The classical Carnot heat engine
Real gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using 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.
van der Waals model
Real gases are often modeled by taking into account their molar weight and molar volume
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:
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
where a and b two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:
Berthelot and modified Berthelot model
The Berthelot equation (named after D. Berthelot is very rarely used,
but the modified version is somewhat more accurate
This model (named after C. Dieterici) fell out of usage in recent years
The Clausius equation (named after Rudolf Clausius) is a very simple three-parameter equation used to model gases.
where Vc is critical volume.
where A, B, C, A′, B′, and C′ are temperature dependent constants.
The Wohl equation (named after A. Wohl) is formulated in terms of critical values, making it useful when real gas constants are not available.
This equation is based on five experimentally determined constants. It is expressed as
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 following table when P is in KPa, v is in , T is in K and R=8.314
|Carbon Dioxide, CO2||507.2836||0.07132||0.10476||0.07235||6.60×10^5|
The BWR equation, sometimes referred to as the BWRS equation,
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.
- 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., and 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 "Investigation of the condition equation", Zeitschrift für Physikalische Chemie (Leipzig) 87 pp. 1–39 (1914)
- Yunus A. Cengel and Michael A. Boles,Thermodynamics: An Engineering Approach 7th Edition, , McGraw-Hill, 2010,ISBN 007-352932-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
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- M. Aznar, and A. Silva Telles, A Data Bank of Parameters for the Attractive Coefficient of the Peng-Robinson Equation of State, Braz. J. Chem. Eng. vol. 14 no. 1 São Paulo Mar. 1997, ISSN 0104-6632
- An introduction to thermodynamics by Y. V. C. Rao
- The corresponding-states principle and its practice: thermodynamic, transport and surface properties of fluids by Hong Wei Xiang