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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Joule-Thomson coefficient on a valid Gibbs chart

Physical scope

Local derivative identity for a closed-composition system on a positive-temperature C2 Gibbs response chart with nonzero constant-pressure heat capacity; it does not assert an open-flow throttle balance, a sign of cooling, or an empirical material coefficient.

Statement

On a C2 Gibbs chart $G (T, p, n)$ at fixed composition $n$ , with $T > 0$ , $V > 0$ , finite $C_{p} > 0$ , and a valid local inverse between $(T, p, n)$ and $(H, p, n)$ , define the Joule-Thomson coefficient by

$μ_{J T} : = {(\frac{\partial T}{\partial p})}_{H, n} .$

Then

$μ_{J T} = \frac{T (\partial V / \partial T)_{p, n} - V}{C_{p}} = \frac{V (T α_{p} - 1)}{C_{p}},$

where $α_{p} = V^{- 1} (\partial V / \partial T)_{p, n}$ and $C_{p} = (\partial H / \partial T)_{p, n}$ are the response functions of Thermodynamic response functions on valid local charts. This is a local derivative identity. It does not prove that a real throttle is isenthalpic or that a particular substance cools under expansion.

Facts & Assumptions

Given: A fixed-composition open C2 chart with valid $(T, p, n)$ and $(H, p, n)$ coordinates, $T, V, C_{p} > 0$ , and the potentials defined in Local thermodynamic Legendre potentials.

[F1]

Differentials of the thermodynamic potentials: On a valid chart, $d H = T d s + V d p + \sum_{j} μ_{j} d n_{j}$ and $d G = - s d T + V d p + \sum_{j} μ_{j} d n_{j}$ .

[F2]

Maxwell relations on valid thermodynamic potential charts: For the Gibbs chart, $s_{p} ∣_{T, n} = - V_{T} ∣_{p, n}$ .

[F3]

Thermodynamic response functions on valid local charts: $C_{p} = H_{T} ∣_{p, n}$ and $α_{p} = V^{- 1} V_{T} ∣_{p, n}$ on a valid response chart.

[F4]

Local thermodynamic Legendre potentials: $H = U + p V$ and $G = H - T s$ on their local natural-variable charts.

[F5]

SI quantities and unit conventions: $p$ is in pascals, $V$ in cubic metres, $U, H, G$ in joules, and $T$ in kelvins, with $P a m^{3} = J$ .

Proof

1.1givenF1F3F4algebra

At fixed $n$ , write $d s = s_{T} ∣_{p, n} d T + s_{p} ∣_{T, n} d p$ in [F1]. The enthalpy $H$ and chart are those of [F4]. Since $C_{p} = H_{T} ∣_{p, n}$ and $d H = T d s + V d p$ , the coefficient of $d T$ gives $C_{p} = T s_{T} ∣_{p, n}$ . Therefore $d H = C_{p} d T + (T s_{p} ∣_{T, n} + V) d p$ .

2.1F2step 1.1algebra

Substitute the Maxwell relation [F2] into step 1.1 to obtain $d H = C_{p} d T + (V - T V_{T} ∣_{p, n}) d p$ .

3.1givenF3step 2.1algebra

Along a curve of constant $H$ and fixed $n$ , step 2.1 gives $0 = C_{p} d T + (V - T V_{T} ∣_{p, n}) d p$ . The assumed $(H, p, n)$ chart and $C_{p} > 0$ therefore give $μ_{J T} = (T V_{T} ∣_{p, n} - V) / C_{p}$ .

4.1F3F5step 3.1algebra∎

Substitute $V_{T} ∣_{p, n} = V α_{p}$ from [F3] to obtain $μ_{J T} = V (T α_{p} - 1) / C_{p}$ . Its units are $K P a^{- 1}$ by [F5].

Remarks

An actual steady insulated throttle has constant upstream/downstream enthalpy only under an additional open-flow energy balance, negligible heat transfer and shaft work, and negligible kinetic and potential energy changes. This theorem is solely the derivative calculation on a specified equilibrium chart.

Joule-Thomson coefficient on a valid Gibbs chart

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources