# Theorem of corresponding states

According to van der Waals, the theorem of corresponding states (or principle/law 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.

The principle 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 ${\displaystyle Z_{c}={\frac {P_{c}v_{c}\mu }{RT_{c}}}}$ , 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 ${\displaystyle 3/8=0.375}$ .

Where:

• ${\displaystyle T_{c}}$ : critical temperature [K]
• ${\displaystyle P_{c}}$ : critical pressure [Pa]
• ${\displaystyle v_{c}}$ : critical specific volume [m3⋅kg−1]
• ${\displaystyle R}$ : gas constant (8.314 JK−1mol−1)
• ${\displaystyle \mu }$ : Molar mass [kg⋅mol−1]

For example:

Substance ${\displaystyle P_{c}}$  [Pa] ${\displaystyle T_{c}}$  [K] ${\displaystyle v_{c}}$  [m3/kg] ${\displaystyle Z_{c}}$
H2O 21.817×106 647.3 3.154×10−3 0.23[4]
4He 0.226×106 5.2 14.43×10−3 0.31[4]
He 0.226×106 5.2 14.43×10−3 0.30[5]
H2 1.279×106 33.2 32.3×10−3 0.30[5]
Ne 2.73×106 44.5 2.066×10−3 0.29[5]
N2 3.354×106 126.2 3.2154×10−3 0.29[5]
Ar 4.861×106 150.7 1.883×10−3 0.29[5]
Xe 5.87×106 289.7 0.9049×10−3 0.29
O2 5.014×106 154.8 2.33×10−3 0.291
CO2 7.290×106 304.2 2.17×10−3 0.275
SO2 7.88×106 430.0 1.900×10−3 0.275
CH4 4.58×106 190.7 6.17×10−3 0.285
C3H8 4.21×106 370.0 4.425×10−3 0.267