Virial equation

Also called the virial expansion, the virial equation of state expresses the compressibility factor (and therefore the pressure) of a real gas in local equilibrium as a power series of molar density.

Notes:

1. The first virial coefficient \(C_1\) is defined to be 1 in order to enforce that the equation reduces to the ideal gas equation as gas density approaches zero.

2. The \(n\)-th virial coefficient represents non-additive \(n\)-body interactions of particles and all mutual interactions of \(2\) up to \((n - 1)\) particles.

3. In general, virial coefficients are functions of temperature.

4. \(O(\dots)\) is the mathematical Big O.

Conditions:

1. Interactions between 4 and more bodies are quite rare to happen, so the expansion is truncated to contain only the second and third virial coefficients. Moreover, the latter have been extensively studied and tabulated for many fluids.

2. In this law the limit \(\rho \to 0\) is assumed.

Links:

1. Wikipedia.

compressibility_factor

compressibility_factor. Also see Compressibility factor is deviation from ideal gas.

Symbol:

Z

Latex:

\(Z\)

Dimension:

dimensionless

second_virial_coefficient

Second virial coefficient correponding to pair interactions between particles.

Symbol:

C_2

Latex:

\(C_{2}\)

Dimension:

volume/amount_of_substance

third_virial_coefficient

Third virial coefficient corresponding to 3-body interaction between particles.

Symbol:

C_3

Latex:

\(C_{3}\)

Dimension:

volume**2/amount_of_substance**2

molar_density

Molar density of the system, or amount_of_substance per unit volume. See Quantity is volumetric density times volume.

Symbol:

rho_m

Latex:

\(\rho_\text{m}\)

Dimension:

amount_of_substance/volume

law

Z = 1 + C_2 * rho_m + C_3 * rho_m^2 + O(rho_m^3)

Latex:

\[Z = 1 + C_{2} \rho_\text{m} + C_{3} \rho_\text{m}^{2} + O\left(\rho_\text{m}^{3}\right)\]