Question

Explain how the Van der Waals equation accounts for critical behavior.

Answer 1

The Van der Waals equation modifies the ideal gas equation (PV = nRT) to account for the effects of intermolecular interactions. Van der Waals equation corrects for both volume and pressure from the ideal gas. Van der Waals equation corrected the ideal gas equation so that V is actually V-b and P is defined as P + (a/V2).

So Van der Waals equation can be written as: P + a(n/V)2 x (V-nb) = nRT

In Van der Waals equation, a and b are the Van der Waals constants and must be determined empirically from experimental data of pressure, temperature and density interdependence.

The critical point is the region where the heat of vaporization is 0 and only one phase exists. It is the inflection point (point at which curve crosses its tangent) in the Pressure- Volume diagram.  At the critical point, the first derivative of the pressure with respect to volume can be written as dP / dV_m = 0. The second derivative with respect to volume can be written as d²P / dVm₂ = 0. At the critical point, the temperature, pressure and volumes are all at their critical temperatures, critical pressures and critical volumes respectively. Therefore, from the equation of state of Van der Waals gas, you can determine the first and second derivatives and connect the critical parameters to the Van der Waals constants. Solving for P, V and T gives the following equations explaining the relationship between the Van der Waal constants and critical parameters:

Volume at Critical Point = 3b

Pressure at Critical Point = (1/27)(a/b2)

Temperature at Critical Point = (8/27) (a/bR)