Real gas

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

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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

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Van der Waals model

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Real gases are often modeled by taking into account their molar weight and molar volume <math display="block">RT = \left(p + \frac{a}{V_\text{m}^2}\right)\left(V_\text{m} - b\right)</math>

or alternatively: <math display="block">p = \frac{RT}{V_m - b} - \frac{a}{V_m^2}</math>

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: <math display="block">\begin{align}

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

\end{align}</math>

The constants at critical point can be expressed as functions of the parameters a, b: <math display="block"> \begin{align} p_c &= \frac{a}{27b^2}, & V_{m,c} &= 3b, \\[2pt] T_c &= \frac{8a}{27bR}, & Z_c &= \frac{3}{8} \end{align} </math>

With the reduced properties <math> p_r = p / p_\text{c} </math>, <math> V_r = V_\text{m} / V_\text{m,c} </math>, <math> T_r = T / T_\text{c} </math> the equation can be written in the reduced form: <math display="block">p_r = \frac{8}{3}\frac{T_r}{V_r - \frac{1}{3}} - \frac{3}{V_r^2}</math>

Redlich–Kwong model

File:Critical isotherm Redlich-Kwong model.png

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 <math display="block">RT = \left(p + \frac{a}{\sqrt{T}V_\text{m}\left(V_\text{m} + b\right)}\right)\left(V_\text{m} - b\right)</math>

or alternatively: <math display="block">p = \frac{RT}{V_\text{m} - b} - \frac{a}{\sqrt{T}V_\text{m}\left(V_\text{m} + b\right)}</math>

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: <math display="block">\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}</math>

The constants at critical point can be expressed as functions of the parameters a, b: <math display="block"> \begin{align} p_c &= {\left[\frac{(\sqrt[3]{2} - 1)^7}{3} \, R \, \frac{a^2}{b^5}\right]}^{1/3}, & V_{m,c} &= \frac{b}{\sqrt[3]{2}-1}, \\[4pt] T_c &= {\left[3 {\left(\sqrt[3]{2} - 1\right)}^2 \frac{a}{bR}\right]}^{2/3}, & Z_c &= \frac{1}{3} \end{align} </math>

Using <math>p_r = p/p_\text{c}</math>, <math>V_r = V_\text{m}/V_\text{m,c}</math>, <math>T_r = T / T_\text{c} </math> the equation of state can be written in the reduced form: <math display="block">p_r = \frac{3 T_r}{V_r - b'} - \frac{1}{b'\sqrt{T_r} V_r \left(V_r + b'\right)}</math> with <math>b' = \sqrt[3]{2} - 1 \approx 0.26</math>

Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot)<ref>D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907)</ref> is very rarely used, <math display="block">p = \frac{RT}{V_\text{m} - b} - \frac{a}{TV_\text{m}^2}</math>

but the modified version is somewhat more accurate <math display="block">p = \frac{RT}{V_\text{m}} \left[1 + \frac{9}{128} \cdot \frac{p}{p_c} \cdot \frac{T_c}{T} \left(1 - 6 \frac{T_\text{c}^2}{T^2}\right)\right]</math>

Dieterici model

This model (named after C. Dieterici<ref>C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)</ref>) fell out of usage in recent years <math display="block">p = \frac{RT}{V_\text{m} - b} \exp\left(-\frac{a}{V_\text{m}RT}\right)</math>

with parameters a, b. These can be normalized by dividing with the critical point stateTemplate:NoteTag:<math display="block">\tilde p = p \frac{(2be)^2}{a}; \quad \tilde T =T \frac{4bR}{a}; \quad \tilde V_m = V_m \frac{1}{2b}</math>which casts the equation into the reduced form:<ref>Template:Cite book</ref><math display="block">\tilde p \left(2\tilde V_m - 1\right) = \tilde T \exp\left(2 - \frac{2}{\tilde T \tilde V_m}\right)</math>

Clausius model

The Clausius equation (named after Rudolf Clausius) is a very simple three-parameter equation used to model gases. <math display="block">RT = \left(p + \frac{a}{T {\left(V_\text{m} + c\right)}^2}\right) \left(V_\text{m} - b\right)</math>

or alternatively: <math display="block">p = \frac{RT}{V_\text{m} - b} - \frac{a}{T\left(V_\text{m} + c\right)^2}</math>

where <math display="block">\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}</math>

where V_c is critical volume.

Virial model

The Virial equation derives from a perturbative treatment of statistical mechanics. <math display="block">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} + \cdots\right]</math>

or alternatively <math display="block">pV_\text{m} = RT \left[1 + B'(T) p + C'(T) p^2 + D'(T) p^3 + \cdots\right]</math>

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<ref>Template:Cite journal</ref>) has the interesting property being useful in modeling some liquids as well as real gases. <math display="block">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)}</math>

Wohl model

File:Isotherm wohl model.png

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)

File:ZPhCh87p9f.png

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

The Wohl equation (named after A. Wohl<ref>Template:Cite journal</ref>) 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. <math display="block">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</math>

or: <math display="block">\left(p - \frac{c}{T^2 V_\text{m}^3}\right)\left(V_\text{m} - b\right) = RT - \frac{a}{TV_\text{m}}</math>

or, alternatively: <math display="block">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)</math>

where <math display="block">\begin{align} a &= 6p_\text{c} T_\text{c} V_\text{m,c}^2, & b &= \frac{V_\text{m,c}}{4}, \\[2pt] c &= 4p_\text{c} T_\text{c}^2 V_\text{m,c}^3 \end{align}</math> where <math>V_\text{m,c} = \frac{4}{15}\frac{RT_c}{p_c}</math>, <math> p_\text{c} </math>, <math>T_c</math> are (respectively) the molar volume, the pressure and the temperature at the critical point.

And with the reduced properties <math>p_r = p/p_\text{c}</math>, <math>V_r = V_\text{m} / V_\text{m,c}</math>, <math>T_r = T / T_\text{c} </math> one can write the first equation in the reduced form: <math display="block">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}</math>

Beattie–Bridgeman model

<ref>Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010, Template:ISBN</ref> This equation is based on five experimentally determined constants. It is expressed as <math display="block">p = \frac{RT}{V_\text{m}^2}\left(1 - \frac{c}{V_\text{m}T^3}\right)(V_\text{m} + B) - \frac{A}{V_\text{m}^2}</math>

where <math display="block">\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}</math>

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

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

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The BWR equation, <math display="block">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)</math>

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 <math> \int_{V_i}^{V_f} \left(\frac{RT}{V_m} - P_\text{real}\right) dV </math>.

References

Template:Reflist Template:Reflist

External links

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

Real gas

Contents

Models

Van der Waals model

Redlich–Kwong model

Berthelot and modified Berthelot model

Dieterici model

Clausius model

Virial model

Peng–Robinson model

Wohl model

Beattie–Bridgeman model

Benedict–Webb–Rubin model

Thermodynamic expansion work

See also

References

Further reading

External links

Navigation menu

Real gas

Models

Van der Waals model

Redlich–Kwong model

Berthelot and modified Berthelot model

Dieterici model

Clausius model

Virial model

Peng–Robinson model

Wohl model

Beattie–Bridgeman model

Benedict–Webb–Rubin model

Thermodynamic expansion work

See also

References

Further reading

External links

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