# Real gas

Real gases are non-hypothetical gases whose molecules occupy space and have interactions; consequently, they adhere to gas laws. To understand the behaviour of real gases, the following must be taken into account:

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

Isotherms of real gas

Dark blue curves – isotherms below the critical temperature. Green sections – metastable states.

The section to the left of point F – normal liquid.
Point F – boiling point.
Line FG – equilibrium of liquid and gaseous phases.
Section FA – superheated liquid.
Section F′A – stretched liquid (p<0).
Section AC – analytic continuation of isotherm, physically impossible.
Section CG – supercooled vapor.
Point G – dew point.
The plot to the right of point G – normal gas.
Areas FAB and GCB are equal.

Red curve – Critical isotherm.
Point K – critical point.

Light blue curves – supercritical isotherms

### Van der Waals model

Real gases are often modeled by taking into account their molar weight and molar volume

${\displaystyle RT=\left(p+{\frac {a}{V_{\text{m}}^{2}}}\right)\left(V_{\text{m}}-b\right)}$

or alternatively:

${\displaystyle p={\frac {RT}{V_{m}-b}}-{\frac {a}{V_{m}^{2}}}}$

Where p is the pressure, T is the temperature, R the ideal gas constant, and Vm the molar volume. a and b are parameters that are determined empirically for each gas, but are sometimes estimated from their critical temperature (Tc) and critical pressure (pc) using these relations:

{\displaystyle {\begin{aligned}a&={\frac {27R^{2}T_{\text{c}}^{2}}{64p_{\text{c}}}}\\b&={\frac {RT_{\text{c}}}{8p_{\text{c}}}}\end{aligned}}}

With the reduced properties ${\displaystyle p_{r}={\frac {p}{p_{\text{c}}}},\ V_{r}={\frac {V_{\text{m}}}{V_{\text{m,c}}}},\ T_{r}={\frac {T}{T_{\text{c}}}}\ }$ the equation can be written in the reduced form:

${\displaystyle p_{r}={\frac {8}{3}}{\frac {T_{r}}{V_{r}-{\frac {1}{3}}}}-{\frac {3}{V_{r}^{2}}}}$

### Redlich–Kwong model

Critical isotherm for Redlich-Kwong model in comparison to van-der-Waals model and ideal gas (with V0=RTc/pc)

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

${\displaystyle RT=\left(p+{\frac {a}{{\sqrt {T}}V_{\text{m}}\left(V_{\text{m}}+b\right)}}\right)\left(V_{\text{m}}-b\right)}$

or alternatively:

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{{\sqrt {T}}V_{\text{m}}\left(V_{\text{m}}+b\right)}}}$

where a and b two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:

{\displaystyle {\begin{aligned}a&=0.42748\,{\frac {R^{2}{T_{\text{c}}}^{\frac {5}{2}}}{p_{\text{c}}}}\\b&=0.08664\,{\frac {RT_{\text{c}}}{p_{\text{c}}}}\end{aligned}}}

Using ${\displaystyle \ p_{r}={\frac {p}{p_{\text{c}}}},\ V_{r}={\frac {V_{\text{m}}}{V_{\text{m,c}}}},\ T_{r}={\frac {T}{T_{\text{c}}}}\ }$ the equation of state can be written in the reduced form:

${\displaystyle p_{r}={\frac {3T_{r}}{V_{r}-b'}}-{\frac {1}{b'{\sqrt {T_{r}}}V_{r}\left(V_{r}+b'\right)}}}$ with ${\displaystyle b'={\sqrt[{3}]{2}}-1\approx 0.26}$

### Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot)[1] is very rarely used,

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{TV_{\text{m}}^{2}}}}$

but the modified version is somewhat more accurate

${\displaystyle p={\frac {RT}{V_{\text{m}}}}\left[1+{\frac {9{\frac {p}{p_{\text{c}}}}}{128{\frac {T}{T_{\text{c}}}}}}\left(1-{\frac {6}{\frac {T^{2}}{T_{\text{c}}^{2}}}}\right)\right]}$

### Dieterici model

This model (named after C. Dieterici[2]) fell out of usage in recent years

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}\exp \left(-{\frac {a}{V_{\text{m}}RT}}\right)}$

with parameters a, b, and

${\displaystyle \exp \left(-{\frac {a}{V_{\text{m}}RT}}\right)=e^{-{\frac {a}{V_{\text{m}}RT}}}=1-{\frac {a}{V_{\text{m}}RT}}+\dots }$

### Clausius model

The Clausius equation (named after Rudolf Clausius) is a very simple three-parameter equation used to model gases.

${\displaystyle RT=\left(p+{\frac {a}{T(V_{\text{m}}+c)^{2}}}\right)\left(V_{\text{m}}-b\right)}$

or alternatively:

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{T\left(V_{\text{m}}+c\right)^{2}}}}$

where

{\displaystyle {\begin{aligned}a&={\frac {27R^{2}T_{\text{c}}^{3}}{64p_{\text{c}}}}\\b&=V_{\text{c}}-{\frac {RT_{\text{c}}}{4p_{\text{c}}}}\\c&={\frac {3RT_{\text{c}}}{8p_{\text{c}}}}-V_{\text{c}}\end{aligned}}}

where Vc is critical volume.

