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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

See also[edit]

References[edit]

  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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