Van der Waals equation of state

From SklogWiki
Revision as of 13:37, 20 October 2009 by Carl McBride (talk | contribs) (Deleted unused Maxwell's equal area construction subsection)
Jump to navigation Jump to search

The van der Waals equation of state, developed by Johannes Diderik van der Waals, takes into account two features that are absent in the ideal gas equation of state; the parameter introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.

Equation of state

The van der Waals equation of state can be written as

where:

  • is the pressure,
  • is the volume,
  • is the number of moles,
  • is the absolute temperature,
  • is the molar gas constant; , with being the Avogadro constant and being the Boltzmann constant.
  • and are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.

Critical point

At the critical point one has , and , leading to



.


and



which then leads to



Dimensionless formulation

If one takes the following reduced quantities

one arrives at

The following image is a plot of the isotherms = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:

Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state
Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state

Interesting reading

References

  • J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).

English translation: