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 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 (T_c) and critical pressure (p_c) using these relations:

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

The constants at critical point can be expressed as functions of the parameters a, b:

${\displaystyle p_{c}={\frac {a}{27b^{2}}},\quad T_{c}={\frac {8a}{27bR}},\qquad V_{m,c}=3b,\qquad Z_{c}={\frac {3}{8}}}$

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=RT_c/p_c)

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

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

The constants at critical point can be expressed as functions of the parameters a, b:

${\displaystyle p_{c}={\frac {({\sqrt[{3}]{2}}-1)^{7/3}}{3^{1/3}}}R^{1/3}{\frac {a^{2/3}}{b^{5/3}}},\quad T_{c}=3^{2/3}({\sqrt[{3}]{2}}-1)^{4/3}({\frac {a}{bR}})^{2/3},\qquad V_{m,c}={\frac {b}{{\sqrt[{3}]{2}}-1}},\qquad Z_{c}={\frac {1}{3}}}$

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. These can be normalized by dividing with the critical point state^{[note 1]}:

$${\displaystyle {\tilde {p}}=p{\frac {(2be)^{2}}{a}};\quad {\tilde {T}}=T{\frac {4bR}{a}};\quad {\tilde {V}}_{m}=V_{m}{\frac {1}{2b}}}$$

which casts the equation into the reduced form:^[3]

$${\displaystyle {\tilde {p}}(2{\tilde {V}}_{m}-1)={\tilde {T}}e^{2-{\frac {2}{{\tilde {T}}{\tilde {V}}_{m}}}}}$$

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 V_c 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^[4]) 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=RT_c/p_c)

Untersuchungen über die Zustandsgleichung, pp. 9,10, Zeitschr. f. Physikal. Chemie 87

The Wohl equation (named after A. Wohl^[5]) 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

^[6] 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 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 is in ${\displaystyle {\frac {{\text{m}}^{3}}{{\text{k}}\,{\text{mol}}}}}$, T is in K and R = 8.314${\displaystyle {\frac {{\text{kPa}}\cdot {\text{m}}^{3}}{{\text{k}}\,{\text{mol}}\cdot {\text{K}}}}}$^[7]

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.