# Virial expansion

The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the number 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 ${\displaystyle N}$ atoms or molecules,

${\displaystyle {\frac {p}{k_{B}T}}=\rho +B_{2}(T)\rho ^{2}+B_{3}(T)\rho ^{3}+\cdots ,}$

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

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

${\displaystyle {\frac {\beta p}{\rho }}=1+\sum _{i=1}^{\infty }B_{i+1}(T)\rho ^{i}}$.

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

## Comparison with Van der Waals equation

The Van der Waals equation can be used to derive the approximation ${\displaystyle B_{2}(T)\approx {\frac {b}{N_{A}}}-{\frac {a}{N_{A}^{2}k_{\rm {B}}T}}}$ with the Van der Waals constants a and b.

And when ${\displaystyle T_{B}={\frac {a}{b\cdot k_{\rm {B}}N_{A}}}={\frac {a}{bR}}}$ then ${\displaystyle B_{2}(T_{B})\approx 0}$, see Boyle temperature.

According to Van der Waals constants (data page) the constants for hydrogen gas are for example a = 0.2476 L2bar/mol2 and b = 0.02661 L/mol and therefore the estimation of the Boyle temperature for hydrogen is ${\displaystyle T_{B}={\frac {0.2476\cdot 10^{-1}{\rm {\frac {m^{6}Pa}{mol^{2}}}}}{0.02661\cdot 10^{-3}{\rm {{\frac {m^{3}}{mol}}\cdot 8.3145{\rm {\frac {J}{mol\cdot K}}}}}}}=112\ {\rm {K}}}$. (The real value for hydrogen is 110 K.[2] In nitrogen the difference is bigger.)