The Generalized Phase Rule, the Extended Definition of the Degree of Freedom, the Component Rule and the Seven Independent Non-Compositional State Variables: To the 150th Anniversary of the Phase Rule of Gibbs
Bibliographic record
Abstract
The phase rule of Gibbs is one of the basic equations in phase equilibria. Although it has been with us for 150 years, discussions, interpretations and extensions have been published. Here, the following new content is provided: (i). the choice of independent components is discussed, and the component rule is introduced, (ii). independent state variables are divided into compositional and non-compositional ones, (iii). the generalized phase rule is derived replacing number two in the original phase rule by the number of independent non-compositional state variables introduced above, (iv). the degree of freedom is decreased by the number of compositional constraints in special points (azeotrope and congruent melting) of phase diagrams, (v). a rule is derived connecting the maximum number of coexisting phases with the dimensions of the phase diagram, (vi). examples show how to apply the phase rule to unary, binary and ternary phase diagrams and their sections, (vii). the same is extended with the discussion of calculable and not calculable phase fractions, (viii). it is shown that the current definition of the degree of freedom is not sufficient in the number of cases, (ix). the current definition of the degree of freedom is extended, (x). the application of the generalized phase rule is demonstrated when other non-compositional state variables are applied for nano-phase diagrams, and/or for phase diagrams under the influence of electric potential difference, external magnetic field, mechanical strain or the gravitational field.
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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.011 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".