Redlich–Kwong equation of state
The Redlich–Kwong equation of state is an empirical cubic equation developed in 1949 by Otto Redlich and Joseph N. S. Kwong to model the pressure-volume-temperature (P-V-T) relationships of real gases and gaseous mixtures, particularly for fugacity calculations in high-pressure systems where only critical temperature and pressure data are available.[1] It takes the formP = \frac{RT}{V - b} - \frac{a}{\sqrt{T} \, V (V + b)},
where P is pressure, V is molar volume, T is temperature, R is the gas constant, and a and b are substance-specific parameters derived from critical properties (a = 0.42748 \frac{R^2 T_c^{2.5}}{P_c} and b = 0.08664 \frac{R T_c}{P_c}, with T_c and P_c as critical temperature and pressure).[1] This equation improves upon the van der Waals model by incorporating a temperature-dependent attractive term (a / \sqrt{T}) to better capture intermolecular forces at elevated temperatures and pressures above the critical point.[1] The Redlich–Kwong equation is fundamentally empirical, justified by its accuracy in approximating experimental P-V-T data for nonpolar gases, while satisfying theoretical limits such as the high-pressure molar volume approaching $0.26 V_c (where V_c is critical volume).[1] It outperforms earlier equations like Berthelot's at supercritical conditions but shows limitations for polar compounds or low-temperature liquid phases due to inaccuracies in vapor pressure predictions.[2] For mixtures, the parameters are combined using mixing rules, with b as a linear mole-fraction average and a incorporating binary interaction terms, enabling applications to multicomponent systems.[1] In chemical engineering, the equation has been widely applied for thermodynamic property estimations, including phase equilibrium calculations, enthalpy departures, and compressibility factors in processes like gas compression and reaction equilibria under high pressure.[2] Its simplicity and reliance on minimal input data (critical constants) made it practical for industrial simulations, though it often requires empirical adjustments for hydrogen or light hydrocarbons.[3] Subsequent modifications have extended its utility; notably, Giorgio Soave's 1972 revision (Soave-Redlich-Kwong or SRK equation) replaced the T^{-0.5} term with a more accurate function of reduced temperature to improve vapor-liquid equilibrium predictions for hydrocarbons.[4] Further developments, such as the Peng-Robinson equation in 1976, addressed liquid density shortcomings, establishing the Redlich–Kwong framework as a cornerstone for modern cubic equations of state in reservoir engineering and process design. Despite these evolutions, the original form remains influential for its balance of accuracy and computational efficiency in supercritical gas modeling.[2]
Mathematical Formulation
Single-Component Equation
The Redlich–Kwong equation of state provides an empirical relation for the pressure of a single-component real gas as a function of temperature and molar volume.[5] It is expressed in the form P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T} \, V_m (V_m + b)}, where P is the pressure, T is the absolute temperature, V_m is the molar volume, and R is the universal gas constant.[5] The parameters a and b are substance-specific constants that account for intermolecular forces.[5] The first term, \frac{RT}{V_m - b}, represents the repulsive contribution from the finite volume occupied by gas molecules, with b serving as the excluded co-volume per mole.[6] The second term, -\frac{a}{\sqrt{T} \, V_m (V_m + b)}, captures the attractive intermolecular forces that reduce the pressure exerted on the container walls, where the temperature dependence in the denominator (a / \sqrt{T}) models the weakening of attractions at higher temperatures.[5] This modification to the attractive parameter, compared to the constant in the van der Waals equation, enhances accuracy for predicting real gas behavior at elevated pressures by better accounting for temperature effects on molecular interactions.[6] In SI units, the parameter a has dimensions of \mathrm{Pa \cdot m^6 \cdot K^{0.5} \cdot mol^{-2}}, reflecting its role in the attractive pressure term, while b is in \mathrm{m^3 \cdot mol^{-1}}, corresponding to a volume per mole.[6] These constants are typically determined from the critical temperature and pressure of the substance.[5]Relation to Critical Constants
