# Virial equation

## Virial equation

An equation of state of gases that has additional terms beyond that for an ideal gas, which account for the interactions between the molecules. The pressure *p* can be expressed in terms of the molar volume *V _{m}* =

*V*/

*n*(where

*n*is the number of moles of gas molecules in a volume

*V*), the absolute temperature

*T*, and the universal gas constant

*R*= 8.3145 J K

^{-1}mol

^{-1}or, in more commonly used practical units, 0.082058 L atm K

^{-1}mol

^{-1}[Eq. (1)].

*B*(

_{n}*T*) are functions only of the temperature and depend on the nature of the gas. In an ideal gas, in which all interactions between the molecules can be neglected because

*V*is sufficiently large, only the first term, unity, survives on the right-hand side.

_{m}The equation is important because there are rigorous relations between the coefficients *B*_{2}, *B*_{3}, and so on, as well as the interactions of the molecules in pairs, triplets, and so forth. It provides a valuable route to a knowledge of the intermolecular forces. Thus if the intermolecular energy of a pair of molecules at a separation *r* is *u*(*r*), then the second virial coefficient can be expressed as Eq. (2).

*N*= 6.0221 × 10

_{A}^{23}mol

^{-1}is the Avogadro constant, and

*k*=

*R*/

*N*. In a gas mixture,

_{A}*B*is a polynomial of order

_{n}*n*in the mole fractions of the components.

*See*Intermolecular forces

The virial equation is useful in practice because it represents the pressure accurately at low and moderate gas densities, for example, up to about 4 mol L^{-1} for nitrogen at room temperature, which corresponds to a pressure of about 100 atm (10 MPa). It is not useful at very high densities, where the series may diverge, and is inapplicable to liquids. It can be rearranged to give the ratio *p**V _{m}*/

*RT*as an expansion in powers of the pressure instead of the density, which is equally useful empirically, but the coefficients of the pressure expansion are not usually called virial coefficients, and lack any simple relation to the intermolecular forces or, in a mixture, to the composition of the gas.

*See*Van der Waals equation