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 was introduced in 1901 by Heike Kamerlingh Onnes as a generalization of the ideal gas law. He wrote 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, $T$ the absolute temperature, and $\rho \equiv N/V$ is the number density of the gas. Note that for a gas containing $N A$ (Avogadro's number) molecules truncation of the virial expansion after the first term leads to $p V = N A k B T = R T$ , which is the ideal gas law.

Writing $β = (k B T) - 1$ , the virial expansion can be written in closed form 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

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