The van der Waals Identity

REVELATION

    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".

EXTENSION

    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.

FOLLOW-THROUGH

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