### Virial model

The Virial equation derives from a perturbative treatment of statistical mechanics.

${\displaystyle pV_{\text{m}}=RT\left[1+{\frac {B(T)}{V_{\text{m}}}}+{\frac {C(T)}{V_{\text{m}}^{2}}}+{\frac {D(T)}{V_{\text{m}}^{3}}}+\ldots \right]}$

or alternatively

${\displaystyle pV_{\text{m}}=RT\left[1+B'(T)p+C'(T)p^{2}+D'(T)p^{3}\ldots \right]}$

where A, B, C, A′, B′, and C′ are temperature dependent constants.

### Peng–Robinson model

Peng–Robinson equation of state (named after D.-Y. Peng and D. B. Robinson[3]) has the interesting property being useful in modeling some liquids as well as real gases.

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a(T)}{V_{\text{m}}\left(V_{\text{m}}+b\right)+b\left(V_{\text{m}}-b\right)}}}$

### Wohl model

Isotherm (V/V0->p_r) at critical temperature for Wohl model, van der Waals model and ideal gas model (with V0=RTc/pc)
Untersuchungen über die Zustandsgleichung, pp. 9,10, Zeitschr. f. Physikal. Chemie 87

The Wohl equation (named after A. Wohl[4]) 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.

${\displaystyle p={\frac {RT}{V_{\text{m}}-b}}-{\frac {a}{TV_{\text{m}}\left(V_{\text{m}}-b\right)}}+{\frac {c}{T^{2}V_{\text{m}}^{3}}}\quad }$

or:

${\displaystyle \left(p-{\frac {c}{T^{2}V_{\text{m}}^{3}}}\right)\left(V_{\text{m}}-b\right)=RT-{\frac {a}{TV_{\text{m}}}}}$

or, alternatively:

${\displaystyle RT=\left(p+{\frac {a}{TV_{\text{m}}(V_{\text{m}}-b)}}-{\frac {c}{T^{2}V_{\text{m}}^{3}}}\right)\left(V_{\text{m}}-b\right)}$

where

${\displaystyle a=6p_{\text{c}}T_{\text{c}}V_{\text{m,c}}^{2}}$
${\displaystyle b={\frac {V_{\text{m,c}}}{4}}}$ with ${\displaystyle V_{\text{m,c}}={\frac {4}{15}}{\frac {RT_{c}}{p_{c}}}}$
${\displaystyle c=4p_{\text{c}}T_{\text{c}}^{2}V_{\text{m,c}}^{3}\ }$, where ${\displaystyle V_{\text{m,c}},\ p_{\text{c}},\ T_{\text{c}}}$ are (respectively) the molar volume, the pressure and the temperature at the critical point.

And with the reduced properties ${\displaystyle \ p_{r}={\frac {p}{p_{\text{c}}}},\ V_{r}={\frac {V_{\text{m}}}{V_{\text{m,c}}}},\ T_{r}={\frac {T}{T_{\text{c}}}}\ }$ one can write the first equation in the reduced form:

${\displaystyle p_{r}={\frac {15}{4}}{\frac {T_{r}}{V_{r}-{\frac {1}{4}}}}-{\frac {6}{T_{r}V_{r}\left(V_{r}-{\frac {1}{4}}\right)}}+{\frac {4}{T_{r}^{2}V_{r}^{3}}}}$

### Beattie–Bridgeman model

[5]This equation is based on five experimentally determined constants. It is expressed as

${\displaystyle p={\frac {RT}{v^{2}}}\left(1-{\frac {c}{vT^{3}}}\right)(v+B)-{\frac {A}{v^{2}}}}$

where

{\displaystyle {\begin{aligned}A&=A_{0}\left(1-{\frac {a}{v}}\right)&B&=B_{0}\left(1-{\frac {b}{v}}\right)\end{aligned}}}

This equation is known to be reasonably accurate for densities up to about 0.8 ρcr, where ρcr 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 is in ${\displaystyle {\frac {{\text{m}}^{3}}{{\text{k}}\,{\text{mol}}}}}$, T is in K and R = 8.314${\displaystyle {\frac {{\text{kPa}}\cdot {\text{m}}^{3}}{{\text{k}}\,{\text{mol}}\cdot {\text{K}}}}}$[6]

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
Helium, He 2.1886 0.05984 0.01400 0.0 40
Hydrogen, H2 20.0117 −0.00506 0.02096 −0.04359 504
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, sometimes referred to as the BWRS equation,

${\displaystyle p=RTd+d^{2}\left(RT(B+bd)-\left(A+ad-a\alpha d^{4}\right)-{\frac {1}{T^{2}}}\left[C-cd\left(1+\gamma d^{2}\right)\exp \left(-\gamma d^{2}\right)\right]\right)}$

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 ${\displaystyle \int _{V_{i}}^{V_{f}}({\frac {RT}{V_{m}}}-P_{real})dV}$.