibits a point with the inflection (concavity) of the curve changes sign. This occurs at the critical point, where we have  

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