The parameters a and b in the Redlich–Kwong equation of state for a pure substance are expressed in terms of the critical temperature T_c, critical pressure P_c, and universal gas constant R. The co-volume parameter is given by b = 0.08664 \frac{R T_c}{P_c}, while the attraction parameter is a = 0.42748 \frac{R^2 T_c^{2.5}}{P_c}. [5] These numerical coefficients are obtained by applying the critical point conditions to the equation of state, namely that at T = T_c and V = V_c, the pressure equals P_c and the slope of the isotherm vanishes: \left( \frac{\partial P}{\partial V} \right)_{T_c, V_c} = 0. Additionally, the form satisfies b \approx 0.26 V_c to align with experimental critical volumes, with the coefficients solved from the resulting system of equations, yielding a critical compressibility factor of 1/3 ≈ 0.333.[5] As a numerical example, consider carbon dioxide (CO₂), for which T_c = 304.13 K and P_c = 7.38 MPa, using R = 8.314 J mol⁻¹ K⁻¹. The co-volume parameter computes to b = 2.97 \times 10^{-5} m³ mol⁻¹, and the attraction parameter to a = 6.46 Pa m⁶ K^{0.5} mol⁻².[7][5]Multicomponent Mixtures
To extend the Redlich–Kwong equation of state to multicomponent mixtures, the parameters a and b for the mixture are calculated using specific mixing rules based on the mole fractions y_i of the components.[1] The co-volume parameter b for the mixture is obtained by a simple linear (arithmetic) average: b_\text{mix} = \sum_i y_i b_i This rule assumes that the repulsive interactions in the mixture are additively combined according to the mole fractions, reflecting the total excluded volume in a random mixing scenario.[1] For the attractive parameter a, the mixing rule employs a quadratic form that incorporates the geometric mean for unlike-pair interactions: a_\text{mix} = \sum_i \sum_j y_i y_j \sqrt{a_i a_j} This corresponds to the Lorentz–Berthelot combining rule, where the cross-term \sqrt{a_i a_j} assumes that the attractive energy between unlike molecules is the geometric mean of the pure-component attractions, again under the assumption of random mixing without specific ordering.[1] The pure-component a_i and b_i values are derived from the critical constants of each species, as defined in prior sections. In practice, the geometric mean term for binary pairs can be adjusted with an optional binary interaction parameter k_{ij} to account for deviations from ideal mixing, particularly in non-ideal systems: \sqrt{a_i a_j} (1 - k_{ij}) where k_{ij} = 0 recovers the original rule, and k_{ij} values are fitted from experimental vapor-liquid equilibrium data for specific pairs. As an illustrative example, consider an equimolar binary mixture of methane and ethane (y_\text{CH}_4 = y_\text{C}_2\text{H}_6 = 0.5), using pure-component parameters in consistent units (atm cm⁶ K^{0.5} mol^{-2} for a, cm³ mol^{-1} for b): a_\text{CH}_4 = 31.59 \times 10^6, b_\text{CH}_4 = 29.6; a_\text{C}_2\text{H}_6 = 97.42 \times 10^6, b_\text{C}_2\text{H}_6 = 45.1. Assuming k_{ij} = 0, b_\text{mix} = 0.5 \times 29.6 + 0.5 \times 45.1 = 37.35 \, \text{cm}^3 \, \text{mol}^{-1} a_\text{mix} = (0.5)^2 (31.59 \times 10^6) + (0.5)^2 (97.42 \times 10^6) + 2 \times 0.5 \times 0.5 \times \sqrt{(31.59 \times 10^6)(97.42 \times 10^6)} = 60.09 \times 10^6 \, \text{atm cm}^6 \, \text{K}^{0.5} \, \text{mol}^{-2}. These mixture parameters can then be substituted into the Redlich–Kwong equation to compute properties like pressure or fugacity for the mixture.[8]Historical Development
Origins and Motivation
By the early 20th century, the van der Waals equation of state, introduced in 1873, had become a foundational model for describing real gas behavior by accounting for molecular volume and intermolecular attractions. However, it exhibited significant limitations at high pressures and temperatures, particularly in predicting liquid densities and compressibility factors. For instance, van der Waals-type cubic equations often failed to accurately model liquid-phase properties, leading to substantial deviations from experimental data in dense fluid regimes. These shortcomings became increasingly apparent in industrial applications involving high-pressure processes, such as petrochemical engineering, where precise volumetric and thermodynamic predictions were essential.[9][1] The principle of corresponding states, originally proposed by van der Waals and further refined in the early 20th century, played a pivotal role in motivating improvements to cubic equations of state during the 1930s and 1940s. This principle posits that substances exhibit similar behavior when expressed in reduced variables relative to their critical points, enabling generalized models with few parameters. In this era, researchers sought to enhance cubic models like van der Waals by incorporating temperature-dependent terms and better aligning with corresponding states to extend applicability to wider pressure and temperature ranges, driven by advances in experimental data for hydrocarbons and the growing demands of chemical engineering. Early efforts focused on semi-empirical adjustments to capture non-ideal behaviors more reliably, setting the stage for more robust thermodynamic modeling.