Almajose-Dalida Equation of State

Background

Equations of state (EOS) are mathematical models relating pressure, volume, and temperature (PVT) for substances. Since van der Waals’ work in the late 19th century, cubic EOS formulations such as the Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR) equations have become standard in chemical engineering due to their balance of computational efficiency and accuracy. However, no single cubic EOS has achieved high predictive accuracy across all pressure and temperature ranges for a wide variety of fluids.

The AD-EOS addresses this gap through a systematic optimization of the attraction term coefficients using a dataset of 65 representative fluids, later validated against an additional 35 compounds.

When tested against SRK, PR, Patel–Teja (PT), and Twu–Coon–Cunningham (TCC) EOS, the AD-EOS showed:

• Liquid density: Competitive with PT and slightly less accurate than TCC.

• Vapor pressure: Lowest percent absolute average deviation among all compared EOS.

• Vaporization enthalpy: Marginally better performance than PR and PT.

• Departure functions: Similar accuracy to established models for enthalpy and entropy across various phases.

Due to its simplicity in fugacity expressions and competitive accuracy, the AD-EOS is suited for:

• Process simulation of pure components and mixtures

• Thermodynamic property prediction where computational efficiency is critical

• Generalized property estimation for fluids lacking experimental data

Equation of state parameters

The equation of state parameters a, b, and c are defined by the equations: 

The parameters are determined at the critical point by minimizing deviations in liquid density and vapor pressure across selected temperatures. Consequently, the Almajose-Dalida equation is expected to yield reliable predictions for liquid densities. Values for the constants ΩA, ΩB, ΩC, and τi are provided below, with a more complete listing of constants provided on this page. 

Should the parameters be unknown for a certain compound, a generalized correlation is also provided below. Although the generalized function for τi accurately predicts the vapor pressures of non-polar compounds, its performance declines with polar substances.

However, it must be noted that the generalized correlations are subject to a limitation. If the attraction-term pole lies beyond the repulsive root, the equation of state becomes discontinuous. Based on the fitted parameters, this occurs whenever the critical compressibility factor satisfies

Since the vast majority of fluids fall within the range ZC ≈ 0.20−0.35, this condition can generally be ignored. Nevertheless, for the very unlikely and few cases where ZC lies below the cutoff, compound-specific EOS parameters must be employed to preserve thermodynamic consistency. Within the present 1100+ compound database, only three fluids fall into this category, listed in the table below for reference.

Vapor pressure modelling

With regards to vapor pressure prediction, improved accuracy can be obtained by applying the two-parameter Heyen alpha function or the more elaborate but more accurate two-parameter Twu alpha function: 

Coefficients for the Heyen and Twu alpha functions are available below, but a more updated and complete listing can be found here. These coefficients are used in conjunction with the previously mentioned quadratic correlations (not the error-minimized constants) for ΩA, ΩB, ΩC. 

The equation of state is continuously being improved and studied, with regular updates and published constants. You can also view the latest set of alpha-function constants for various vapor pressure functions here. Generalized alpha correlations are also made available here. 

Pure fluid thermodynamic property calculations

The following expressions have been derived from the Almajose-Dalida equation of state. These formulas are used to calculate key thermodynamic properties such as compressibility factor, fugacity coefficient, and departure functions, enabling analysis of fluid phase behavior and volumetric properties.  The constants A, B, and C; essential for applying the introduced formulas, are dimensionless parameters of the equation of state:

Pure fluid formulas

Mixture thermodynamic property calculations

In multicomponent systems, the parameters of the Almajose–Dalida equation of state (AD-EOS) are obtained from pure-component parameters using mixing rules. Two approaches are used: a van der Waals–type one-fluid mixing rule and a Huron–Vidal–type mixing rule coupled with an activity coefficient model.

Van der Waals mixing rule formulas

The van der Waals (vdW) mixing rule applies to all three AD-EOS parameters a, b, and c. The attraction parameter amix is evaluated using a quadratic mole-fraction average with a geometric mean and a binary interaction coefficient kij​: 

The co-volume parameter b and the quadratic denominator parameter c are evaluated as mole-fraction–weighted linear averages:

The use of the same additive rule for bmix and cmix is based on their shared interpretation as size parameters. Size parameters are extensive properties in mole fraction and therefore combine linearly.

Huron-Vidal type mixing rule formulas

The Huron–Vidal (HV)–type mixing rule replaces the direct vdW evaluation of amix​ with a value derived from an excess Gibbs energy model. In this approach, the molar excess Helmholtz energy at infinite pressure from the EOS is equated to that from an activity coefficient model such as NRTL or UNIFAC. The resulting mixing rule is formulated as:

This ensures that the EOS reflects the composition-dependent non-ideality of the chosen activity coefficient model. As in the vdW mixing rule, the size parameters are calculated as:

The HV-type rule improves predictive capability for highly non-ideal and asymmetric mixtures by incorporating activity-coefficient-based liquid-phase behavior directly into the EOS. 

Mixture formulas

Similar to those of the pure components, the constants Amix, Bmix, and Cmix; essential for applying the introduced formulas, are dimensionless parameters of the equation of state:

One of the main advantages of the AD-EOS is that, despite being a three-parameter cubic equation of state, the fugacity expression for species in solution is almost identical in algebraic form to those of the SRK and PR EOS. This makes the evaluation of fugacity coefficients computationally efficient while offering increased accuracy for mixtures.

For the van der Waals mixing rule, the derivative of the attraction parameter with respect to the number of moles can be easily derived:

For the Huron–Vidal type of mixing rule derived for the AD-EOS, it is suggested to perform the differentiations with respect to mole fraction numerically, since the complexity mainly arises from the derivatives of the activity coefficient model itself. However, the analytical derivative has been made available if use is desired:

References

• Almajose, A. P. L., & Dalida, M. L. P. (2025). Prediction of pure and mixture thermodynamic properties and phase equilibria using an optimized equation of state – Part 2: Vapor pressure modelling and extension to mixtures. Fluid Phase Equilibria. https://doi.org/10.1016/j.fluid.2025.114345

• Almajose, A. P. L., & Dalida, M. L. P. (2025). Prediction of pure and mixture thermodynamic properties and phase equilibria using an optimized equation of state – Part 1: Parameter estimation. Fluid Phase Equilibria. https://doi.org/10.1016/j.fluid.2024.114240