Redlich–Kwong equation of state

It was formulated by Otto Redlich and Joseph Neng Shun Kwong in 1949.

[3] Although this equation is not currently employed in practical applications,[4] modifications derived from this mathematical model like the Soave Redlich-Kwong (SWK), and Peng Robinson have been improved and currently used in simulation and research of vapor–liquid equilibria.

[3][5] The Redlich–Kwong equation is formulated as:[6][7] where: The constants are different depending on which gas is being analyzed.

The constants can be calculated from the critical point data of the gas:[6] where: The Redlich–Kwong equation can also be represented as an equation for the compressibility factor of gas, as a function of temperature and pressure:[8] where: Or more simply: This equation only implicitly gives Z as a function of pressure and temperature, but is easily solved numerically, originally by graphical interpolation, and now more easily by computer.

Moreover, analytic solutions to cubic functions have been known for centuries and are even faster for computers.

The Redlich-Kwong equation of state may also be expressed as a cubic function of the molar volume.

is an often empirically fitted parameter accounting for asymmetry in the molecular cross-interactions.

These manners of creating a and b parameters for a mixture from the parameters of the pure fluids are commonly known as the van der Waals one-fluid mixing and combining rules.

[9] The Van der Waals equation, formulated in 1873 by Johannes Diderik van der Waals, is generally regarded as the first somewhat realistic equation of state (beyond the ideal gas law): However, its modeling of real behavior is not sufficient for many applications, and by 1949, had fallen out of favor, with the Beattie–Bridgeman and Benedict–Webb–Rubin equations of state being used preferentially, both of which contain more parameters than the Van der Waals equation.

Kwong had begun working at Shell in 1944, where he met Otto Redlich when he joined the group in 1945.

The equation arose out of their work at Shell - they wanted an easy, algebraic way to relate the pressures, volumes, and temperatures of the gasses they were working with - mostly non-polar and slightly polar hydrocarbons (the Redlich–Kwong equation is less accurate for hydrogen-bonding gases).

It was presented jointly in Portland, Oregon at the Symposium on Thermodynamics and Molecular Structure of Solutions in 1948, as part of the 14th Meeting of the American Chemical Society.

[11] The success of the Redlich–Kwong equation in modeling many real gases accurately demonstrate that a cubic, two-parameter equation of state can give adequate results, if it is properly constructed.

The Redlich–Kwong equation is very similar to the Van der Waals equation, with only a slight modification being made to the attractive term, giving that term a temperature dependence.

This approximation is quite good for many small, non-polar compounds – the value ranges between about 0.24Vc and 0.28Vc.

The functional form of a with respect to the critical temperature and pressure is empirically chosen to give the best fit at moderate pressures for most relatively non-polar gasses.

[11] The values of a and b are completely determined by the equation's shape and cannot be empirically chosen.

, enforcing the thermodynamic criteria for a critical point, and without loss of generality defining

yields 3 constraints, Simultaneously solving these while requiring b' and Zc to be positive yields only one solution: The Redlich–Kwong equation was designed largely to predict the properties of small, non-polar molecules in the vapor phase, which it generally does well.

It was recognized by the mid 1960s that to significantly improve the equation, the parameters, especially a, would need to become temperature dependent.

As early as 1966, Barner noted that the Redlich–Kwong equation worked best for molecules with an acentric factor (ω) close to zero.

He therefore proposed a modification to the attractive term: where It soon became desirable to obtain an equation that would also model well the Vapor–liquid equilibrium (VLE) properties of fluids, in addition to the vapor-phase properties.

[10] Perhaps the best known application of the Redlich–Kwong equation was in calculating gas fugacities of hydrocarbon mixtures, which it does well, that was then used in the VLE model developed by Chao and Seader in 1961.

[10][14] However, in order for the Redlich–Kwong equation to stand on its own in modeling vapor–liquid equilibria, more substantial modifications needed to be made.

[15] Soave's modification involved replacing the T1/2 power found in the denominator attractive term of the original equation with a more complicated temperature-dependent expression.

[10] Several modifications have been made that attempt to more accurately represent the first term, related to the molecular size.

The first significant modification of the repulsive term beyond the Van der Waals equation's (where Phs represents a hard spheres equation of state term.)

was developed in 1963 by Thiele:[17] where This expression was improved by Carnahan and Starling to give [18] The Carnahan-Starling hard-sphere equation of state has term been used extensively in developing other equations of state,[10] and tends to give very good approximations for the repulsive term.

[19] Beyond improved two-parameter equations of state, a number of three parameter equations have been developed, often with the third parameter depending on either Zc, the compressibility factor at the critical point, or ω, the acentric factor.

Schmidt and Wenzel proposed an equation of state with an attractive term that incorporates the acentric factor:[20]