Real gas

Models

Main article: Equation of state

Van der Waals model

Main article: van der Waals equation

Real gases are often modeled by taking into account their molar weight and molar volume RT = \left(p + \frac{a}{V_\text{m}^2}\right)\left(V_\text{m} - b\right)

or alternatively: 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: \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: \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}

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

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

or alternatively: 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: \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: \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}

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

Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot) is very rarely used, p = \frac{RT}{V_\text{m} - b} - \frac{a}{TV_\text{m}^2}

but the modified version is somewhat more accurate 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]

Dieterici model

This model (named after C. Dieterici) fell out of usage in recent years 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:\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:\tilde p \left(2\tilde V_m - 1\right) = \tilde T \exp\left(2 - \frac{2}{\tilde T \tilde V_m}\right)

Clausius model

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

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

where \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 Vc is critical volume.

Virial model

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

or alternatively pV_\text{m} = RT \left[1 + B'(T) p + C'(T) p^2 + D'(T) p^3 + \cdots\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) has the interesting property being useful in modeling some liquids as well as real gases. 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

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 = \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: \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: 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 \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} where V_\text{m,c} = \frac{4}{15}\frac{RT_c}{p_c}, p_\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_\text{c}, V_r = V_\text{m} / V_\text{m,c}, T_r = T / T_\text{c} one can write the first equation in the reduced form: 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

This equation is based on five experimentally determined constants. It is expressed as 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}

where \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 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, Vm is in \frac{\text{m}^3}{\text{k},\text{mol}}, T is in K and R = 8.314 \mathrm{\frac{kPa \cdot m^3}{kmol \cdot K}}

Benedict–Webb–Rubin model

Main article: Benedict–Webb–Rubin equation

The BWR equation, 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 \int_{V_i}^{V_f} \left(\frac{RT}{V_m} - P_\text{real}\right) dV .

References

References

1. D. Berthelot in ''Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII'' (Paris: Gauthier-Villars, 1907)
2. C. Dieterici, ''Ann. Phys. Chem. Wiedemanns Ann.'' 69, 685 (1899)
3. Pippard, Alfred B.. (1981). "Elements of classical thermodynamics: for advanced students of physics". Univ. Pr.
4. (1976). "A New Two-Constant Equation of State". Industrial and Engineering Chemistry: Fundamentals.
5. (1914). "Investigation of the condition equation". Zeitschrift für Physikalische Chemie.
6. Yunus A. Cengel and Michael A. Boles, ''Thermodynamics: An Engineering Approach'' 7th Edition, McGraw-Hill, 2010, {{ISBN. 007-352932-X
7. Gordan J. Van Wylen and Richard E. Sonntage, ''Fundamental of Classical Thermodynamics'', 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3