Real gas

 ( Gases ) 

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.

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Models of real gas

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

The section to the left of point F – normal liquid.
Point F – boiling point.
Line FG – equilibrium of liquid and gaseous phases.
Section FA – Superheating.
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]]

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Van der Waals model Real gases are often modeled by taking into account their molar weight and molar volume $$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{V\_\backslash text\{m\}^2\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively: $$p\; =\; \backslash frac\{RT\}\{V\_m\; -\; b\}\; -\; \backslash frac\{a\}\{V\_m^2\}$$

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: $$\backslash begin\{align\}$$

 a &= \frac{27R^2 T_\text{c}^2}{64p_\text{c}}, &
 b &= \frac{RT_\text{c}}{8p_\text{c}}
     

\end{align}

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

With the reduced properties $p\_r\; =\; p\; /\; p\_\backslash text\{c\}$, $V\_r\; =\; V\_\backslash text\{m\}\; /\; V\_\backslash text\{m,c\}$, $T\_r\; =\; T\; /\; T\_\backslash text\{c\}$ the equation can be written in the reduced form: $$p\_r\; =\; \backslash frac\{8\}\{3\}\backslash frac\{T\_r\}\{V\_r\; -\; \backslash frac\{1\}\{3\}\}\; -\; \backslash frac\{3\}\{V\_r^2\}$$

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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 $$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{\backslash sqrt\{T\}V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively: $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{\backslash sqrt\{T\}V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\}$$

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: $$\backslash begin\{align\}$$

 a &= 0.42748\, \frac{R^2{T_\text{c}}^\frac{5}{2}}{p_\text{c}}, \\[2pt]
 b &= 0.08664\, \frac{RT_\text{c}}{p_\text{c}}
     

\end{align}

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

Using $p\_r\; =\; p/p\_\backslash text\{c\}$, $V\_r\; =\; V\_\backslash text\{m\}/V\_\backslash text\{m,c\}$, $T\_r\; =\; T\; /\; T\_\backslash text\{c\}$ the equation of state can be written in the reduced form: $$p\_r\; =\; \backslash frac\{3\; T\_r\}\{V\_r\; -\; b\text{'}\}\; -\; \backslash frac\{1\}\{b\text{'}\backslash sqrt\{T\_r\}\; V\_r\; \backslash left(V\_r\; +\; b\text{'}\backslash right)\}$$ with $b\text{'}\; =\; \backslash sqrt3\{2\}\; -\; 1\; \backslash approx\; 0.26$

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Berthelot and modified Berthelot model The Berthelot equation (named after D. Berthelot)D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907) is very rarely used, $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}^2\}$$

but the modified version is somewhat more accurate $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\}\; \backslash left1$$

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Dieterici model This model (named after C. DietericiC. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)) fell out of usage in recent years $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; \backslash exp\backslash left(-\backslash frac\{a\}\{V\_\backslash text\{m\}RT\}\backslash right)$$

with parameters a, b. These can be normalized by dividing with the critical point state, and taking the derivative with respect to $V\_m$. The equation $(\backslash partial\_\{V\_m\}p)\_T\; =\; 0$ is a quadratic equation in $V\_m$, and it has a double root precisely when $V\_m\; =\; V\_c;\; T=T\_c$.}}:$$\backslash tilde\; p\; =\; p\; \backslash frac\{(2be)^2\}\{a\};\; \backslash quad\; \backslash tilde\; T\; =T\; \backslash frac\{4bR\}\{a\};\; \backslash quad\; \backslash tilde\; V\_m\; =\; V\_m\; \backslash frac\{1\}\{2b\}$$which casts the equation into the reduced form:

$$\backslash tilde\; p\; \backslash left(2\backslash tilde\; V\_m\; -\; 1\backslash right)\; =\; \backslash tilde\; T\; \backslash exp\backslash left(2\; -\; \backslash frac\{2\}\{\backslash tilde\; T\; \backslash tilde\; V\_m\}\backslash right)$$

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Clausius model The Clausius equation (named after Rudolf Clausius) is a very simple three-parameter equation used to model gases. $$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{T\; \{\backslash left(V\_\backslash text\{m\}\; +\; c\backslash right)\}^2\}\backslash right)\; \backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

or alternatively: $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{T\backslash left(V\_\backslash text\{m\}\; +\; c\backslash right)^2\}$$

where $$\backslash begin\{align\}$$

 a &= \frac{27R^2 T_\text{c}^3}{64p_\text{c}}, \\[4pt]
 b &= V_\text{c} - \frac{RT_\text{c}}{4p_\text{c}}, \\[4pt]
 c &= \frac{3RT_\text{c}}{8p_\text{c}} - V_\text{c}
     

\end{align}

where V_c is critical volume.

