Law of corresponding states

The law of corresponding states is an empirical law encapsulates the finding that the equations of state for many real gases are remarkably similar when they are expressed in terms of reduced temperatures ($T_{r}=T/T_{c}$), pressures, ($p_{r}=p/p_{c}$) and volumes ($V_{r}=V/V_{c}$), where the subscript $c$ represents the value of the property at the critical point. This law was first described by Johannes Diderik van der Waals in his 1873 thesis, and forms the subject of a paper by him in 1913 (Ref. 1).

For argon, krypton, nitrogen, oxygen, carbon dioxide and methane one has (Ref. 4)

${\frac {p_{c}V_{c}}{RT_{c}}}\approx 0.292$

(for pressure measured in atmospheres, and volume in cm³mole^-1)

For neon, argon, and oxygen one has (Ref. 4)

${\frac {T_{B}}{T_{c}}}\approx 2.7$

where $T_{B}$ is the Boyle temperature.

For neon, argon, krypton,and xenon one has (Ref. 4)

${\frac {T_{tp}}{T_{c}}}\approx 0.555$

where $T_{tp}$ is the triple point.

Assumptions

Pitzer (Ref. 3) produced a list of assumptions in order for the law of corresponding states to apply. This list was later modified by Guggenheim (Ref. 4). These are:

1. There is negligible difference between Fermi–Dirac statistics and Bose–Einstein statistics for the system (i.e. the system behaves classically).
2. The effect of quantisation of the translational degrees of freedom is negligible (i.e. the system behaves classically).
3. The molecules are spherically symmetrical, either actually or by virtue of rapid and free rotation.
4. The intramolecular degrees of freedom are assumed to be completely independent of the volume per molecule.
5. The potential energy will be taken as a function only of the various intermolecular distances.
6. The potential energy for a pair of molecules can be written as $A\Phi (r/r_{0})$ where $r$ is the intermolecular distance, and $A$ and $r_{0}$ are characteristic constants, and $\Phi$ is a universal function.

References

1. Johannes Diderik van der Waals "The law of corresponding states for different substances", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 15 II pp. 971-981 (1913)
2. J. de Boer and A. Michels "Contribution to the quantum-mechanical theory of the equation of state and the law of corresponding states. Determination of the law of force of helium", Physica 5 pp. 945-957 (1938)
3. Kenneth S. Pitzer "Corresponding States for Perfect Liquids", Journal of Chemical Physics 7 pp. 583-590 (1939)
4. E. A. Guggenheim "The Principle of Corresponding States", Journal of Chemical Physics 13 pp. 253-261 (1945)
5. Patrick Grosfils and James F. Lutsko "Dependence of the liquid-vapor surface tension on the range of interaction: A test of the law of corresponding states", Journal of Chemical Physics 130 054703 (2009)