# Theorem of corresponding states

According to van der Waals, the theorem of corresponding states (or principle of corresponding states) indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree.[1][2]

Material constants that vary for each type of material are eliminated, in a recast reduced form of a constitutive equation. The reduced variables are defined in terms of critical variables.

It originated with the work of Johannes Diderik van der Waals in about 1873[3] when he used the critical temperature and critical pressure to characterize a fluid.

The most prominent example is the van der Waals equation of state, the reduced form of which applies to all fluids.

## Compressibility factor at the critical point

The compressibility factor at the critical point, which is defined as ${\displaystyle Z_{c}={\frac {p_{c}V_{c}}{n_{c}k_{B}T_{c}}}}$, where the subscript ${\displaystyle c}$ indicates the critical point is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of ${\displaystyle 3/8=0.375}$.

Substance Value
H2O 0.23[4]
4He 0.31[4]
He 0.30[5]
H2 0.30[5]
Ne 0.29[5]
N2 0.29[5]
Ar 0.29[5]

## References

1. ^ Tester, Jefferson W. & Modell, Michael (1997). Thermodynamics and its applications. Prentice Hall. ISBN 0-13-915356-X.
2. ^ Çengel Y.A.; Boles M.A. (2007). Thermodynamics: An Engineering Approach (Sixth ed.). McGraw Hill. ISBN 9780071257718. page 141
3. ^ A Four-Parameter Corresponding States Correlation for Fluid Compressibility Factors by Walter M. Kalback and Kenneth E. Starling, Chemical Engineering Department, University of Oklahoma.
4. ^ a b Goodstein, David (1985) [1975]. "6" [Critical Phenomena and Phase Transitions]. States of Matter (1st ed.). Toronto, Canada: General Publishing Company, Ltd. p. 452. ISBN 0-486-64927-X.
5. de Boer, J. (April 1948). "Quantum theory of condensed permanent gases I the law of corresponding states". Physica. Utrecht, Netherlands: Elsevier. 14: 139–148. Bibcode:1948Phy....14..139D. doi:10.1016/0031-8914(48)90032-9.