Real gas

Real gases are non-ideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law.
To understand the behaviour of real gases, the following must be taken into account:

compressibility effects;
variable specific heat capacity;
van der Waals forces;
non-equilibrium thermodynamic effects;
issues with molecular dissociation and elementary reactions with variable composition

For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, real-gas models have to be used near the condensation point of gases, near critical points, at very high pressures, to explain the Joule–Thomson effect, and in other less usual cases. The deviation from ideality can be described by the compressibility factor Z.

Models

Van der Waals model

Real gases are often modeled by taking into account their molar weight and molar volume
or alternatively:
Where p is the pressure, T is the temperature, R the ideal gas constant, and V_m the molar volume. a and b are parameters that are determined empirically for each gas, but are sometimes estimated from their critical temperature and critical pressure using these relations:
The constants at critical point can be expressed as functions of the parameters a, b:
With the reduced properties,, the equation can be written in the reduced form:

Redlich–Kwong model

The Redlich–Kwong equation is another two-parameter equation that is used to model real gases. It is almost always more accurate than the van der Waals equation, and often more accurate than some equations with more than two parameters. The equation is
or alternatively:
where a and b are two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:
The constants at critical point can be expressed as functions of the parameters a, b:
Using,, the equation of state can be written in the reduced form:
with

Berthelot and modified Berthelot model

The Berthelot equation is very rarely used,
but the modified version is somewhat more accurate

Dieterici model

This model fell out of usage in recent years
with parameters a, b. These can be normalized by dividing with the critical point state:which casts the equation into the reduced form:

Clausius model

The Clausius equation is a very simple three-parameter equation used to model gases.
or alternatively:
where
where V_c is critical volume.

Virial model

The Virial equation derives from a perturbative treatment of statistical mechanics.
or alternatively
where A, B, C, A′, B′, and C′ are temperature dependent constants.

Peng–Robinson model

has the interesting property being useful in modeling some liquids as well as real gases.

Wohl model

The Wohl equation is formulated in terms of critical values, making it useful when real gas constants are not available, but it cannot be used for high densities, as for example the critical isotherm shows a drastic decrease of pressure when the volume is contracted beyond the critical volume.
or:
or, alternatively:
where
where,, are the molar volume, the pressure and the temperature at the critical point.
And with the reduced properties,, one can write the first equation in the reduced form:

Beattie–Bridgeman model

This equation is based on five experimentally determined constants. It is expressed as
where
This equation is known to be reasonably accurate for densities up to about 0.8 ρ_cr, where ρ_cr is the density of the substance at its critical point. The constants appearing in the above equation are available in the following table when p is in kPa, V_m is in, T is in K and

Gas	A₀	a	B₀	b	c
Air	131.8441	0.01931	0.04611	−0.001101	4.34×10⁴
Argon, Ar	130.7802	0.02328	0.03931	0.0	5.99×10⁴
Carbon dioxide, CO₂	507.2836	0.07132	0.10476	0.07235	6.60×10⁵
Ethane, C₂H₆	595.791	0.05861	0.09400	0.01915	90.00×10⁴
Helium, He	2.1886	0.05984	0.01400	0.0	40
Hydrogen, H₂	20.0117	−0.00506	0.02096	−0.04359	504
Methane, CH₄	230.7069	0.01855	0.05587	-0.01587	12.83×10⁴
Nitrogen, N₂	136.2315	0.02617	0.05046	−0.00691	4.20×10⁴
Oxygen, O₂	151.0857	0.02562	0.04624	0.004208	4.80×10⁴

Benedict–Webb–Rubin model

The BWR equation,
where d is the molar density and where a, b, c, A, B, C, α, and γ are empirical constants. Note that the γ constant is a derivative of constant α and therefore almost identical to 1.

Thermodynamic expansion work

The expansion work of the real gas is different than that of the ideal gas by the quantity.