The ideal gas law ( PV = nRT), relating molar volume of a gas to pressure and temperature, is predicated on two assumptions: 1) that the volume occupied by the gas molecules is negligible compared to the volume of the vessel; and 2) that no intermolecular attractive forces are present. In reality, these assumptions limit the applicability of the ideal gas law, especially at higher pressures and densities, and low temperatures. For accurate calculations of PVT behavior of real gases in industrial settings, engineers have developed many equations of state (EoS) for various conditions. This reference provides information on several key cubic equations of state that attempt to accurately predict real gas behavior.
Van der Waals equation
In 1873, Dutch physicist Johannes D. Van der Waals noted the non-ideality of gases and proposed modifications to the ideal gas law that addressed the size of gas molecules and accounted for the strength of the mutual attraction among them. He introduced the two constants (a and b) to the ideal gas law to correct for intermolecular attractive forces (a) and for finite molecular size (b) (Equation (1)). The coefficients a and b are characteristic of each individual gas and were determined experimentally from the critical temperature and pressure.
The Van der Waals (VdW) EoS is not sufficiently accurate for most industrial applications, but has provided a platform for many important refinements that better model fluids at different P and T conditions.
The VdW equation, along with its subsequent modifications, are referred to as cubic EoS, because they can be written as cubic (as opposed to linear or quadratic) functions of molar volume. Cubic EoS have the advantages of simplicity and tunability, but also have disadvantages, so other types of EoS, such as molecular-based and virial (not discussed here) have also been developed.
No single equation of state can correctly predict pressure, volume and temperature behavior for all types of gases, including mixtures, under all possible conditions.
In 1949, Otto Redlich and J. Kwong developed what would become a widely used modification to the VdW equation (Equation (2)), using the coefficients for a and b found in VdW.
The Redlich-Kwong (RK) equation can be expressed in terms of the compressibility factor (Z), a dimensionless ratio of the product of pressure and specific volume to the product of gas constant and temperature (Equation (3)).
The compressibility factor is a measure of deviation from the ideal-gas behavior. For ideal gases, Z is equal to one, and can be either greater or less than one for real gases. The further the value of Z lies away from one, the greater the degree of deviation from ideal-gas behavior.
The RK equation has itself been modified by many subsequent researchers, as they sought to widen its applicability or improve its accuracy at specific conditions. A highly used modification of the RK equation was developed by G. Soave in 1972. Soave’s contribution was to express the attractive term from the VdW equation not only as a function of temperature, but also as a function of the sphericity of the molecule. This is accomplished by including a term α that includes an acentric factor for the species of gas (SRK equation; Equation (4)). The acentric factor α is a measure of non-sphericity and is specific to each gas species.
The SRK equation was further modified by other researchers, and one of the most useful modifications was developed in 1976 at the University of Alberta by Ding-yu Peng and Donald Robinson. The Peng-Robinson (PR) equation recalculated the α function and modified the volume dependency of the attractive term.
1. Valderrama, J.O., The state of the cubic equations of state, Ind. Eng. Chem. Res., 42, pp. 1,603-1,618, 2003.
2. Aleksandrov, A.A., PVT Relationships, Thermopedia: A-Z guide to thermodynamics, February 2011, accessed at http://www.thermopedia.com/content/1067/
3. Huang, M. and Gramoll, K., Ideal Gas Equation of State, University of Oklahoma, Thermodynamics ebook, chapter 2, www.ecourses.ou.edu/
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