Excess-energy-coupled EoS mixing rule¶

A cubic-equation-of-state mixture rule incorporates an activity or excess-energy model into its mixture parameters.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14142
Domain group: Natural Sciences
Origin domain: Chemistry & Materials Science
Subdomains: Phase Equilibrium Thermodynamics, Cubic Equations of State → Chemistry & Materials Science
Aliases: EoS/G^E mixing rule, Excess-free-energy EoS mixing rule

Core Idea¶

An excess-energy-coupled equation-of-state mixing rule constructs mixture parameters for a cubic equation of state (EoS) using information from a specified activity-coefficient or excess-free-energy model. The EoS supplies the mixture-parameter carrier; the external model supplies nonideal-mixture information; and a stated construction makes that information affect at least one EoS mixture parameter. Calculating the models side by side without changing the EoS parameter rule does not instantiate this family.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^ref-bb252ca9efa0]

Many members match an EoS excess Gibbs or Helmholtz energy to a model quantity at a reference pressure. This is a common construction, not a universal one: LCVM combines Vidal and Michelsen rules coupled with UNIFAC and has no single reference pressure for the composite. The selected Wong–Sandler rule uses infinite-pressure excess-Helmholtz matching and adds a quadratic-composition second-virial condition; that additional condition does not define the whole family.[^{ref-c0d0abca2867][}ref-bb252ca9efa0][^ref-2674639ce05b]

Use Within the Domain¶

Identify four roles in an actual rule: a cubic EoS carrier, a specified external activity or excess-energy model, a relation incorporating that model, and the EoS mixture parameter it determines or constrains. Huron–Vidal's local-composition branch uses an excess-energy relation for the EoS energy parameter; Wong–Sandler links attraction and covolume under its Helmholtz and virial conditions; LCVM constructs an attractive-term rule by combining two model-coupled routes.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^ref-bb252ca9efa0]

The reference pressure, potential, EoS form and activity model are choices of particular members. Huron and Vidal's separate empirical polynomial smoothing rule is not automatically part of this family merely because it appears in their 1979 paper. Michelsen's modified Huron–Vidal rule uses a zero-pressure route to incorporate existing excess-Gibbs correlations. Dahl and Michelsen's later high-pressure VLE study applies such rules; it does not supply a new necessary family role.[^{ref-c0ce15c7e939][}ref-8364dfb25596][^ref-91adc7017b95]

Clarity¶

The word Mixing names a parameter rule as composition varies; it does not require physical redistribution of constituents. A conventional EoS-only quadratic rule lacks the chosen external model that materially informs this family. An activity model by itself lacks the EoS parameter output.[^{ref-c0ce15c7e939][}ref-c0d0abca2867]

The EoS/G^E label can also hide differences. Wong–Sandler's construction matches excess Helmholtz energy, and LCVM's combined rule has no single reference pressure. A definition limited to Gibbs energy or infinite-pressure matching would exclude attested members.[^{ref-c0d0abca2867][}ref-bb252ca9efa0][^ref-2674639ce05b]

Manages Complexity¶

The four-role test separates the rule's identity from details of a calculation: fluid system, pure-component data, EoS form, reference state, interaction parameters and phase-equilibrium results. It permits comparison of Huron–Vidal, Wong–Sandler, Michelsen and LCVM without treating their distinct potentials, covolume choices or consistency conditions as interchangeable.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^{ref-8364dfb25596][}ref-bb252ca9efa0]

This grouping does not certify accuracy or parameter transfer. The original Wong–Sandler paper tested specified binary and ternary systems, and later review work discusses limits for size-asymmetric mixtures. Performance remains a question for the chosen rule, model and mixture.[^{ref-c0d0abca2867][}ref-2674639ce05b]

Abstract Reasoning¶

Remove the external model as a counterfactual. If the EoS parameter rule remains the same, the case is outside this family. If the model affects the parameter construction, identify the route: a direct excess-energy equality or a combination of already model-dependent rules. This distinguishes a genuine LCVM-style composite from an EoS-only formula that happens to look similar.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^ref-bb252ca9efa0]

Then separate family roles from member-specific conditions. Removing Wong–Sandler's quadratic second-virial condition changes that member, but a different rule can remain in the broader model-coupled family. Likewise, LCVM need not acquire a single reference pressure to qualify. The current catalog admits this entry as an unparented domain-specific root: no tested live Prime has a proved strict whole-object, prerequisite or constituent relation across every member. A future source-backed intermediate could change that placement.[^{ref-c0d0abca2867][}ref-bb252ca9efa0][^ref-2674639ce05b]

Knowledge Transfer¶

The same four questions can classify rules across cubic-EoS studies: Which EoS carries the parameters? Which external model supplies information? How is that information incorporated? Which parameter is affected? They keep model construction separate from downstream phase-equilibrium tests.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^ref-91adc7017b95]

