Van der Waals equation of state

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The van der Waals equation of state, developed by Johannes Diderik van der Waals [1] [2], takes into account two features that are absent in the ideal gas equation of state; the parameter  b introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas  a 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

\left(p + \frac{an^2}{V^2}\right)\left(V-nb\right) = nRT

where:

Critical point

At the critical point one has \left.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 , and \left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 , leading to

T_c= \frac{8a}{27bR}


p_c=\frac{a}{27b^2}


\left.v_c\right.=3b


along with a critical point compressibility factor of


\frac{p_c v_c}{RT_c}= \frac{3}{8} = 0.375


which then leads to


a= \frac{27}{64}\frac{R^2T_c^2}{p_c}


b= \frac{RT_c}{8p_c}

Virial form

One can re-write the van der Waals equation given above as a virial equation of state as follows:

Z :=  \frac{pV}{nRT} = \frac{1}{1- \frac{bn}{V}} - \frac{an}{RTV}

Using the well known series expansion (1-x)^{-1} = 1 + x + x^2 + x^3 + ... one can write the first term of the right hand side as [3]:

\frac{1}{1- \frac{bn}{V}} = 1 +  \frac{bn}{V} +  \left( \frac{bn}{V} \right)^2 +  \left( \frac{bn}{V} \right)^3 + ...

Incorporating the second term of the right hand side in its due place leads to:

Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} +  \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 +   ....

From the above one can see that the second virial coefficient corresponds to

B_{2}(T)=  b -\frac{a}{RT}

and the third virial coefficient is given by

B_{3}(T)=  b^2

Boyle temperature

The Boyle temperature of the van der Waals equation is given by

B_2\vert_{T=T_B}=0 =  b -\frac{a}{RT_B}

leading to

T_B = \frac{a}{bR}

Dimensionless formulation

If one takes the following reduced quantities

\tilde{p} = \frac{p}{p_c};~ \tilde{V} = \frac{V}{V_c}; ~\tilde{t} = \frac{T}{T_c};

one arrives at

\tilde{p} = \frac{8\tilde{t}}{3\tilde{V} -1} -\frac{3}{\tilde{V}^2}

The following image is a plot of the isotherms T/T_c = 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

Critical exponents

The critical exponents of the Van der Waals equation of state place it in the mean field universality class.

See also

References

  1. J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873)
  2. English translation: J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930
  3. This expansion is valid as long as -1 < x < 1, which is indeed the case for bn/V

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