Compressibility factor via intermolecular force potential

The virial equation describes the deviation of a real gas from ideal gas behaviour. The virial coefficients in the virial expansion account for interactions between successively larger groups of molecules. Since interactions between large numbers of molecules are rare, the virial equation is usually truncated at the third term onwards. Under the assumption that only pair interactions are present, the compressibility factor can be linked to the intermolecular force potential.

Notation:

1. \(k_\text{B}\) (k_B) is boltzmann_constant.

Conditions:

1. The virial expansion is done up to the second virial coefficient inclusively.

Links:

1. Wikipedia, second formula.

compressibility_factor

compressibility_factor of the gas. Also see Compressibility factor.

Symbol:

Z

Latex:

\(Z\)

Dimension:

dimensionless

particle_count

particle_count of the system.

Symbol:

N

Latex:

\(N\)

Dimension:

dimensionless

volume

volume of the system.

Symbol:

V

Latex:

\(V\)

Dimension:

volume

intermolecular_distance

euclidean_distance between gas molecules.

Symbol:

r

Latex:

\(r\)

Dimension:

length

intermolecular_force_potential

Intermolecular force potential as a function of intermolecular_distance. See potential_energy.

Symbol:

U(r)

Latex:

\(U{\left(r \right)}\)

Dimension:

energy

temperature

temperature of the system.

Symbol:

T

Latex:

\(T\)

Dimension:

temperature

law

Z = 1 + 2 * pi * N / V * Integral((1 - exp(-U(r) / (k_B * T))) * r^2, (r, 0, oo))

Latex:

\[Z = 1 + \frac{2 \pi N}{V} \int\limits_{0}^{\infty} \left(1 - \exp{\left(- \frac{U{\left(r \right)}}{k_\text{B} T} \right)}\right) r^{2}\, dr\]