Fact-checked by Grok 2 weeks ago

Cubic equations of state

Cubic equations of state (EOS) are a class of thermodynamic models that describe the relationship between pressure (P), volume (V), and temperature (T) for real fluids, expressed as cubic polynomials in molar volume or the compressibility factor (Z).^[1] These equations extend the ideal gas law by incorporating corrections for finite molecular size and intermolecular attractions, enabling predictions of phase behavior, including liquid-vapor equilibria, across a wide range of conditions.^[1] Originating in the late 19th century, they remain foundational in chemical engineering due to their simplicity, computational efficiency, and applicability to nonpolar hydrocarbons and mixtures.^[1] The development of cubic EOS began with Johannes Diderik van der Waals's 1873 doctoral thesis, which introduced the first such equation to account for the continuity between gaseous and liquid states:
P = \frac{RT}{V_m - b} - \frac{a}{V_m^2},
where R is the gas constant, V_m is molar volume, a represents attractive forces, and b accounts for molecular volume exclusions.^[2] In 1949, Otto Redlich and J. N. S. Kwong modified this form to improve high-pressure predictions by making the attraction term temperature-dependent:
P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T} V_m (V_m + b)},
with parameters a and b derived from critical properties.^[3] This evolution continued in 1972 when Giorgio Soave refined the Redlich-Kwong equation for better vapor pressure correlations in hydrocarbons, introducing a temperature function for the attraction parameter α(T).^[4] The Peng-Robinson EOS, proposed in 1976 by Ding-Yu Peng and Donald B. Robinson, further enhanced liquid density accuracy and applicability to natural gas systems through an adjusted form of the repulsion term.^[5] In general, cubic EOS share a common structure, often generalized as
P = \frac{RT}{V_m - b} - \frac{a(T)}{V_m(V_m + u b) + w b (V_m - b)},
where u and w are constants specific to each variant (e.g., u = 1, w = 0 for Redlich-Kwong; u = 2, w = −b for Peng-Robinson), and parameters are typically functions of critical temperature (T_c), critical pressure (P_c), and acentric factor (ω).^[1] These models are extensively applied in process simulation software for vapor-liquid equilibrium calculations, reservoir engineering, and the design of separation processes involving petroleum fractions and CO₂ systems.^[1] Their advantages include ease of parameterization and ability to capture subcritical phase transitions via multiple real roots, though they exhibit limitations in predicting properties of polar or associating compounds and near-critical densities without modifications.^[1]

Fundamentals

Definition and General Form

Cubic equations of state (EOS) are empirical thermodynamic models designed to describe the behavior of non-ideal gases and fluids, particularly for predicting pressure-volume-temperature relationships beyond the ideal gas law. These models account for intermolecular attractions and repulsions by incorporating correction terms derived from experimental data at critical conditions. Unlike more complex statistical mechanical approaches, cubic EOS offer a balance of simplicity and accuracy for phase equilibrium calculations in engineering applications, such as hydrocarbon processing and chemical separations. The mathematical foundation of these equations lies in their cubic polynomial nature when expressed in terms of molar volume V or the compressibility factor Z = PV/RT. Solving for volume typically yields a cubic equation of the form

Z^3 + a Z^2 + b Z + c = 0,

where a, b, and c are coefficients dependent on reduced temperature T_r = T/T_c, reduced pressure P_r = P/P_c, and fluid-specific parameters related to critical temperature T_c and pressure P_c. This cubic form arises from truncating higher-order terms in the virial expansion of the equation of state, P/RT = 1/V + B/V^2 + C/V^3 + \cdots, where B, C represent second and third virial coefficients capturing pairwise and three-body interactions, respectively. By limiting the expansion to the $1/V^3 term, the model captures essential non-idealities while remaining analytically solvable. A general pressure-explicit form for cubic EOS, encompassing many variants including the van der Waals equation as the prototypical example, is

P = \frac{RT}{V - b} - \frac{a(T)}{V(V + u b) + w b (V - b)},

where R is the universal gas constant, b corrects for the finite molecular volume (covolume), a(T) represents temperature-dependent attractive forces, and u and w are constants specific to each variant (e.g., u = 0, w = 0 for van der Waals; u = 1, w = 0 for Redlich-Kwong; u = 2, w = -1 for Peng-Robinson). The parameters a(T), b, u, and w are typically derived from the principle of corresponding states, which posits that reduced thermodynamic properties are universal functions for similar substances when scaled by critical properties. At the critical point, the isotherm exhibits an inflection (\partial P / \partial V = 0 and \partial^2 P / \partial V^2 = 0), providing constraints to fit these parameters, often augmented by the acentric factor for non-spherical molecules. This approach originated in the late 19th and early 20th centuries. For subcritical conditions (T < T_c), the cubic equation in Z (or V) generally yields three real positive roots corresponding to distinct physical interpretations: the smallest root represents the liquid-like phase (compressed volume), the largest the vapor-like phase (expanded volume), and the intermediate root an unstable configuration between them. Phase selection follows Maxwell's equal-area criterion or stability analysis, ensuring the physically relevant root aligns with experimental densities; for supercritical conditions, a single real root suffices. A key advantage of cubic EOS is their ability to perform analytical vapor-liquid equilibrium (VLE) calculations by equating chemical potentials or fugacities across phases, \hat{f}_i^L = \hat{f}_i^V, where fugacity coefficients \phi_i = \hat{f}_i / (x_i P) are derived directly from the EOS via integration of (\partial \ln \phi / \partial P)_T = (Z - 1)/P. This facilitates efficient flash calculations and phase diagrams without numerical regression of binary interaction parameters in many cases.

