A revisit of the vapor-liquid equilibrium calculation with cubic equations of state
Bibliographic record
Abstract
Abstract Based on the roots-coefficient relations for a cubic function, quadratic functions are constructed that strictly relate the saturated volumes of liquid and vapor phases and the third solution from a cubic equation of state (EoS). The vapor-liquid equilibrium (VLE) calculation with a cubic EoS is thus reduced to solving a single nonlinear equation. In light of a recent finding that the “unphysical” third solution, namely the Maxwell crossover or the M-line, plays a central role as the dividing interface in the density gradient theory, here we show that it can also be used to derive analytically approximate solutions to a VLE problem. The van der Waals EoS and the Soave-Redlich-Kwong (SRK) EoS are discussed as examples. The method proposed in this work simplifies the calculations of the traditional VLE calculations with a cubic EoS. With one-time-only effort for a given system, simple analytical solutions can be obtained to avoid the repetitively iterative computations for a VLE problem. Finally, the relationship between the Widom line in the supercritical region and the M-line is briefly discussed with the SRK EoS.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".