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:
Breedveld's Integral Causal Form