Historical Context

In the early 19th century, scientists began recognizing significant deviations from ideal gas behavior, particularly through experimental observations of real gas properties. The Joule-Thomson effect, experimentally investigated by James Prescott Joule and William Thomson (Lord Kelvin) in the 1850s, demonstrated that real gases cool upon free expansion under certain conditions, highlighting the limitations of the ideal gas law in accounting for intermolecular forces and finite molecular volumes. These findings underscored the need for more realistic models to describe gas behavior under varying pressures and temperatures. The foundational cubic equation of state emerged in 1873 with Johannes Diderik van der Waals' doctoral thesis, which introduced corrections for the finite volume occupied by molecules and attractive forces between them, marking a seminal advancement in thermodynamic modeling.^[6] This work laid the groundwork for subsequent cubic formulations by providing a mathematical framework that captured phase transitions and critical phenomena more accurately than prior attempts. In 1949, Otto Redlich and Joseph N. S. Kwong proposed an improved cubic equation, refining the temperature dependence of the attractive term to enhance predictions at high pressures and temperatures, addressing shortcomings of the van der Waals model for industrial applications.^[7] The 1970s saw a surge in developments driven by the petroleum industry's demand for precise vapor-liquid equilibrium (VLE) calculations in hydrocarbon systems; Giorgio Soave modified the Redlich-Kwong equation in 1972 by incorporating an acentric factor to better represent non-spherical molecules.^[8] Similarly, in 1976, Ding-Yu Peng and Donald B. Robinson introduced their equation, optimizing vapor pressure predictions near the critical point for complex mixtures.^[5] Further extensions in the 1980s and 1990s expanded the applicability of cubic equations to more challenging systems. In 1986, Roman Stryjek and Juan H. Vera developed the Peng-Robinson-Stryjek-Vera (PRSV) modification, introducing a compound-specific parameter to improve accuracy for polar and asymmetric compounds.^[9] By 1996, Georgios M. Kontogeorgis and colleagues proposed the Cubic-Plus-Association (CPA) equation, combining a cubic term with an association contribution from Wertheim's perturbation theory to handle hydrogen-bonding fluids like water and alcohols.^[10] Throughout this evolution, advancements were predominantly motivated by the petroleum sector's requirements for reliable phase behavior predictions in reservoir fluids and processing operations.

Core Cubic Equations

Van der Waals Equation

The van der Waals equation of state, introduced by Johannes Diderik van der Waals in his 1873 doctoral thesis, represents the foundational cubic model for describing the behavior of real gases by incorporating corrections for molecular volume and intermolecular attractions beyond the ideal gas law. This equation marked a significant advance in understanding the continuity between gaseous and liquid states, earning van der Waals the Nobel Prize in Physics in 1910 for his contributions to gas equations of state.^[11] For one mole of gas, the van der Waals equation is given by

\left( P + \frac{a}{V^2} \right) (V - b) = RT,

where P is the pressure, V is the molar volume, T is the absolute temperature, R is the universal gas constant, a quantifies the attractive forces between molecules, and b represents the excluded volume or covolume occupied by the molecules themselves.^[11] The parameter a arises from the reduction in pressure exerted on the container walls due to attractive intermolecular forces pulling molecules inward, while b accounts for the finite size of molecules, preventing the volume from approaching zero as in the ideal gas model.^[11] The constants a and b can be expressed in terms of the critical temperature T_c and critical pressure P_c of the substance. Specifically, a = \frac{27 R^2 T_c^2}{64 P_c} and b = \frac{R T_c}{8 P_c}, derived from the conditions at the critical point where the gas-liquid distinction vanishes.^[11] The critical point itself is determined by the inflection point of the isotherm, satisfying \left( \frac{\partial P}{\partial V} \right)_T = 0 and \left( \frac{\partial^2 P}{\partial V^2} \right)_T = 0. Substituting the van der Waals equation into these conditions yields the critical molar volume V_c = 3b, critical temperature T_c = \frac{8a}{27 R b}, and critical pressure P_c = \frac{a}{27 b^2}, allowing prediction of phase behavior near criticality.^[11] Despite its pioneering role, the van der Waals equation exhibits limitations, particularly in accuracy at high pressures and near the critical point, where it fails to capture the complex density dependencies of real fluids due to the absence of temperature dependence in the attraction parameter a.^[6] This static a leads to overestimation of attractive effects at lower temperatures, resulting in deviations from experimental compressibility factors. For phase equilibria, the equation enables fugacity calculations via thermodynamic relations; equal fugacities between phases indicate equilibrium.^[11] The cubic nature of the equation in V provides up to three real roots, from which the physically relevant volume (e.g., the largest for vapor) can be selected for stable phase identification.^[11]

Redlich–Kwong Equation

The Redlich–Kwong equation of state, proposed in 1949, represents a significant refinement of the van der Waals model by incorporating temperature dependence in the attractive forces term, enabling more accurate descriptions of real gas behavior over a broader range of conditions. This equation builds on the van der Waals framework as a precursor but adjusts the intermolecular attraction parameter to decrease with increasing temperature, addressing limitations in predicting properties near and above the critical point. The resulting model is particularly suited for non-polar substances and provides a cubic relationship in molar volume that facilitates thermodynamic calculations. The equation takes the form

P = \frac{RT}{V - b} - \frac{a}{\sqrt{T} \, V (V + b)},

where P is pressure, T is temperature, V is molar volume, R is the universal gas constant, and a and b are substance-specific parameters. The constants are defined in terms of critical properties as a = 0.42748 \, R^2 T_c^{2.5} / P_c and b = 0.08664 \, R T_c / P_c, with T_c and P_c denoting the critical temperature and pressure, respectively. The key innovation lies in the temperature-dependent attractive term a(T) = a / \sqrt{T}, which captures the weakening of intermolecular attractions at higher temperatures, unlike the constant a in the van der Waals equation. This formulation derives from modifying the van der Waals attraction term to better match experimental critical behavior, yielding a critical compressibility factor Z_c \approx 0.333, closer to observed values for many gases than the van der Waals prediction of $0.375. The equation is cubic in V, but below the critical temperature, it often yields three real roots, with the intermediate one typically representing an unstable state. Strengths include improved vapor pressure predictions for non-polar gases, such as light hydrocarbons, up to moderate pressures, making it valuable for high-pressure gas-phase applications. However, weaknesses persist, notably in overpredicting liquid densities and limited accuracy for polar or heavy compounds, where further modifications proved necessary.

