Physics theoremStatement: AI-adaptedProof: 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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Ideal-bath refrigerator coefficient-of-performance bound

Physical scope

Conditional cooling coefficient-of-performance bound for an admitted closed refrigerator cycle between positive-temperature ideal baths inside an isolated compound with an ideal work store; finite reservoirs, unlisted transfers, and empirical appliance performance are excluded.

Statement

Let an admitted refrigerator cycle in the sense of Ideal-reservoir thermodynamic cycles operate between fixed ideal-bath temperatures $0 < T_{c} < T_{h}$ , extract $Q_{c} > 0$ from the cold bath, reject $Q_{h} > 0$ to the hot bath, and consume $W_{i n} > 0$ . Define its cooling coefficient of performance by ${C O P}_{R} = Q_{c} / W_{i n}$ . Then ${C O P}_{R} \leq \frac{T_{c}}{T_{h} - T_{c}} .$ Equality holds exactly when the total entropy change of the isolated compound is zero. A reversible cycle in the stated boundary model saturates the bound; no converse about reversibility is asserted.

Facts & Assumptions

Given: A cyclic working system with $0 < T_{c} < T_{h}$ , $Q_{c} > 0$ , $Q_{h} > 0$ , $W_{i n} > 0$ , no unlisted transfers, and the full working system, baths, and work store inside the declared isolated compound.

[F1]

Ideal-reservoir thermodynamic cycles: In refrigerator mode $Q [P] = Q_{c} - Q_{h}$ , $W_{o n} [P] = W_{i n}$ , the cold and hot bath energy changes are $- Q_{c}$ and $+ Q_{h}$ , their endpoint entropy changes are $- Q_{c} / T_{c}$ and $+ Q_{h} / T_{h}$ , and the work store supplies this work with $Δ E_{w} = - W_{i n}$ and zero entropy change. The admitted total-compound endpoint tuples contain those same working-system endpoints and bath/store energy coordinates, lie in one feasible set for the second-law postulate, and use the declared additive entropy split.

[F2]

First-law energy balance for closed process records: For an admitted closed fixed-constitution process with zero endpoint changes in separately tracked center-of-mass kinetic and external-potential energies, $Δ U = Q [P] + W_{o n} [P]$ .

[F3]

Macroscopic second law for isolated compounds: For an admitted isolated compound process with equilibrium endpoints and the declared additive total entropy, $Δ S_{t o t} \geq 0$ ; an admitted reversible operation has $Δ S_{t o t} = 0$ .

Proof

Given: The fixed-bath cycle hypotheses in the Statement and the heat/work signs in [F1].

1.1givenF1F2algebra

The working system returns to its initial state, so $Δ U = 0$ . Applying [F2] with [F1] gives $0 = Q_{c} - Q_{h} + W_{i n}$ , hence $Q_{h} = Q_{c} + W_{i n}$ .

1.2givenF1F3algebra

The cold bath loses $Q_{c}$ , the hot bath gains $Q_{h}$ , and their entropy slopes are constant. Thus their entropy changes are $- Q_{c} / T_{c}$ and $Q_{h} / T_{h}$ . The working system and work store have zero entropy change, so [F3] gives $- \frac{Q_{c}}{T_{c}} + \frac{Q_{h}}{T_{h}} = Δ S_{t o t} \geq 0.$

2.1givenstep 1.1step 1.2algebra

Since $T_{c}, T_{h} > 0$ , step 1.2 gives $Q_{h} \geq Q_{c} T_{h} / T_{c}$ . Substituting $Q_{h} = Q_{c} + W_{i n}$ from step 1.1 yields $W_{i n} \geq Q_{c} \frac{T_{h} - T_{c}}{T_{c}} .$

3.1givenF3step 2.1algebra∎

Divide by $W_{i n} > 0$ and use $T_{h} - T_{c} > 0$ to obtain ${C O P}_{R} \leq T_{c} / (T_{h} - T_{c})$ . Equality is equivalent to $Δ S_{t o t} = 0$ ; a reversible cycle has this property by [F3].

Ideal-bath refrigerator coefficient-of-performance bound

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources