Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-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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Local Clapeyron slope on a regular pure-phase coexistence branch

Physical scope

A one-component, positive-temperature, C1 coexistence branch with bilateral energy, volume, and species exchange, nonzero molar-volume jump, and the explicit local same-tag homogeneity and reversible-heat assumptions stated here, with homogeneity either independently stipulated or supplied on each designated common-law bulk branch through the local Euler theorem; the slope formula requires nonzero molar-volume jump and C1 branch charts and is not asserted when either condition fails.

Statement

Let $α, β$ be two one-component phase branches from Phase branches and coexistence domains on a common open chart $O \subset (0, \infty) \times R$ with coordinates $(T, p)$ . Assume each branch has a local $C^{2}$ energy representation $U_{i} (s, V, n)$ , $i \in {α, β}$ , and that within each fixed branch tag it satisfies the same-tag homogeneity

U_{i} (λ s, λ V, λ n) = λ U_{i} (s, V, n)

for all $λ$ in a neighborhood of $1$ for which the scaled state remains in the chart. This is a phase-branch hypothesis: the general tagged law of Additive separated systems and tagged bulk scaling does not imply it. It may be stipulated independently on each local branch, or supplied by the common-law bulk subclass of Euler and Gibbs-Duhem identities under same-tag homogeneity when each phase's local inverse states lie in its designated entropy cone with fixed species and intensive parameters and common SI units and energy/entropy reference conventions. That supplied subclass uses local inverse uniqueness and does not require a global energy inverse.

Suppose a state $(T_{0}, p_{0}) \in O$ is an attained interior coexistence equilibrium with bilateral energy, volume, and species exchange between the two branches, and both branches admit the valid local Gibbs charts of Local thermodynamic Legendre potentials. Write $g_{i} = G_{i} / n_{i}$ , $s_{i} = S_{i} / n_{i}$ , and $v_{i} = V_{i} / n_{i}$ in the units specified by SI quantities and unit conventions. If

Δ v (T_{0}, p_{0}) : = v_{β} (T_{0}, p_{0}) - v_{α} (T_{0}, p_{0}) \neq 0,

then, on some open interval about $T_{0}$ , the equal-Gibbs-per-mole coexistence condition $g_{α} (T, p (T)) = g_{β} (T, p (T))$ defines a unique local $C^{1}$ curve and

\frac{d p}{d T} = \frac{s_{β} - s_{α}}{v_{β} - v_{α}} .

If, in addition, conversion from $α$ to $β$ is admitted as a reversible process in contact with an ideal thermal reservoir at the same fixed $T$ , and the full finite isolated compound (converted phases, reservoir, work stores and all auxiliaries) has equilibrium endpoints with additive component entropy, with every component other than the converted phases and that reservoir having zero entropy change, and the admitted forward and reverse complete-compound records have nonnegative total endpoint entropy changes for that same additive split with their full endpoint tuples interchanged, then its molar latent heat absorbed by the system is $ℓ_{α \to β} = T (s_{β} - s_{α})$ . These are conditional model deductions. The slope conclusion is not asserted when $Δ v = 0$ or where the branch charts are not $C^{1}$ .

Facts & Assumptions

Given: Two pure phase branches on a common open $(T, p)$ chart; positive one-component amounts; the C2 local energy and Gibbs maps and C1 phase-branch maps stated above; the same-tag homogeneity assumption for each phase; an attained interior maximum with bilateral energy, volume, and species exchange; $Δ v (T_{0}, p_{0}) \neq 0$ ; and, for the latent-heat clause, a reversible conversion with a fixed-temperature ideal reservoir. Use SI units: $T$ in kelvins, $p$ in pascals, $G$ in joules, $n$ in moles, $s$ in $J m o l^{- 1} K^{- 1}$ , and $v$ in $m^{3} m o l^{- 1}$ .

[A1]

Each phase's fixed-tag local C2 energy representation has the degree-one homogeneity stated in the Statement. This may be an independent local constitutive hypothesis or the supplied common-law bulk subclass of Euler and Gibbs-Duhem identities under same-tag homogeneity under its exact inverse-chart and reference conventions. The general tagged-scaling postulate alone does not supply it.

[A2]

The admitted reversible conversion and its reverse are records of the complete finite isolated compound with aggregate equilibrium endpoints represented by full constituent and auxiliary state tuples. Its components include the converted phases, the ideal thermal reservoir at fixed $T$ , and every work store or auxiliary, with an adopted additive total endpoint entropy map $S_{t o t}$ . Each other component has zero endpoint entropy change, and the reservoir contributes $- Q / T$ when heat $Q$ is absorbed by the system. The reverse record interchanges the two full endpoint tuples and reverses the signed transfer, as required by Reversible thermodynamic process record. Adopt explicitly $Δ S_{t o t} [P] \geq 0$ and $Δ S_{t o t} [P^{r e v}] \geq 0$ for these two records. Since their entropy changes are negatives, $Δ S_{t o t} [P] = 0$ . The postulate Macroscopic second law for isolated compounds supplies these inequalities when its admitted core-record, complete endpoint-association and additive-split hypotheses hold; without that additional discharge, the two inequalities are separate adopted model premises of this latent-heat clause. No entropy premise is needed for the slope-only clause.

