Definitions

# Real gas

Real gas effects refers to an assumption base where the following are taken into account:

• Compressibility effects
• Variable 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 "over-kill" and the ideal gas approximation is used. Real-gas models have to be used near condensation point of gases, near critical point, at very high pressures, and in several other less usual cases.

## Modelisation

### Van der Waals modelisation

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

$RT=\left(P+frac\left\{a\right\}\left\{V_m^2\right\}\right)\left(V_m-b\right)$

Where P is the pressure, T is the temperature, R the ideal gas constant, and Vm the molar mass. 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:

$a=frac\left\{27R^2T_c\right\}\left\{64P_c\right\}$

$b=frac\left\{RT_c\right\}\left\{8P_c\right\}$

### Redlich-Kwong modelisation

The Redlich-Kwong equation is another two-parameters equation that is used to modelize real gases. It is almost always more accurate than the Van der Waals equation, and ofter more accurate than some equation with more than two parameters. The equation is

$RT=P+frac\left\{a\right\}\left\{V_m\left(V_m+b\right)T^frac\left\{1\right\}\left\{2\right\}\right\}\left(V_m-b\right)$

where a and b two empirical parameters that are not the same parameters as in the Van der Waals equation.

### Berthelot and modified Berthelot modelisation

The Berthelot Equation is very rarely used,

$P=frac\left\{RT\right\}\left\{V-b\right\}-frac\left\{a\right\}\left\{TV^2\right\}$

but the modified version is somewhat more accurate

$P=frac\left\{RT\right\}\left\{V\right\}left\left(1+frac\left\{9PT_c\right\}\left\{128P_cT\right\}frac\left\{\left(1-6T_c^2\right)\right\}\left\{T^2\right\}right\right)$

### Dieterici modelisation

This modelisation fell out of usage in recent years

$P=RTfrac\left\{exp\left\{\left(frac\left\{-a\right\}\left\{V_mRT\right\}\right)\right\}\right\}\left\{V_m-b\right\}$

### Clausius modelisation

The Clausius equation is a very simply three-parameter equation used to modelize gases.

$RT=left\left(P+frac\left\{a\right\}\left\{T\left(V_m+c\right)^2\right\}right\right)\left(V_m-b\right)$

where

$a=frac\left\{V_c-RT_c\right\}\left\{4P_c\right\}$

$b=frac\left\{3RT_c\right\}\left\{8P_c\right\}-V_c$

$c=frac\left\{27R^2T_c^3\right\}\left\{64P_c\right\}$

### Virial Modelisation

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

$PV_m=RTleft\left(1+frac\left\{B\left(T\right)\right\}\left\{V_m\right\}+frac\left\{C\left(T\right)\right\}\left\{V_m^2\right\}+frac\left\{D\left(T\right)\right\}\left\{V_m^3\right\}+...right\right)$

or alternatively

$PV_m=RTleft\left(1+frac\left\{B^prime\left(T\right)\right\}\left\{P\right\}+frac\left\{C^prime\left(T\right)\right\}\left\{P^2\right\}+frac\left\{D^prime\left(T\right)\right\}\left\{P^3\right\}+...right\right)$

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

### Peng-Robinson Modelisation

This two parameter equation has the interesting property being useful in modelizing some liquids as well as real gases.

$P=frac\left\{RT\right\}\left\{V_m-b\right\}-frac\left\{a\left(T\right)\right\}\left\{V_m\left(V_m+b\right)+b\left(Vm-b\right)\right\}$

### Wohl modelisation

The Wohl equation is formulated in terms of critial values, making it useful when real gas constants are not available.

$RT=left\left(P+frac\left\{a\right\}\left\{TV_m\left(V_m-b\right)\right\}-frac\left\{c\right\}\left\{T^2V_m^3\right\}right\right)\left(V_m-b\right)$

where

$a=6P_cT_cV_c^2$

$b=frac\left\{V_c\right\}\left\{4\right\}$

$c=4P_cT_c^2V_c^3$

### Beatte-Bridgeman Modelisation

The Beattie-Bridgeman equation

$P=RTd+\left(BRT-A-frac\left\{Rc\right\}\left\{T^2\right\}\right)d^2+\left(-BbRT+Aa-frac\left\{RBc\right\}\left\{T^2\right\}\right)d^3+frac\left\{RBbcd^4\right\}\left\{T^2\right\}$

where d is the molal density and a, b, c, A, and B are empirical parameters.

### Benedict-Webb-Rubin Modelisation

The BWR equation, sometimes referred to as the BWRS equation

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

Where d is the molal density and where a, b, c, A, B, C, α, and γ are empirical constants.

## References

http://www.ccl.net/cca/documents/dyoung/topics-orig/eq_state.html

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