Redlich–Kwong Modifications

Soave-Redlich–Kwong Equation

The Soave-Redlich–Kwong (SRK) equation of state, proposed in 1972, modifies the original Redlich–Kwong equation by introducing a temperature-dependent correction to the attractive parameter, enabling more accurate predictions of vapor-liquid equilibrium (VLE) for hydrocarbons.80096-4) The general form is given by

P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b)},

where P is pressure, T is temperature, V is molar volume, R is the gas constant, b is the co-volume parameter, and a(T) is the temperature-dependent attractive parameter.80096-4) This structure retains the repulsive term from the Redlich–Kwong model while enhancing the attractive term's sensitivity to temperature and molecular shape. The attractive parameter is expressed as a(T) = 0.42747 \frac{R^2 T_c^2}{P_c} \alpha(T), where T_c and P_c are the critical temperature and pressure, respectively, and \alpha(T) is a dimensionless function defined as

\alpha(T) = \left[1 + (0.48508 + 1.55171\omega - 0.15613\omega^2)(1 - \sqrt{T/T_c})\right]^2.

Here, \omega is the acentric factor, a compound-specific parameter that accounts for deviations from spherical molecular shape and improves the model's tunability for non-spherical hydrocarbons.80096-4) The co-volume b remains unchanged from the Redlich–Kwong equation, calculated as b = 0.08664 \frac{R T_c}{P_c}, ensuring consistency in the excluded volume representation. Both a (at the critical point) and b are estimated directly from critical properties without additional experimental data beyond T_c, P_c, and \omega.80096-4) This formulation offers significant advantages in predicting saturated vapor pressures for non-polar molecules, achieving close reproduction of experimental vapor pressures across a range of hydrocarbons with average deviations under 2% for many systems.80096-4) Its simplicity and effectiveness have made it widely adopted in oil and gas simulations for phase behavior modeling of hydrocarbon mixtures, particularly in reservoir engineering applications where VLE calculations are critical. For practical implementation, the equation is solved as a cubic in volume to obtain real roots corresponding to vapor, liquid, or supercritical phases; in two-phase regions, the Maxwell construction is applied to enforce equal fugacity between phases, ensuring thermodynamic consistency.80096-4)

Volume Translation Method

The volume translation method, developed by Péneloux, Rauzy, and Fréze in 1982, provides a post-hoc correction to improve the volumetric predictions of cubic equations of state, particularly addressing the systematic underprediction of saturated liquid densities. This approach is especially useful for hydrocarbons, where uncorrected models like the Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson equations can underestimate liquid volumes by 10-20% at subcritical conditions.85003-1)^[12] The core of the method involves a simple shift in the volume term: the translated volume is defined as V' = V - c, where V is the volume obtained from the cubic equation of state and c is a substance-specific translation parameter. To compute pressure, the equation of state is evaluated using the translated volume, yielding P = P(V'). However, to avoid disrupting vapor-liquid equilibrium calculations, the fugacity coefficients are derived from the original unshifted volume V. This selective application ensures that phase boundaries and compositions remain unchanged while enhancing density accuracy in the liquid phase.85003-1)00185-4) The translation parameter c is determined by regressing against experimental saturated liquid density data, typically at a reduced temperature of T_r = 0.7, which corresponds to conditions representative of liquid behavior away from the critical point. For predictive use with the Soave-Redlich-Kwong or Peng-Robinson equations, c can be estimated via the correlation

c = 0.40768 (0.29441 - Z_{RA}) \frac{R T_c}{P_c},

where Z_{RA} is the Rackett compressibility factor (often approximated near the critical compressibility Z_c), R is the universal gas constant, T_c is the critical temperature, and P_c is the critical pressure. This correlation facilitates application without experimental data for many non-polar compounds.85003-1) Subsequent developments have introduced temperature-dependent forms of c to extend accuracy across broader ranges, particularly for polar or associating fluids where constant shifts prove insufficient. For instance, linear or polynomial dependencies on reduced temperature have been proposed, improving deviations to within 1-2% over wide thermodynamic conditions while preserving the method's simplicity. These variants maintain the original framework's advantages in computational efficiency and compatibility with existing cubic models.00185-4)

Peng–Robinson Family

Original Peng–Robinson Equation

The original Peng–Robinson equation of state (PR EOS), introduced in 1976, represents a significant advancement in cubic equations for modeling the thermodynamic behavior of hydrocarbons and nonpolar fluids, particularly in improving vapor-liquid equilibrium (VLE) calculations and liquid density predictions. Developed as a modification to earlier Redlich–Kwong-based models, it addresses limitations in near-critical and saturation properties by incorporating temperature-dependent attraction parameters and a refined repulsive term.^[5] The PR EOS is expressed in the pressure-explicit form:

