SCIENCE · INVESTIGATE AND EXPLAIN

Reject a two-reservoir claim using entropy as well as energy

A cyclic engine absorbs 1000 J from a fixed hot reservoir at Th and rejects heat only to a 300 K reservoir. A claimed work fraction determines W and Qc=1000−W. This first-law relation is necessary but insufficient: a mathematically closed energy ledger can still require a forbidden entropy decrease.

Learning outcome: Compare a rejected, exact-boundary and below-bound claim. Retain energy and entropy ledgers with explicit kelvin conversion; specify which unlisted input would change the model rather than rescue the original claim.

Starting recap

Review logarithmic integrals, equilibrium states, absolute temperature and the first-law work convention. State the domain and distinguish calibration data from unused validation.

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Try

For 600 K and a 60% claim, W=600 J, Qc=400 J. Reservoir change 400/300−1000/600=−1/3 J/K rejects the claim. The maximum possible work under the declared boundary is 500 J.

Open the experiment

Predict

An energy-balanced cyclic two-reservoir proposal has negative total reservoir entropy change. What follows?

Your prediction and reason

Experiment

Compare a rejected, exact-boundary and below-bound claim. Retain energy and entropy ledgers with explicit kelvin conversion; specify which unlisted input would change the model rather than rescue the original claim.

Reject an energy-consistent but entropy-inconsistent claim

Worked teaching model · synthetic values · Cyclic device exchanges heat only with fixed reservoirs Th and Tc=300 K. It absorbs 1000 J each cycle and claims the selected work fraction. First-law rejection heat is inferred. Kelvin temperatures and complete cyclic system boundary are required.

Hot reservoir temperature: 600 K Claimed work fraction: 60 %

Reset keeps your prediction, captures and practice history.

The same first-law ledger can cross a second-law bound

Graph axes rescale independently; read units and values, not screen angles. 1: two fixed reservoirs (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for The same first-law ledger can cross a second-law bound

two fixed reservoirs

claimed efficiency (%)	reservoir entropy change (J/K)
20	1
20.75	0.975
21.5	0.95
22.25	0.925
23	0.9
23.75	0.875
24.5	0.85
25.25	0.825
26	0.8
26.75	0.775
27.5	0.75
28.25	0.725
29	0.7
29.75	0.675
30.5	0.65
31.25	0.625
32	0.6
32.75	0.575
33.5	0.55
34.25	0.525
35	0.5
35.75	0.475
36.5	0.45
37.25	0.425
38	0.4
38.75	0.375
39.5	0.35
40.25	0.325
41	0.3
41.75	0.275
42.5	0.25
43.25	0.225
44	0.2
44.75	0.175
45.5	0.15
46.25	0.125
47	0.1
47.75	0.075
48.5	0.05
49.25	0.025
50	0
50.75	-0.025
51.5	-0.05
52.25	-0.075
53	-0.1
53.75	-0.125
54.5	-0.15
55.25	-0.175
56	-0.2
56.75	-0.225
57.5	-0.25
58.25	-0.275
59	-0.3
59.75	-0.325
60.5	-0.35
61.25	-0.375
62	-0.4
62.75	-0.425
63.5	-0.45
64.25	-0.475
65	-0.5
65.75	-0.525
66.5	-0.55
67.25	-0.575
68	-0.6
68.75	-0.625
69.5	-0.65
70.25	-0.675
71	-0.7
71.75	-0.725
72.5	-0.75
73.25	-0.775
74	-0.8
74.75	-0.825
75.5	-0.85
76.25	-0.875
77	-0.9
77.75	-0.925
78.5	-0.95
79.25	-0.975
80	-1

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Claimed work	600	J
Required rejected heat	400	J
Carnot maximum work	500	J
Reservoir entropy change	-0.3333333	J/K
Second-law admissibility flag	0	1 admissible; 0 rejected

This cyclic two-reservoir claim violates the second law despite closing energy.

The cyclic working substance has zero state-entropy change. Reservoir change is Qc/Tc−Qh/Th≥0, implying W/Qh≤1−Tc/Th. The exact numerator comparison preserves supported equality states. Celsius ratios are invalid; unlisted fuels, storage changes or additional reservoirs would change the selected model rather than evade its bound.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

This cyclic two-reservoir claim violates the second law despite closing energy.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Declare the system and evidence

Derive the state or rate account

The cyclic working substance has zero state-entropy change. Reservoir changes are −1000/Th and Qc/300. Their sum must be nonnegative for the complete isolated collection, giving W/1000≤1−300/Th. Equality is the reversible limiting case, not experimental evidence of zero losses or a finite-power implementation.

Audit the constraint and model

Temperatures in the bound are absolute kelvin temperatures. Ratios of Celsius readings are not invariant under an arbitrary zero shift and cannot be substituted. A proposed device that also consumes fuel or changes stored energy is a different system; list those inputs rather than claiming that the same two-reservoir theorem has been broken.

Predict a changed boundary

Predict the exact boundary for Th=600 K: W=500 J and Qc=500 J give zero reservoir entropy change. Raising the claimed work to 600 J makes the change negative. Lowering it to 400 J leaves a positive budget, but passing this necessary thermodynamic test does not establish mechanical feasibility, safety or an actually operating device.

Worked original case

For 600 K and a 60% claim, W=600 J, Qc=400 J. Reservoir change 400/300−1000/600=−1/3 J/K rejects the claim. The maximum possible work under the declared boundary is 500 J.

Investigation

Changed-case practice

Try the changed case before feedback

For hot 800 K, cold 300 K and heat input 1000 J, find maximum reversible work in joules.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic cases checked 31 October 2026 against the primary teaching reference. No source exercises or diagrams are reproduced. All thermal operations are mathematical proposals or supplied records, not physical heating, pressure-vessel or exposure instructions. Public feedback and retained prose do not certify independent reasoning or actual practical participation. Preserve first work, critique, revision and a new assigned response in the separate course workspace.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which system boundary, reference path, constitutive law or independently calibrated record supports this entropy inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Audit a closed-cycle heat input before dividing for efficiency Next: Derive a work limit when both source temperatures change

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