This named family remains within fluid-mixture thermodynamics. General talk of “model coupling” in another field does not establish the EoS carrier, external nonideality model and thermodynamic parameter output required here. The live Mixing Prime concerns redistribution; Constraint and Function Mapping have fuller signatures than the inspected sources establish for the whole family; Coupling requires dynamic subsystem interdependence.[^{ref-c0ce15c7e939][}ref-c0d0abca2867][^ref-bb252ca9efa0]

Example¶

Huron–Vidal local-composition branch. A cubic EoS is the carrier; a local-composition excess-energy model supplies nonideality information; an infinite-pressure excess-energy relation carries that information into the mixture energy parameter. Its separate empirical polynomial rule must be examined on its own terms.[^ref-c0ce15c7e939]

Wong–Sandler. A cubic EoS and an activity model are linked through infinite-pressure excess Helmholtz energy. The construction determines attraction and covolume together and imposes a quadratic-composition second-virial condition. That last condition identifies this member, not the whole family.[^ref-c0d0abca2867]

LCVM. The attractive-term rule linearly combines Vidal and Michelsen rules and couples the result to original UNIFAC with a translated-modified Peng–Robinson EoS. The composite has no single reference pressure; the accessible original abstract does not justify printing its coefficients here.[^{ref-bb252ca9efa0][}ref-2674639ce05b]

Neighborhood in Abstraction Space¶

Excess-energy-coupled EoS mixing rule sits in a sparse region of the domain-specific corpus (94^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Molecular Structure & Chemical Bonding (35 abstractions)

Nearest neighbors

UNIQUAC — 0.79
Eötvös rule — 0.78
Ewald summation — 0.78
Cubic function — 0.78
Dependent component analysis — 0.77

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

Physical mixing: redistribution of material rather than calculation of mixture-model parameters.
An EoS-only quadratic rule: no chosen external activity or excess-energy model materially enters its construction.[^ref-c0ce15c7e939]
A stand-alone activity model: a potential information source, but no cubic-EoS parameter rule by itself.[^ref-8364dfb25596]
Universal reference-pressure matching: common among members, but absent as one reference pressure for LCVM's composite.[^{ref-bb252ca9efa0][}ref-2674639ce05b]
Universal second-virial condition: the quadratic-composition condition is specific to Wong–Sandler.[^ref-c0d0abca2867]
A successful VLE result: evidence for an application, not part of the rule's identity or a guarantee for other mixtures.[^ref-91adc7017b95]

References¶

[^ref-c0ce15c7e939]: Marie-José Huron and Jean Vidal. (1979). New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria 3(4), 255–271. Original publisher abstract inspected; original-paper prose and printed pp. 257–263 checked in a third-party OCR reading copy at https://www.scribd.com/document/276009792/Huron-vidal-mixing-rules. OCR mathematical glyphs are unreliable; no exact equation is transcribed here. [^ref-c0d0abca2867]: David Shan Hill Wong and Stanley I. Sandler. (1992). A theoretically correct mixing rule for cubic equations of state. AIChE Journal 38(5), 671–680. Original publisher abstract inspected; original-paper prose and printed pp. 671–674 checked in a third-party OCR reading copy at https://www.scribd.com/document/237295042/Wong-Sandler-1992. OCR equations were not used for exact transcription. [^ref-8364dfb25596]: Michael L. Michelsen. (1990). A modified Huron-Vidal mixing rule for cubic equations of state. Fluid Phase Equilibria 60, 213–219. Original publisher abstract inspected; original-paper prose at printed pp. 213–215 and 218 checked in a third-party OCR reading copy at https://www.scribd.com/document/399782969/michelsen1990-pdf. This is distinct from Michelsen's separate 1990 method paper at pp. 47–58. [^ref-bb252ca9efa0]: C. Boukouvalas, N. Spiliotis, P. Coutsikos, N. Tzouvaras and D. Tassios. (1994). Prediction of vapor-liquid equilibrium with the LCVM model, a linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIFAC and the t-mPR equation of state. Fluid Phase Equilibria 92, 75–106. The printed title uses a colon after “model”; the linked title uses a comma for reference-key parsing. Original publisher abstract inspected; full original article unavailable. [^ref-2674639ce05b]: Georgios M. Kontogeorgis and Philippos Coutsikos. (2012). Thirty Years with EoS GE Models, What Have We Learned?. Industrial & Engineering Chemistry Research 51(11), 4119–4142, especially printed pp. 4119–4120 and 4130. Full publisher-version review hosted by Technical University of Denmark; comparative synthesis, not an original member derivation. The printed title uses “EoS/G^E Models—What Have We Learned?”; the linked title simplifies punctuation for reference-key parsing. [^ref-91adc7017b95]: Søren Dahl and Michael L. Michelsen. (1990). High-pressure vapor-liquid equilibrium with a UNIFAC-based equation of state. AIChE Journal 36(12), 1829–1836. Original publisher abstract inspected. This tests Michelsen's rules with modified UNIFAC and is not a distinct rule-family member.