Theorem of corresponding states

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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 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

See also[]

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

References[]

External links[]

• 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.

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