P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b) + b(V - b)}

where the temperature-dependent attraction parameter is a(T) = 0.45724 \frac{R^2 T_c^2}{P_c} \alpha(T), with \alpha(T) = \left[ 1 + (0.37464 + 1.54226 \omega - 0.26992 \omega^2) (1 - \sqrt{T / T_c}) \right]^2, and the co-volume parameter is b = 0.07780 \frac{R T_c}{P_c}. Here, R is the universal gas constant, T_c and P_c are the critical temperature and pressure, V is the molar volume, T is the temperature, and \omega is the acentric factor. These constants were selected to match critical point conditions while optimizing performance for hydrocarbons.^[5]^[13] A primary innovation is the denominator in the attractive term, V(V + b) + b(V - b) = V^2 + 2bV - b^2, which approximates the hard-sphere repulsive contribution more effectively than the Soave–Redlich–Kwong (SRK) form V(V + b), especially at low reduced temperatures and near the critical region. This adjustment yields a critical compressibility factor Z_c \approx 0.307, aligning closely with experimental values for many nonpolar compounds and enhancing the equation's reliability for phase behavior in petroleum systems.^[5]^[13] Compared to the SRK equation, the original PR EOS provides superior accuracy in saturated liquid densities and vapor pressure predictions, with average absolute deviations of approximately 8% for saturated liquid densities of n-alkanes up to C20 (an improvement over SRK's typical 10-12% overprediction) and often below 2% for vapor pressures. This makes PR particularly advantageous for applications in natural gas processing and reservoir engineering, where precise VLE data for complex mixtures is crucial.^[5]^[14]^[15] The PR EOS also facilitates computation of departure functions for key thermodynamic properties, such as enthalpy and entropy, derived from the Helmholtz energy and integrated along isotherms and isobars to ideal-gas references. These functions enable consistent predictions of calorific properties essential for process simulations.^[5]

Peng–Robinson–Stryjek–Vera Variants

The Peng–Robinson–Stryjek–Vera (PRSV) variants represent refinements to the Peng–Robinson equation of state, specifically targeting improved vapor-liquid equilibrium predictions for substances where the original model exhibits limitations, such as quantum gases like methane and slightly polar compounds. These modifications introduce compound-specific parameters to enhance the accuracy of the attractive term's temperature dependence, enabling better reproduction of experimental vapor pressure data across a wide range of reduced temperatures. Developed in response to observed inaccuracies in the base model for light hydrocarbons and polar molecules, the PRSV approaches prioritize fitting to saturation properties while maintaining the cubic structure's simplicity for phase equilibrium calculations.^[16] The initial PRSV1 formulation, proposed in 1986, modifies the α(T) function to incorporate adjustable κ parameters that allow precise matching of pure-component vapor pressures. The expression is given by

\alpha(T) = \left[ 1 + \kappa_0 (1 - \sqrt{T_r}) + \kappa_1 (1 + \sqrt{T_r}) (0.7 - T_r) (1 - \sqrt{T_r}) \right]^2,

where T_r = T / T_c, \kappa_0 = 0.378893 + 1.4897153\omega - 0.17131848\omega^2 + 0.0196554\omega^3, ω is the acentric factor, and κ₁ is an empirically fitted parameter (often near zero for nonpolar compounds). This adjustment addresses deficiencies in the original model's behavior for quantum gases, such as methane, by reducing deviations in saturation pressure predictions to typically below 1% for nonpolar compounds, and it extends applicability to slightly polar substances like acetone without requiring additional mixing rules for binaries. In the same year, the PRSV2 variant was introduced as an extension of PRSV1, refining the temperature dependence for reduced temperatures (T_r) greater than 1 to improve high-temperature vapor pressure accuracy. This refinement involves an additional κ₃ term in the α(T) expression, applied conditionally when T_r > 1, along with κ₂ for further fitting:

\alpha(T) = \left[ 1 + \kappa_0 (1 - \sqrt{T_r}) + [\kappa_1 + \kappa_2 (1 - \sqrt{T_r}) ( \kappa_3 - T_r ) ] (1 + \sqrt{T_r}) (0.7 - T_r) (1 - \sqrt{T_r}) \right]^2 \quad (T_r \leq 1),

with a modified form for T_r > 1 to prevent divergence. For instance, PRSV2 yields superior results for light gases like hydrogen and helium, where PRSV1 may underperform due to insufficient flexibility at supercritical regimes, achieving average absolute deviations in vapor pressure as low as 0.5% for tested hydrocarbons. The κ parameters in both variants are obtained through nonlinear regression to experimental saturation data from sources like the Design Institute for Physical Properties (DIPPR), ensuring the model's predictive power for multicomponent systems relies on minimal binary interaction adjustments.^[17]

Peng–Robinson–Babalola-Susu Equation

The Peng–Robinson–Babalola-Susu (PRBS) equation of state, developed as a modification within the Peng–Robinson family, addresses limitations in modeling complex hydrocarbon mixtures under extreme conditions by incorporating pressure-dependent adjustments tailored to high-pressure petroleum systems.^[18] This variant optimizes predictions for phase behavior in deep reservoirs, particularly those involving heavy oils and asphaltenes, where standard cubic equations often fail due to inadequate handling of molecular associations and density variations.^[18] Central to the PRBS formulation are adjustments to the co-volume parameter b and the attraction term a in the original Peng–Robinson framework, with density-dependent mixing rules specifically designed for asphaltenes and heavy oils. The parameter b_i for individual components is parameterized directly from critical volumes, ensuring consistency with experimental data on pure compounds. The attraction term is rendered pressure-dependent as a = a_1 P + a_0, where a_1 and a_0 are fitted coefficients, allowing the equation to capture non-ideal interactions in dense phases. Key features include the incorporation of asphaltene association through a modified temperature-dependent function \alpha(T), which accounts for self-association in heavy fractions, alongside binary interaction parameters derived from swelling tests on crude oil components to refine multicomponent mixing. These elements enable the PRBS equation to model asphaltene precipitation and solubility more accurately in multicomponent systems. The full modified equation takes the form:

P = \frac{RT}{v - b} - \frac{(a_1 P + a_0) \alpha(T)}{v(v + b) + b(v - b)}

where v is the molar volume, R is the gas constant, T is temperature, and the other terms follow standard Peng–Robinson conventions with the specified modifications.^[18] The PRBS equation demonstrates advantages in predicting phase envelopes for deep reservoir conditions exceeding 100 MPa, where it provides superior accuracy over the standard Peng–Robinson model by better resolving compressibility and phase stability in asphaltene-rich fluids. Validation studies on crude oil systems show improved determination of asphaltene onset pressures, with deviations reduced by up to 20% compared to baseline predictions, highlighting its utility for reservoir simulation and flow assurance in high-pressure environments.^[18]

