Bibliographic record
Abstract
Abstract The well-known Maxwell construction1 (the equal-area rule, EAR) was devised for vapor liquid equilibrium (VLE) calculation with the van der Waals (vdW) equation of state (EoS)2. The EAR generates an intermediate volume between the saturated liquid and vapor volumes. The trajectory of the intermediate volume over the coexistence region is defined here as the Maxwell crossover, denoted as the M-line, which is independent of EoS. For the vdW or any cubic3 EoS, the intermediate volume corresponds to the “unphysical” root, while other two corresponding to the saturated volumes of vapor and liquid phases, respectively. Due to it’s “unphysical” nature, the intermediate volume has always been discarded. Here we show that the M-line, which turns out to be strictly related to the diameter4 of the coexistence curve, holds the key to solving several major issues. Traditionally the coexistence curve with two branches is considered as the extension of the Widom line5,6-9. This assertion causes an inconsistency in three planes of temperature, pressure and volume. It is found that the M-line is the natural extension of the Widom line into the vapor-liquid coexistence region. As a result, the united single line coherently divides the entire phase space, including the coexistence and supercritical fluid regions, into gas-like and liquid-like regimes in all the planes. Moreover, along the M-line the vdW EoS finds a new perspective to access the second-order transition in a way better aligning with observations and modern theory10. Lastly, by using the feature of the M-line, we are able to derive a highly accurate and analytical proximate solution to the VLE problem with the vdW 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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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".