From his Corresponding States Law, van der Waals, himself, showed that the critically reduced, dimensionless form of his equation of state became:
where p = P / Pc, v = V / Vc, t = T / Tc, with Pc,Vc,Tc the respective critical constants for pressure, volume and temperature.
The critical isotherm then occurs for t = 1, giving p = q, where
Replacing v by reduced density w = 1/v yields
However, this last expression is found to be identical to
The equivalence q = q* may be called "The van der Waals Identity".
The identity above may be extended and generalized to the form
where the van der Waals original values are b = 1/3, c = 0, d = 1. However, values of b near 1/4 and c near 1 yield much better fits to the critical isotherms of various pure substances.
Finally, the original reduced van der Waals equation may be written alternatively as
suggesting the extended veridical form
with a-values closer to 6 than to 3.
Recognizing the remarkable value of this identity and its generalization has led to a series of papers by this writer and others, the most direct of which are found via the following links:
Paynter's Equation of State
Breedveld's Integral Causal Form
van der Waals' Nobel Lecture