Specialized Cubic Models

Elliott–Suresh–Donohue Equation

The Elliott–Suresh–Donohue (ESD) equation of state, introduced in 1990, represents a hybrid cubic model designed to handle polar and associating fluids by integrating a repulsive term akin to the Peng–Robinson equation with an explicit association contribution derived from perturbation theory and a square-well potential. This approach addresses limitations in standard cubic equations for non-ideal behaviors in polarizable substances, such as alcohols and amines, where hydrogen bonding and dipole interactions dominate. The model maintains the cubic nature for computational efficiency while enhancing accuracy for phase equilibria in such systems.^[19] The pressure-explicit form of the ESD equation is given by

P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b)} + \frac{A}{2V} \ln\left[\frac{V + \eta b}{V - \eta b}\right],

where the first two terms capture the repulsive and dispersive attractions, a(T) is the temperature-dependent attraction parameter expressed as in the Peng–Robinson model (a(T) = a_c \alpha(T), with \alpha(T) based on acentric factor), and b is the co-volume parameter both fitted to critical temperature, pressure, and volume. The third term accounts for association, with A denoting the association strength related to molecular bonding sites, and \eta = 0.25 fixed for the square-well potential to model short-range attractive interactions. Parameters a(T) and b are derived from pure-component critical data, while A is typically regressed from experimental dipole moments, vapor pressures, or solubility measurements for polar compounds.^[19] In applications, the ESD equation demonstrates strengths in predicting liquid-liquid equilibria and compressibility in binary mixtures of polar fluids, such as water-ethanol, where it outperforms basic cubic models by explicitly treating association effects that lead to phase splitting and non-ideal mixing. For instance, it accurately reproduces upper consolute boundaries and density profiles in alcohol-water systems under moderate pressures. However, the model's mixing rules for multicomponent systems are more involved than those of pure cubic equations, requiring specialized combining rules for the association parameter A (often using geometric means adjusted for cross-association) to ensure thermodynamic consistency, which increases parameter estimation complexity.^[19]

Patel–Teja Equation

The Patel–Teja equation of state, introduced in 1982, represents a three-parameter extension of cubic equations of state designed to enhance accuracy in predicting volumetric and phase equilibrium properties, especially for light hydrocarbons and some polar compounds. It builds on the repulsive term of the Redlich–Kwong equation while incorporating an additional parameter to better capture behavior near the critical point. The equation is particularly noted for its ability to correlate vapor-liquid equilibria without requiring extensive data fitting beyond critical properties and the acentric factor. The functional form of the Patel–Teja equation is given by

P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b) + c(V - b)},

where a(T) = a \, \alpha(T), \alpha(T) = \left[1 + \sqrt{\frac{A}{B}} \left(1 - \sqrt{\frac{T}{T_c}}\right)\right]^2, R is the universal gas constant, T is temperature, V is molar volume, T_c is the critical temperature, and a, b, and c are substance-specific parameters. The constant a is calculated as a = 0.452123 \frac{(R T_c)^2}{P_c}, where P_c is the critical pressure. The co-volume parameter b is determined from b = 0.329032 \frac{R T_c}{P_c} - \frac{c}{3}, and the third parameter c is c = (0.452 + 1.309 \omega - 0.295 \omega^2) \frac{R T_c}{P_c}, with \omega being the acentric factor; here, \sqrt{A/B} in the \alpha(T) function corresponds to a similar expression involving \omega. These relations ensure the equation satisfies critical point conditions, including the second and third derivatives of pressure with respect to volume being zero at T_c and P_c. A key innovation of the Patel–Teja equation is the inclusion of the parameter c in the denominator of the attractive term, which allows it to exactly reproduce a critical compressibility factor Z_c = 0.329, an empirical average value that aligns closely with experimental data for many substances and improves upon the fixed Z_c values in two-parameter models (e.g., 0.333 for Soave-Redlich–Kwong and 0.307 for Peng–Robinson). This adjustment enhances the equation's performance for light hydrocarbons by providing better representation of repulsive interactions near criticality without altering the overall cubic structure. For polar fluids, substance-specific values of the parameters are often fitted to experimental saturated liquid densities and vapor pressures to further optimize accuracy. The Patel–Teja equation demonstrates advantages in predicting vapor pressures and liquid densities over the Peng–Robinson and Soave-Redlich–Kwong equations, achieving average deviations of around 3% for saturated liquid densities in nonpolar fluids without needing volume translation corrections. It excels in phase equilibrium calculations for binary mixtures of light hydrocarbons, where interaction parameters are typically close to unity, and offers improved critical region behavior compared to its predecessors. These features make it a valuable tool for applications in petroleum and natural gas processing, though parameter fitting may be required for highly polar systems to maintain precision.

Hybrid Cubic Approaches

Cubic-Plus-Association Model

The Cubic-Plus-Association (CPA) equation of state was developed in 1996 to model fluids exhibiting hydrogen bonding by combining a cubic equation of state for physical interactions with an association term based on Wertheim's perturbation theory.^[20] This approach addresses limitations of pure cubic models in handling self-association and cross-association effects in polar compounds. The physical contribution derives from the Soave-Redlich-Kwong (SRK) or Peng-Robinson equation, providing a simple framework for repulsive and dispersion forces, while the association term explicitly accounts for specific interactions like hydrogen bonds.^[20] The residual Helmholtz energy density in the CPA model is expressed as:

\frac{A^r}{RT} = \frac{A^{phys}}{RT} + \frac{A^{assoc}}{RT}

where the association contribution is:

\frac{A^{assoc}}{RT} = \sum_i x_i \sum_{A_i} \left[ \ln X_{A_i} - \frac{X_{A_i}}{2} + \frac{1}{2} \right]

Here, x_i is the mole fraction of component i, and X_{A_i} represents the fraction of molecules of component i not bonded at association site A_i, solved iteratively from:

