ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Phase fractions from an additive molar volume

Physical scope

A constructed one-component, two-phase separated mixture with additive volume and fixed total amount; no coexistence data or measured material values are asserted.

Example

At a stipulated common $T, p$ , let one mole of a component occupy phase $α$ with molar volume $v_{α} = 1.0 \times 10^{- 3} m^{3} m o l^{- 1}$ or phase $β$ with $v_{β} = 4.0 \times 10^{- 3} m^{3} m o l^{- 1}$ . For this constructed model, additionally adopt amount-independent molar volumes at this fixed $(T, p)$ : $V_{i} = n_{i} v_{i}$ for every admitted amount of phase $i$ , with $v_{i}$ equal to the displayed one-mole value. This is an explicit constitutive scaling assumption, separate from additivity between phases. Consider a separated mixture of total amount $N = 2 m o l$ and total volume $V = 5.0 \times 10^{- 3} m^{3}$ ; use the volume convention of Phase branches and coexistence domains and additivity of Additive separated systems and tagged bulk scaling.

Facts & Assumptions

Given: Both phases are present, their molar volumes and the mixture totals are as in the Example, and volume and amount are additive for this separated compound, and the additional constitutive assumption $V_{i} = n_{i} v_{i}$ holds at the fixed $(T, p)$ for each admitted positive phase amount.

[F1]

Phase branches and coexistence domains and Additive separated systems and tagged bulk scaling: For one component, the phase amounts add to $N$ , and the separated phase volumes add to $V$ .

Verification

1.1givenF1algebra

Let $λ_{β} : = n_{β} / N$ and $λ_{α} : = n_{α} / N$ . Positivity and amount additivity give $λ_{α} + λ_{β} = 1$ , with both fractions in $(0, 1)$ . The adopted $V_{i} = n_{i} v_{i}$ relation and volume additivity give

\overset{ˉ}{v} : = \frac{V}{N} = λ_{α} v_{α} + λ_{β} v_{β} .

All terms have units $m^{3} m o l^{- 1}$ .

2.1givenstep 1.1algebra∎

Here $\overset{ˉ}{v} = 2.5 \times 10^{- 3} m^{3} m o l^{- 1}$ . Solving the affine relation in step 1.1 yields

λ_{β} = \frac{\overset{ˉ}{v} - v_{α}}{v_{β} - v_{α}} = \frac{1}{2}, λ_{α} = \frac{v_{β} - \overset{ˉ}{v}}{v_{β} - v_{α}} = \frac{1}{2} .

The fractions are positive and sum to one, so $n_{α} = n_{β} = 1 m o l$ satisfies the given total amount and volume.

Phase fractions from an additive molar volume

Example

Facts & Assumptions

Verification

Depends on

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