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Virial model The Virial equation derives from a perturbative treatment of statistical mechanics. $$pV\_\backslash text\{m\}\; =\; RT\backslash left1$$

or alternatively $$pV\_\backslash text\{m\}\; =\; RT\; \backslash left1$$

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

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Peng–Robinson model Peng–Robinson equation of state (named after D.-Y. Peng and D. B. Robinson) has the interesting property being useful in modeling some liquids as well as real gases. $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a(T)\}\{V\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; +\; b\backslash right)\; +\; b\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\}$$

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Wohl model The Wohl equation (named after A. Wohl) 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. $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}\; -\; b\}\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\}\; +\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash quad$$

or: $$\backslash left(p\; -\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)\; =\; RT\; -\; \backslash frac\{a\}\{TV\_\backslash text\{m\}\}$$

or, alternatively: $$RT\; =\; \backslash left(p\; +\; \backslash frac\{a\}\{TV\_\backslash text\{m\}(V\_\backslash text\{m\}\; -\; b)\}\; -\; \backslash frac\{c\}\{T^2\; V\_\backslash text\{m\}^3\}\backslash right)\backslash left(V\_\backslash text\{m\}\; -\; b\backslash right)$$

where where $V\_\backslash text\{m,c\}\; =\; \backslash frac\{4\}\{15\}\backslash frac\{RT\_c\}\{p\_c\}$, $p\_\backslash text\{c\}$, $T\_c$ are (respectively) the molar volume, the pressure and the temperature at the critical point.

And with the reduced properties $p\_r\; =\; p/p\_\backslash text\{c\}$, $V\_r\; =\; V\_\backslash text\{m\}\; /\; V\_\backslash text\{m,c\}$, $T\_r\; =\; T\; /\; T\_\backslash text\{c\}$ one can write the first equation in the reduced form: $$p\_r\; =\; \backslash frac\{15\}\{4\}\backslash frac\{T\_r\}\{V\_r\; -\; \backslash frac\{1\}\{4\}\}\; -\; \backslash frac\{6\}\{T\_r\; V\_r\backslash left(V\_r\; -\; \backslash frac\{1\}\{4\}\backslash right)\}\; +\; \backslash frac\{4\}\{T\_r^2\; V\_r^3\}$$

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Beattie–Bridgeman model Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010, This equation is based on five experimentally determined constants. It is expressed as $$p\; =\; \backslash frac\{RT\}\{V\_\backslash text\{m\}^2\}\backslash left(1\; -\; \backslash frac\{c\}\{V\_\backslash text\{m\}T^3\}\backslash right)(V\_\backslash text\{m\}\; +\; B)\; -\; \backslash frac\{A\}\{V\_\backslash text\{m\}^2\}$$

where $$\backslash begin\{align\}$$

 A &= A_0 \left(1 - \frac{a}{V_\text{m}}\right), &
 B &= B_0 \left(1 - \frac{b}{V_\text{m}}\right)
     

\end{align}

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 $\backslash frac\{\backslash text\{m\}^3\}\{\backslash text\{k\}\backslash ,\backslash text\{mol\}\}$, T is in K and $R\; =\; 8.314\; \backslash mathrm\{\backslash frac\{kPa\; \backslash cdot\; m^3\}\{kmol\; \backslash cdot\; K\}\}$Gordan J. Van Wylen and Richard E. Sonntage, Fundamental of Classical Thermodynamics, 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3

4>

Air	4.34×104
Argon, Ar	5.99×104
Carbon dioxide, CO2	6.60×105
Ethane, C2H6	90.00×104
Helium, He	40
Hydrogen, H2	504
Methane, CH4
Nitrogen, N2	4.20×104
Oxygen, O2	4.80×104

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Benedict–Webb–Rubin model The BWR equation, $$p\; =\; RTd\; +\; d^2\backslash left(RT(B\; +\; bd)\; -\; \backslash left(A\; +\; ad\; -\; a\backslash alpha\; d^4\backslash right)\; -\; \backslash frac\{1\}\{T^2\}\backslash leftC\backslash right)$$

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.

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Thermodynamic expansion work The expansion work of the real gas is different than that of the ideal gas by the quantity $\backslash int\_\{V\_i\}^\{V\_f\}\; \backslash left(\backslash frac\{RT\}\{V\_m\}\; -\; P\_\backslash text\{real\}\backslash right)\; dV$.

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See also

• Compressibility factor
• Equation of state
• Ideal gas law: Boyle's law and Gay-Lussac's law

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Further reading

• (1998). 9780471973935, John Wiley & Sons. ISBN 9780471973935
• (1993). 9780132757027, Prentice-Hall. ISBN 9780132757027
• (1985). 9780409951622, Butterworth Publishers. ISBN 9780409951622
• (2025). 9788173714610, Universities Press. ISBN 9788173714610
• (2025). 9780080459042, Elsevier. ISBN 9780080459042

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External links

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

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