X_{A_i} = \left[ 1 + \sum_j \rho_j \sum_{B_j} X_{B_j} \Delta^{A_i B_j} \right]^{-1}

with \rho_j as the molar density of component j. The association strength \Delta^{A_i B_j} is defined as:

\Delta^{A_i B_j} = g(\rho) \cdot \kappa^{A_i B_j} \cdot \left[ \exp\left( \frac{\epsilon^{A_i B_j}}{RT} \right) - 1 \right] \cdot b_i

The radial distribution function at contact, g(\rho), approximates the hard-sphere value as g(\rho) = 1 / (1 - 1.9\eta), where \eta = b \rho / 4 is the reduced density, b_i is the co-volume parameter from the cubic term, \epsilon^{A_i B_j} is the association energy, and \kappa^{A_i B_j} is the association volume parameter.^[20] For mixtures, cross-association parameters are obtained via combining rules, typically the geometric mean: \epsilon^{A_i B_j} = \sqrt{\epsilon^{A_i B_i} \epsilon^{B_j B_j}} and \kappa^{A_i B_j} = \sqrt{\kappa^{A_i B_i} \kappa^{B_j B_j}}, which assume symmetric interactions between sites of different components.^[21] Common association schemes include the 4C model for water, representing four cross-associating sites (two donors and two acceptors), and the 2B model for alcohols, featuring two sites (one donor and one acceptor) to capture linear chain formation.^[21] These schemes are selected based on molecular structure to reflect bonding patterns, such as three-dimensional networks in water or oligomerization in alcohols. The CPA model excels in applications to vapor-liquid equilibria (VLE) and solubility predictions for hydrogen-bonding systems like water and alcohol mixtures, often requiring only one adjustable binary interaction parameter for the physical term. It effectively correlates phase behavior in binaries and multicomponent systems involving alcohols, glycols, and water-hydrocarbon mixtures by integrating association effects with cubic simplicity.^[21] Key advantages include accurate representation of second virial coefficients for pure associating compounds, which cubic models alone cannot achieve, and reliable predictions of liquid-liquid equilibria (LLE) in associating mixtures without additional tuning.

Cubic-Plus-Chain Model

The cubic-plus-chain (CPC) model represents a hybrid equation of state developed in 2019 to extend the applicability of classical cubic equations to chain-like molecules, particularly nonpolar polymers and oligomers, by incorporating a chain connectivity term derived from perturbed-chain statistical associating fluid theory (PC-SAFT).^[22] This approach addresses limitations of pure cubic models, such as the Peng–Robinson equation, in capturing the structural effects of long chains on thermodynamic properties.^[22] The CPC formulation combines a cubic equation of state, typically the Redlich–Kwong or Peng–Robinson form, to describe the repulsive and attractive interactions of monomeric segments with an additional chain term from PC-SAFT that accounts for covalent bonding between segments. The residual Helmholtz energy is expressed as:

\frac{A^{\text{CPC}}}{RT} = m \left( \frac{A^{\text{rep}}}{RT} + \frac{A^{\text{att}}}{RT} \right) + \frac{A^{\text{chain}}}{RT}

where m is the number of segments per chain, A^{\text{rep}}/RT and A^{\text{att}}/RT are the repulsive and attractive contributions from the cubic equation scaled by m, and A^{\text{chain}}/RT is the PC-SAFT chain term given by m [g^{\text{hs}}(\xi^3) - 1], with g^{\text{hs}} as the hard-sphere radial distribution function and \xi^3 as the reduced density packing fraction.^[22] Key molecular parameters include m (chain length), \sigma (segment diameter), and u/k_B (dispersion energy parameter divided by Boltzmann's constant), which are fitted to experimental data; the cubic repulsion is adapted by treating the chain as m bonded segments with modified co-volume and energy terms.^[22] This structure hybridizes the computational simplicity of cubics with SAFT's ability to model chain flexibility. Recent extensions, such as the 2023 CPC IV framework, incorporate association terms alongside chain effects for broader applicability to systems with both connectivity and hydrogen bonding.^[23] The model is particularly suited for high-molecular-weight fluids, such as polyethylene and asphaltenes, where standard cubics fail to accurately predict phase behavior due to neglect of chain connectivity. For instance, CPC has been applied to model vapor–liquid and liquid–liquid equilibria in polyethylene–solvent systems, demonstrating improved accuracy in upper cloud point pressures compared to Peng–Robinson predictions. In asphaltene modeling, it better represents precipitation onset in crude oil mixtures by accounting for chain-like aggregation.^[22] Mixing rules in CPC follow van der Waals one-fluid theory, with cross-parameters for segment size and energy determined via combining rules (e.g., Lorentz–Berthelot for unlike interactions), often refined by fitting to experimental cloud point data for polymer solutions.^[22] This enables predictive capabilities for multicomponent systems without excessive parameterization. Compared to the standard Peng–Robinson equation, CPC excels in capturing compressibility factors and phase envelopes for polymer solutions, offering accuracy comparable to PC-SAFT but with reduced computational demand due to the cubic backbone.^[22] It builds on earlier hybrid approaches like the cubic-plus-association model by focusing on chain effects rather than hydrogen bonding.^[22]

