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 derive a universal property of all fluids that follow the van der Waals equation of state. It predicts a value of ${\displaystyle 3/8=0.375}$ that is found to be an overestimate when compared to real gases.

Edward A. Guggenheim used the phrase "Principle of Corresponding States" in an opt-cited paper to describe the phenomenon where different systems have very similar behaviors when near a critical point.^[4]

There are many examples of non-ideal gas models which satisfy this theorem, such as the van der Waals model, the Dieterici model, and so on, that can be found on the page on real gases.

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 physical quantities measured at the critical point, is predicted to be a constant independent of substance by many equations of state.

The table below for a selection of gases uses the following conventions:

• ${\displaystyle T_{c}}$: 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]

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[5]
4He	0.226×106	5.2	14.43×10−3	0.31[5]
He	0.226×106	5.2	14.43×10−3	0.30[6]
H2	1.279×106	33.2	32.3×10−3	0.30[6]
Ne	2.73×106	44.5	2.066×10−3	0.29[6]
N2	3.354×106	126.2	3.2154×10−3	0.29[6]
Ar	4.861×106	150.7	1.883×10−3	0.29[6]
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

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 Archived 2007-03-17 at the Wayback Machine by Walter M. Kalback and Kenneth E. Starling, Chemical Engineering Department, University of Oklahoma.
4. Guggenheim, E. A. (1945-07-01). "The Principle of Corresponding States". The Journal of Chemical Physics. 13 (7): 253–261. doi:10.1063/1.1724033. ISSN 0021-9606.
5. Goodstein, David (1985) [1975]. "6" [Critical Phenomena and Phase Transitions]. States of Matter (1st ed.). Toronto, Ontario, Canada: General Publishing Company, Ltd. p. 452. ISBN 0-486-64927-X.
6. de Boer, J. (April 1948). "Quantum theory of condensed permanent gases I the law of corresponding states". Physica. 14 (2–3). Utrecht, Netherlands: Elsevier: 139–148. Bibcode:1948Phy....14..139D. doi:10.1016/0031-8914(48)90032-9.

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.