[1][10] Otto Redlich, born in 1896 in Vienna, brought a strong foundation in physical chemistry to these challenges, having earned his doctorate in 1922 from the University of Vienna with work on nitric acid equilibria and later publishing extensively on thermodynamics and strong electrolytes during the 1920s and 1930s at the Vienna Institute of Technology. Emigrating to the United States in 1938 due to political persecution, he continued research on non-ideal solutions at institutions like Washington State College before joining Shell Development Company in 1945, where industrial needs for accurate fugacity calculations in high-pressure systems influenced his thermodynamic pursuits. Joseph N. S. Kwong, born in 1916 in China and educated in the U.S. with a Ph.D. in chemical engineering from the University of Minnesota in 1942, contributed expertise in volumetric data analysis and hydrocarbon thermodynamics; after early work on mineral crushing energetics, he joined Shell in 1944, focusing on practical modeling for process optimization. Their collaboration at Shell was motivated by the necessity to develop an equation that better represented P-V-T relations for pure substances and mixtures under extreme conditions, addressing gaps in prior models for engineering applications.[11][12][1]Publication and Early Impact
The Redlich–Kwong equation of state was introduced in the 1949 paper "On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions" by Otto Redlich and J. N. S. Kwong, published in Chemical Reviews, volume 44, issue 1, pages 233–248.[1] The work, originating from the Shell Development Company, presented the equation as a two-parameter cubic model designed to enhance predictions of real gas behavior, particularly for fugacity calculations in gaseous solutions above the critical temperature.[1][5] Upon publication, the equation was immediately applied to forecast gas-phase properties and fugacities in hydrocarbon mixtures, addressing limitations in prior models like the van der Waals equation for high-pressure systems common in petroleum processing.[1] Its temperature-dependent attractive term allowed for more accurate representation of intermolecular forces in non-polar fluids such as methane and ethane, yielding satisfactory results for pressures up to several hundred atmospheres.[1] In the 1950s, the Redlich–Kwong equation gained traction within the chemical engineering community, with early citations appearing in studies on thermodynamic properties of industrial gases and phase equilibria.[13] It was adopted for process simulations involving hydrocarbon systems, facilitating calculations of compressibility factors and fugacity coefficients in vapor-liquid separations and reactor design.[14] By the mid-1950s, its cubic form enabled straightforward analytical solutions for volume, contributing to its integration into generalized corresponding-states methods for multicomponent predictions.[14]Theoretical Derivation
Modification of van der Waals Equation
The van der Waals equation of state, given byP = \frac{RT}{V_m - b} - \frac{a}{V_m^2},
serves as the foundational model for the Redlich–Kwong equation, where P is pressure, V_m is molar volume, T is temperature, R is the gas constant, and a and b are substance-specific parameters accounting for intermolecular attractions and excluded volume, respectively. To address limitations in representing real gas behavior, particularly the temperature dependence of attractive forces, Redlich and Kwong modified the attractive term by replacing \frac{a}{V_m^2} with \frac{a}{\sqrt{T} V_m (V_m + b)}, yielding the form
P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T} V_m (V_m + b)}. This adjustment incorporates a volume exclusion factor (V_m + b) in the denominator to reduce the attractive correction at high densities, while the \sqrt{T} factor introduces an inverse square-root temperature dependence that empirically improves agreement with experimental vapor pressure data over a wider temperature range compared to the constant a in the van der Waals model. The repulsive term \frac{RT}{V_m - b} and the excluded volume parameter b are retained unchanged in structure, though the values of a and b are recalibrated to better match critical point properties such as critical temperature and pressure.