Applications and Comparisons

Thermodynamic Applications

Cubic equations of state, particularly the Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) variants, are extensively employed in vapor-liquid equilibrium (VLE) calculations for phase behavior predictions in multicomponent systems. These models facilitate flash computations by equating fugacity coefficients across phases to determine equilibrium compositions at specified temperature and pressure. The fugacity coefficient \phi_i for component i in a mixture is calculated as \phi_i = \exp\left( \int \frac{Z-1}{P} dP - \ln(Z - B) + \cdots \right), where Z is the compressibility factor, B is the second virial coefficient term from the EOS, and the integral accounts for departure from ideality along an isotherm. This approach enables accurate determination of bubble points, dew points, and phase fractions in processes involving hydrocarbons and associated gases.^[24] In pressure-volume-temperature (PVT) analysis, cubic EOS model isotherms and compressibility factors essential for reservoir simulation, capturing fluid behavior under varying conditions from gas condensates to heavy oils. For instance, PR and SRK EOS predict saturation pressures with average absolute deviations around 5.75-5.76% when tuned with pseudocomponent parameters derived from molecular weight and specific gravity. Compressibility factors Z = PV/RT are derived by solving the cubic form in Z, aiding in volumetric predictions for injection and production scenarios. Volume translations further refine liquid densities, achieving deviations as low as 1.67% for SRK-based models.^[25] Caloric properties, such as enthalpy and entropy, are computed via departure functions that quantify deviations from ideal gas behavior. The enthalpy departure is given by H - H_{ig} = RT(Z-1) + \int terms involving EOS parameters, with specific forms for PR: H - H_{ig} = RT(Z-1) + \frac{RT(1 + \kappa) A}{2\sqrt{2} B} \ln\left(\frac{Z + 2.414B}{Z - 0.414B}\right), where \kappa = 0.37464 + 1.54226\omega - 0.26992\omega^2 and \omega is the acentric factor. For SRK, it simplifies to H - H_{ig} = RT(Z-1) + RT(1 + m) \frac{A}{B} \ln\left(\frac{Z + B}{Z}\right), with m = 0.480 + 1.574\omega - 0.176\omega^2. Entropy departures follow analogous integrations, S - S_{ig} = R \ln(Z - B) + temperature-dependent terms. PR generally outperforms SRK in vaporization enthalpy predictions, with deviations under 2% for many hydrocarbons across subcritical and supercritical regions.^[26] For multicomponent systems, conventional mixing rules extend pure-component parameters to mixtures, particularly for binaries. The attractive parameter is mixed as a_m = \sum_i \sum_j x_i x_j \sqrt{a_i a_j} (1 - k_{ij}), where x_i and x_j are mole fractions, a_i and a_j are pure-component values, and k_{ij} is the binary interaction parameter (often near zero for hydrocarbons but adjusted for non-ideal pairs like CO2-hydrocarbon). The co-volume b_m = \sum_i x_i b_i uses linear mixing. These rules, rooted in van der Waals theory, ensure thermodynamic consistency and are applied with PR or SRK for phase equilibria in binary and ternary systems.^[27] In industrial contexts, cubic EOS underpin natural gas processing, where PR models phase equilibria for dew point calculations in synthetic natural gas mixtures, achieving good agreement with experiments up to the cricondentherm (deviations <5% for many compositions). For CO2 sequestration, modified PR EOS predict compressibility factors in high-CO2 natural gas streams (up to 100% CO2) with average deviations of 2.7%, outperforming standard PR by 9.6% and supporting pipeline transport and injection simulations in fields like Huangqiao. SRK complements these in lighter mixtures, though PR is preferred for denser phases.^[28]^[29]

Comparative Performance

Cubic equations of state (EOS) are evaluated using metrics such as average absolute deviation (AAD) in saturation pressure (P_sat), vapor-liquid equilibrium (VLE) compositions, and liquid densities, often benchmarked against experimental data for hydrocarbons, polar compounds, and associating fluids. For non-polar hydrocarbons like alkanes, the Peng-Robinson (PR) EOS typically achieves an AAD of less than 2% for P_sat predictions across a wide temperature range, outperforming the Soave-Redlich-Kwong (SRK) EOS, which yields 2-5% AAD due to its less accurate representation of attractive forces near the critical point.^[30]^[31] Both PR and SRK excel in VLE calculations for hydrocarbon mixtures, with deviations under 1% in mole fractions for binary systems, but they underperform in liquid density predictions, often exceeding 5% AAD without volume translation corrections.^[32] Early cubic EOS families, such as van der Waals and Redlich-Kwong, exhibit poor performance for densities (AAD >10%) and are largely obsolete for modern applications, while PR and SRK remain robust for VLE in non-polar systems but falter in polar or associating fluids. Hybrids like the Cubic-Plus-Association (CPA) EOS improve VLE predictions for water and alcohols, achieving lower errors (e.g., around 7% for polar phases) compared to pure cubics (often >10%), by explicitly accounting for hydrogen bonding.^[33] In contrast, statistical associating fluid theory (SAFT) models, such as PC-SAFT, provide superior accuracy for polymers and strongly associating systems (AAD <2% in densities), though they are computationally more intensive than cubics, which solve faster for routine reservoir simulations.^[34] All cubic EOS share limitations in describing quantum gases like helium, where quantum effects lead to deviations exceeding 20% in VLE without specialized corrections, and they inadequately handle strong associations in pure form, necessitating hybrid extensions.^[35] For selection, PR is recommended for non-polar hydrocarbons due to its balance of accuracy and simplicity; CPA or similar hybrids for polar/associating compounds like glycols; and variants like Peng-Robinson-Babalola-Susu (PRBS) for high-pressure oil systems, where it achieves lower AAD (under 3%) in phase envelopes up to 160 MPa compared to standard PR.^[36] Recent 2020s benchmarks indicate that advanced variants like PRSV2 improve polar VLE predictions over original PR in binary mixtures involving alcohols, enhancing reliability for process design.^[37]

