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Real gas
Non-hypothetical gases whose molecules occupy space and have interactions
Real gases are non-hypothetical gases whose molecules occupy space and have interactions; consequently, they adhere to gas laws.
To understand the behaviour of real gases, the following must be taken into account:
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.
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.
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:
With the reduced properties$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:
Critical isotherm for Redlich-Kwong model in comparison to van-der-Waals model and ideal gas (with V_{0}=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
Using $\ 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:
$p_{r}={\frac {3T_{r}}{V_{r}-b'}}-{\frac {1}{b'{\sqrt {T_{r}}}V_{r}\left(V_{r}+b'\right)}}$ with $b'={\sqrt[{3}]{2}}-1\approx 0.26$
Berthelot and modified Berthelot model
The Berthelot equation (named after D. Berthelot)^{[1]} is very rarely used,
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^{[3]}) has the interesting property being useful in modeling some liquids as well as real gases.
Isotherm (V/V_{0}->p_r) at critical temperature for Wohl model, van der Waals model and ideal gas model (with V_{0}=RT_{c}/p_{c})
Untersuchungen über die Zustandsgleichung, pp. 9,10, Zeitschr. f. Physikal. Chemie 87
The Wohl equation (named after A. Wohl^{[4]}) 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.
$b={\frac {V_{\text{m,c}}}{4}}$ with $V_{\text{m,c}}={\frac {4}{15}}{\frac {RT_{c}}{p_{c}}}$
$c=4p_{\text{c}}T_{\text{c}}^{2}V_{\text{m,c}}^{3}\$, where $V_{\text{m,c}},\ p_{\text{c}},\ T_{\text{c}}$ are (respectively) the molar volume, the pressure and the temperature at the critical point.
And with the reduced properties$\ 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:
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 ${\frac {{\text{m}}^{3}}{{\text{k}}\,{\text{mol}}}}$, T is in K and R = 8.314${\frac {{\text{kPa}}\cdot {\text{m}}^{3}}{{\text{k}}\,{\text{mol}}\cdot {\text{K}}}}$^{[6]}
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 $\int _{V_{i}}^{V_{f}}({\frac {RT}{V_{m}}}-P_{real})dV$.
^D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: Gauthier-Villars, 1907)
^C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)
^Peng, D. Y. & Robinson, D. B. (1976). "A New Two-Constant Equation of State". Industrial and Engineering Chemistry: Fundamentals. 15: 59–64. doi:10.1021/i160057a011.
^A. Wohl (1914). "Investigation of the condition equation". Zeitschrift für Physikalische Chemie. 87: 1–39.
^Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGraw-Hill, 2010, ISBN007-352932-X
^Gordan J. Van Wylen and Richard E. Sonntage, Fundamental of Classical Thermodynamics, 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3
Rao, Y. V. C (2004). An introduction to thermodynamics. Universities Press. ISBN978-81-7371-461-0.
Xiang, H. W. (2005). The Corresponding-States Principle and its Practice: Thermodynamic, Transport and Surface Properties of Fluids. Elsevier. ISBN978-0-08-045904-2.