Virial expansion

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The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the density. The virial expansion, introduced in 1901 by Heike Kamerlingh Onnes, is a generalization of the ideal gas law. He wrote that for a gas containing N atoms or molecules,


   \frac{p}{k_BT} = \rho + B_2(T) \rho^2 +B_3(T) \rho^3+ \cdots,

where p is the pressure, k_B is the Boltzmann constant, T is the absolute temperature, and \rho \equiv N/V is the number density of the gas. Note that for a gas containing a fraction n of N_A (Avogadro's number) molecules, truncation of the virial expansion after the first term leads to pV = n N_A k_B T = nRT, which is the ideal gas law.

Writing \beta=(k_{B}T)^{-1}, the virial expansion can be written as

\frac{\beta p}{\rho}=1+\sum_{i=1}^{\infty}B_{i+1}(T)\rho^{i}.

The virial coefficients B_i(T) are characteristic of the interactions between the particles in the system and in general depend on the temperature T. Virial expansion can also be applied to aqueous ionic solutions, as shown by Harold Friedman.

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