References

[1]
Cubic Equation of State - an overview | ScienceDirect Topics
Cubic equations of state are mathematical models used to describe the relationship between pressure, volume, and temperature in fluids, ...
[2]
Van der waals equation of state - AIP Publishing
Nov 1, 2022 · Van der Waals (vdW) invented his equation of state about 150 year ago. This equation described the liquidgas phase transition and predicted an existence of the ...
[3]
[PDF] EQUATION OF STATE. FUGACITIES OF GASEOUS SOLUTIONS
OTTO REDLICH AND J. N. S. KWONG. Shell Development Company, Emeryville, California. Received October 15, 1948. The calculation of fugacities from P-V-T data ...
[4]
Equilibrium constants from a modified Redlich-Kwong equation of ...
A modified Redlich-Kwong equation of state is proposed. Vapor pressures of pure compounds can be closely reproduced by assuming the parameter a in the original ...
[5]
A New Two-Constant Equation of State | Industrial & Engineering ...
Traditional Fossil Fuels February 1, 1976. A New Two-Constant Equation ... Peng–Robinson Equation of State. Industrial & Engineering Chemistry Research ...
[6]
Taking Another Look at the van der Waals Equation of State–Almost ...
Aug 2, 2019 · The van der Waals (vdW) equation of state has long fascinated researchers and engineers, largely because of its simplicity and engineering flexibility.
[7]
On the Thermodynamics of Solutions. V. An Equation of State ...
A classical thermodynamic model for dispersed nanophase stability and its application for investigating the stability of air nanobubbles in water.
[8]
Equilibrium constants from a modified Redlich-Kwong equation of ...
The modified Redlich-Kwong equation makes the parameter 'a' temperature-dependent, uses an acentric factor, and can be used for multicomponent VLE calculations ...
[9]
Cubic Equations of State-Which? - ACS Publications
Density and viscosity modeling of three deep eutectic solvents using a volume-shifted cubic equation of state coupled with the friction theory.
[10]
New volume translated PR equation of state for pure compounds ...
The purpose of this paper is to improve the Peng–Robinson equation of state (PR EOS 1978) prediction of saturated liquid density of pure compounds and gas ...
[11]
Peng-Robinson EOS (1976) | PNG 520: Phase Behavior of Natural ...
The Peng-Robinson EOS is a popular equation of state for natural gas systems, developed by D. Peng and D.B. Robinson, and is suited for gas/condensate systems.Missing: paper | Show results with:paper
[12]
Comparative Assessment of RK, SRK, and PR EOS | PNG 520
Serve similar functions as the Redlich Kwong EOS but require more parameters. · PR obtains better liquid densities than SRK. · Overall, PR does a better job ( ...
[13]
An improved volume-translated SRK EOS dedicated to more ...
Oct 15, 2020 · In this study, we propose an improved volume translated Soave-Redlich-Kwong (VT-SRK) EOS to achieve more accurate volumetric calculations for various pure ...Missing: formula | Show results with:formula
[14]
https://courses.ems.psu.edu/png520/m11_p3.html
[15]
https://www.sciencedirect.com/science/article/abs/pii/S0378381220302703
[16]
https://doi.org/10.1002/cjce.5450640224
[17]
https://doi.org/10.1002/cjce.5450640515
[18]
https://doi.org/10.5541/ijot.419923
[19]
Method of calculation of fugacity coefficient from cubic equations of ...
In this paper a general theory for calculating the fugacity coefficients of real gases has been developed for the case when the equation of state (EOS) is an ...
[20]
General approach to characterizing reservoir fluids for EoS models using a large PVT database
### Summary of Cubic EOS in PVT Behavior for Reservoir Simulation
[21]
[PDF] THERMODYNAMIC PROPERTIES FROM CUBIC EQUATIONS OF ...
For temperature prediction, all three equations of state give similar results in the subcritical and supercritical regions. None of the three equations is ...
[22]
Mixing Rules for Cubic Equations of State - ResearchGate
Aug 10, 2025 · The resulting mixing rules are applied to the van der Waals, Redlich-Kwong, and Peng-Robinson equations of state as the three representative ...<|separator|>
[23]
Calculation of Phase Equilibrium of Natural Gases with the Peng ...
Our purpose is to compare the quality of predictions of phase equilibrium and critical points calculations in natural gas mixtures using a simple EOS and a ...
[24]
An improved PR equation of state for CO2-containing gas ...
Peng-Robinson equation of state (PR EoS) is a reliable and widely used cubic EoS to predict natural gases thermodynamic properties.
[25]
Vapor pressures of pure compounds using the Peng–Robinson ...
Equation of state. The PR [11] EOS can be expressed as P= RT v−b − a v(v+b)+b(v−b) where a is called the attractive term and it is a function of temperature.Missing: AAD | Show results with:AAD
[26]
A New Cubic Equation of State for Predicting Phase Behavior of ...
It was also found that the new cubic equation of state predicts the saturated vapor pressure of the pure hydrocarbons much better than the PR and the SRK ...Missing: original | Show results with:original<|control11|><|separator|>
[27]
Critical parameters optimized for accurate phase behavior modeling ...
In Section 3, we developed a new set of critical parameters for the PR EOS that can accurately predict liquid densities and vapor pressures of n-alkanes up to C ...
[28]
A comparative performance evaluation of cubic equations of state for ...
May 1, 2021 · CPA and SAFT EoS were found to be very accurate especially for hydrate prediction systems given their ability to account for the association ...
[29]
Comparisons of PC-SAFT and cubic EOS simulations - ScienceDirect
Conclusion. The PC-SAFT equation is superior to the cubic EOS for prediction of gas phase compressibility factors and oil phase compressibilities. In this work ...
[30]
Accurate quantum-corrected cubic equations of state for helium ...
Dec 1, 2020 · The quantum corrections yield a vast improvement in the accuracy of cubic EoS, both for single-component fluids and mixtures. The VLE of all ...2. Theory · 2.4. The Quantum-Corrected... · 4. Results And Discussion<|separator|>
[31]
Promising Accurate Equation of State Models for High-Pressure ...
Nov 1, 2024 · The study examined four fluid phase behavior Equation of State (EOS) ... Peng-Robinson-Babalola-Susu (PRBS). Their strengths, weaknesses, and ...
[32]
On the mixing rules matter: The VLE predictions for binary systems
Apr 1, 2019 · In this work, the original Peng-Robinson EOS [20] and two of its variants, PRSV [21] and PRSV2 [22], are coupled to activity coefficient models.