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