Virial equation of state

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The virial equation of state is used to describe the behavior of diluted gases. It is usually written as an expansion of the compressibility factor,  Z , in terms of either the density or the pressure. Such an expansion was first introduced in 1885 by Thiesen [1] and extensively studied by Heike Kamerlingh Onnes [2] [3], and mathematically by Ursell [4]. One has

 \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}.

where

Virial coefficients[edit]

The second virial coefficient represents the initial departure from ideal-gas behaviour

B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2

where N_A is Avogadros number and d\tau_1 and d\tau_2 are volume elements of two different molecules in configuration space.

One can write the third virial coefficient as

B_{3}(T)= - \frac{1}{3V} \int \int \int f_{12} f_{13} f_{23}  dr_1 dr_2 dr_3

where f is the Mayer f-function (see also: Cluster integrals). See also [5]

Convergence[edit]

For a commentary on the convergence of the virial equation of state see [6] and section 3 of [7].

Quantum virial coefficients[edit]

Using the path integral formulation one can also calculate the virial coefficients of quantum systems [8].

References[edit]

Related reading