Van der Waals equation of state: Difference between revisions

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* <math> R  </math> is the so-called gas constant; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]].
* <math> R  </math> is the so-called gas constant; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]].


The van der Waals equation of state takes into account two features that are absent in the [[Equation of State: Ideal Gas | ideal Gas]] equation of state:
The van der Waals equation of state takes into account two features that are absent in the [[Equation of State: Ideal Gas | ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,  
The parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,  
it represents the minimum molar volume of the system, whereas <math> a </math> 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.
it represents the minimum molar volume of the system, whereas <math> a </math> 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.



Revision as of 15:53, 24 September 2007

The van der Waals equation of state, developed by Johannes Diderik van der Waals, can be written as

.

where:

  • is the volume,
  • is the number of moles,
  • is the so-called gas constant; , with being the Avogadro constant and being the Boltzmann constant.

The van der Waals equation of state 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.


Critical point

The critical point for the van der Waals equation of state can be found at

,

and at

.

Dimensionless formulation

If one takes the following 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