[F1]

Phase branches and coexistence domains: a one-component branch supplies $g = G / n$ , $s = S / n$ , and $v = V / n$ on its valid positive-temperature chart; the definition does not assert $g = μ$ without same-tag homogeneity.

[F2]

Local thermodynamic Legendre potentials: on its valid local natural-variable chart, $G = U - T s + p V$ .

[F3]

Intensive equilibrium conditions for coexisting phases: with energy, volume, and species transfer feasible in both directions at an attained interior entropy maximum, the two branches have equal positive temperatures, equal pressures, and equal chemical potentials.

[F4]

Differentials of the thermodynamic potentials: on a valid local C2 energy-representation chart, $d G = - S d T + V d p + μ d n$ . Its current batch-9 carrier is registered with draft status and its proof is conditional on the actual local-energy and partial-Legendre suppliers; their interfaces and proof statuses are recorded in batch-13 cross-batch evidence.

[F5]

Euler and Gibbs-Duhem identities under same-tag homogeneity: if the same-tag homogeneity [A1] holds on a valid local C2 energy chart, its current Statement gives $U = T s - p V + μ n$ , hence $G / n = μ$ . Its general conditional implication requires [A1], and its separately designated common-law bulk subclass derives that local premise from adopted entropy transport and local inverse uniqueness. Neither route silently equates distinct physical system tags.

[L1]

Local graph form at a nonsingular derivative minor: for a C1 scalar equation $F (T, p) = 0$ on an open subset of $R^{2}$ , if $F (T_{0}, p_{0}) = 0$ and $\partial_{p} F (T_{0}, p_{0}) \neq 0$ , the local zero set is uniquely a C1 graph $p = p (T)$ with $p^{'} = - F_{T} / F_{p}$ . The current carrier is registered in batch 4; it remains a draft mathematical item, not an independent acceptance.

[L2]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : differentiating a composition of differentiable maps gives the composed derivative.

Proof

1.1givenA1F3F5algebra

By [F3], bilateral energy, volume, and species exchange give common $T_{0}, p_{0}$ and $μ_{α} = μ_{β}$ at the attained interior coexistence point. Apply [F5] separately to the two phase branches using [A1]: $G_{i} / n_{i} = μ_{i}$ for $i = α, β$ . Thus $F (T_{0}, p_{0}) : = g_{β} (T_{0}, p_{0}) - g_{α} (T_{0}, p_{0}) = 0$ .

1.2givenF1F2F4algebra

On each branch, hold the amount $n_{i}$ fixed while varying $(T, p)$ . By [F2] and [F4], $d (G_{i} / n_{i}) = - s_{i} d T + v_{i} d p$ . Consequently $F_{T} = - (s_{β} - s_{α})$ and $F_{p} = v_{β} - v_{α}$ at $(T_{0}, p_{0})$ . The nonzero-volume-jump hypothesis makes $F_{p} \neq 0$ .

2.1L1L2step 1.1step 1.2algebra

Apply [L1] to $F = g_{β} - g_{α}$ on $O$ . It gives a unique local C1 graph $p = p (T)$ satisfying $F (T, p (T)) = 0$ . Differentiating this identity using [L2] gives $0 = F_{T} + F_{p} p^{'} (T)$ . Substitution of the derivatives from step 1.2 yields

p^{'} (T_{0}) = \frac{s_{β} - s_{α}}{v_{β} - v_{α}} .

The same calculation applies at every point of the shrunken coexistence interval with nonzero $Δ v$ .

3.1givenA2F1algebra∎

For the reversible latent-heat clause, the system entropy change when $n$ moles are converted is $n (s_{β} - s_{α})$ . By [A2], the full isolated compound has zero total entropy change; all other components have zero entropy change, and the reservoir contributes $- Q / T$ , so $Q / T = n (s_{β} - s_{α})$ . Dividing by $n > 0$ gives $ℓ_{α \to β} : = Q / n = T (s_{β} - s_{α})$ . This conclusion is conditional on the stated reversible operation and ideal reservoir; it is not a property of an arbitrary irreversible conversion path.

Remarks

The statement's same-tag homogeneity is deliberately local to each branch. The Euler supplier preserves the general implication from independently stipulated local homogeneity and also supplies it on the separately adopted common-law bulk branch. The general tagged postulate still allows $c^{(λ)} \neq c$ ; neither constitutive route is a proof of exact extensivity for every finite body, surface-sensitive or long-range system, or gravitating system. Valid Gibbs charts, admitted bilateral exchanges, positive amounts, the nonzero molar-volume jump, and the reversible ideal-reservoir qualification remain essential.

Local Clapeyron slope on a regular pure-phase coexistence branch

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources