Almajose-Dalida EOS | Mixing Rules

The vdW one-fluid mixing rule treats a mixture as if it were a single “pseudo-fluid.” The mixture parameters are calculated from the pure-component parameters using mole-fraction averages and pairwise interaction terms. Binary interaction parameters are often added to better match experimental mixture behavior. Because it is simple and efficient, this mixing rule is commonly used with cubic equations of state.

Binary assumption for van der Waals one-fluid

For a binary mixture, the mixing rule becomes:

In some mixtures, the interaction between two different components may be treated as asymmetric. This means the interaction parameter from component 1 to component 2 (k₁₂) does not have to be equal to the interaction parameter from component 2 to component 1 (k₂₁). Allowing k₁₂ ≠ k₂₁ provides additional flexibility when fitting experimental mixture data. In this case, the expression for the attractive mixture parameter becomes:

Van der Waals one-fluid mixing rule (vdW1F) derivatives for thermodynamic formulas

The following expressions give the derivatives of the van der Waals one-fluid mixing rule with respect to number of moles and temperature. The composition terms are written using z to keep the formulation general for either liquid or vapor mixtures in AD-EOS.

with

hence

For a binary mixture, the derivative formulas reduce to the following expressions:

where

The Panagiotopoulos–Reid mixing rule extends the classical van der Waals one-fluid formulation by allowing asymmetric cross interactions between components. In this formulation the interaction parameter from component i toward component j does not have to equal the interaction from component j toward component i. This allows the model to represent systems where the unlike interactions are direction-dependent while still preserving the quadratic mixing structure used in cubic equations of state. 

The mixture attractive parameter is written as a double summation over all component pairs, but the cross interaction terms depend explicitly on composition. This additional dependence prevents the interaction terms from collapsing into a symmetric form and therefore maintains the asymmetric behavior of the model.

Binary assumption for Panagiotopoulos-Reid

For a binary mixture, the mixing rule becomes:

Combining:

Panagiotopoulos-Reid mixing rule derivatives for thermodynamic formulas

The following expressions give the derivatives of the Panagiotopoulos–Reid mixing rule with respect to number of moles and temperature. The composition terms are written using z so the formulation remains valid for either liquid or vapor mixtures.

with

hence

For a binary mixture, the derivative formulas reduce to the following expressions:

where

The Mathias–Klotz–Prausnitz mixing rule was developed to address the lack of composition invariance in asymmetric higher-order mixing rules used with cubic equations of state. In non-invariant formulations, the predicted thermodynamic properties of a mixture may change if a component is artificially divided into identical pseudo-components, which is physically unacceptable.

To resolve this, the MKP formulation restructures the attractive parameter as a hierarchical expansion in which nonlinear contributions are applied to composition-weighted sums, rather than to individual pairwise terms. This ensures that the resulting expression remains invariant with respect to composition splitting, while still allowing asymmetric interaction parameters between components.

The mixture attractive parameter is written as:

The two parameter and three-parameter MKP mixing rule can be written as:

The interaction parameters used in the MKP formulation follow specific symmetry relationships. The baseline parameter is symmetric, such that:

while the higher-order parameters are antisymmetric,

Binary assumption for Mathias-Klotz-Prausnitz

For a binary system, the MKP two-parameter and three-parameter mixing rules are expanded:

As such, the MKP two-parameter and three-parameter mixing rules applied to a binary system can be written as:

It is important to note that the values in the database are reported in terms of transformed parameters, Lij and Qij, which are related to lij and qij as:

Mathias-Klotz-Prausnitz mixing rule derivatives for thermodynamic formulas

The following expressions give the derivatives of the Mathias-Klotz-Prausnitz mixing rule with respect to number of moles and temperature. The composition terms are written using z so the formulation remains valid for either liquid or vapor mixtures. 

For a binary mixture applying the two-parameter MKP mixing rule, the derivative formulas reduce to the following expressions:

Likewise, a binary mixture applying the three-parameter MKP mixing rule results to the following expressions:

The Almajose–Dalida / Huron–Vidal–Orbey–Sandler (AD-HVOS) mixing rule reformulates the attractive term of the AD-EOS by embedding an excess Gibbs energy contribution directly into the mixture parameter. This approach departs from conventional mixing rules that rely on empirically fitted binary interaction coefficients, and instead represents intermolecular interactions using thermodynamically consistent Gᴱ/RT models.

By incorporating composition-dependent excess Gibbs energy expressions into the attractive parameter, the formulation provides a physically meaningful description of non-ideal behavior while preserving the mathematical structure of the AD-EOS. This enables the direct use of interaction parameters from established activity coefficient models, allowing mixture behavior to be described without introducing EOS-specific fitting constants.

The original HVOS formulation was developed for two-parameter equations of state, where the characteristic constant is obtained by assuming V = ub with u = 1. This corresponds to an infinite-pressure reference state (P → ∞, V = b), and the resulting CEOS is therefore defined at this limit. While thermodynamically consistent, this creates a mismatch with most activity coefficient models, which are formulated at low-pressure conditions.

In contrast, the Modified Huron–Vidal (MHV1) framework used in PSRK and VTPR is based on excess Gibbs energies evaluated at the low-pressure limit, yielding a CEOS consistent with near-atmospheric conditions. The corresponding value of u is determined by matching the infinite-pressure CEOS of HVOS to the low-pressure CEOS of MHV1, effectively mapping the two formulations.

This inverse evaluation gives u ≈ 1.23 – 1.27, and for simplicity, u = 1.25 is adopted. This ensures that the AD-HVOS formulation remains compatible with low-pressure activity coefficient models without requiring transformation to an infinite-pressure reference state.

Binary assumption for AD-HVOS

The AD-HVOS can be applied to binary systems, resulting to the following formulation:

AD-HVOS mixing rule derivatives for thermodynamic formulas

The following expressions present the derivatives of the AD-HVOS mixing rule with respect to the number of moles and temperature. Composition is expressed in terms of z, allowing the formulation to be applied consistently to both liquid and vapor mixtures. 

The term ∂Λmix/∂ni represents the sensitivity of the geometric correction factor Λmix to changes in the number of moles of component i, accounting for how mixture nonlinearity arising from the c/b ratio affects the attractive term. Meanwhile, ξi serves as a compositional correction that links the excess Gibbs energy contribution to the equation of state, incorporating both activity coefficient effects and size-related (co-volume) deviations within the mixture. 

For a binary system, the partial derivatives may be written as:

The partial derivatives of the geometric and compositional corrections in binary form may be written as: