Theorem of corresponding states

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 J⋅K⁻¹⋅mol⁻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

• Van der Waals equation
• Equation of state
• Compressibility factors
• Johannes Diderik van der Waals equation
• Noro-Frenkel law of corresponding states

• Properties of Natural Gases. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document)
• Theorem of corresponding states on SklogWiki.