SCIENCE · INVESTIGATE AND EXPLAIN

Reject an energy-balanced engine that violates the second law

A cyclic heat engine returns its working substance to its initial state. Over a cycle ΔU=0, so absorbed heat Qh equals work output W plus rejected heat Qc, using positive magnitudes. Efficiency W/Qh describes conversion to work. An energy balance is necessary but does not by itself establish a physically possible engine.

Learning outcome: Compare one entropy-increasing, one reversible-limit and one rejected proposal. Retain work, rejected heat and both reservoir entropy terms, then build a separate refrigerator energy account without reusing engine-efficiency terminology.

Starting recap

Declare temperature and time references, system boundaries and model regimes. Use absolute temperatures for ratios; distinguish state, path, oscillator phase and wave propagation.

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A proposed 800 J heat input with 0.4 work fraction would reject 480 J and yield entropy change −2+1.6=−0.4 J/K. Reject it despite its correct first-law arithmetic.

Open the experiment

Predict

A two-reservoir cyclic engine proposal with negative total reservoir entropy change is:

Your prediction and reason

Experiment

Compare one entropy-increasing, one reversible-limit and one rejected proposal. Retain work, rejected heat and both reservoir entropy terms, then build a separate refrigerator energy account without reusing engine-efficiency terminology.

Check both energy and reservoir entropy for a proposed engine

Worked teaching model · synthetic values · Cyclic engine exchanges heat only with ideal reservoirs at 400 K and 300 K. Work fraction proposes W/Qhot; energy fixes rejected heat. A negative total reservoir entropy change rejects the proposal even if energy closes. No real engine is constructed.

Heat absorbed from hot reservoir: 600 J/cycle Proposed work fraction: 0.3 ratio

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Reservoir entropy change versus proposed efficiency

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: cycle reservoir ledger (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Reservoir entropy change versus proposed efficiency

cycle reservoir ledger

proposed work fraction	reservoir entropy change (J/K)
0.1	0.3
0.10375	0.2925
0.1075	0.285
0.11125	0.2775
0.115	0.27
0.11875	0.2625
0.1225	0.255
0.12625	0.2475
0.13	0.24
0.13375	0.2325
0.1375	0.225
0.14125	0.2175
0.145	0.21
0.14875	0.2025
0.1525	0.195
0.15625	0.1875
0.16	0.18
0.16375	0.1725
0.1675	0.165
0.17125	0.1575
0.175	0.15
0.17875	0.1425
0.1825	0.135
0.18625	0.1275
0.19	0.12
0.19375	0.1125
0.1975	0.105
0.20125	0.0975
0.205	0.09
0.20875	0.0825
0.2125	0.075
0.21625	0.0675
0.22	0.06
0.22375	0.0525
0.2275	0.045
0.23125	0.0375
0.235	0.03
0.23875	0.0225
0.2425	0.015
0.24625	0.0075
0.25	0
0.25375	-0.0075
0.2575	-0.015
0.26125	-0.0225
0.265	-0.03
0.26875	-0.0375
0.2725	-0.045
0.27625	-0.0525
0.28	-0.06
0.28375	-0.0675
0.2875	-0.075
0.29125	-0.0825
0.295	-0.09
0.29875	-0.0975
0.3025	-0.105
0.30625	-0.1125
0.31	-0.12
0.31375	-0.1275
0.3175	-0.135
0.32125	-0.1425
0.325	-0.15
0.32875	-0.1575
0.3325	-0.165
0.33625	-0.1725
0.34	-0.18
0.34375	-0.1875
0.3475	-0.195
0.35125	-0.2025
0.355	-0.21
0.35875	-0.2175
0.3625	-0.225
0.36625	-0.2325
0.37	-0.24
0.37375	-0.2475
0.3775	-0.255
0.38125	-0.2625
0.385	-0.27
0.38875	-0.2775
0.3925	-0.285
0.39625	-0.2925
0.4	-0.3

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
Proposed work output	180	J
Rejected heat fixed by energy balance	420	J
Proposed efficiency	0.3	ratio
Reversible efficiency ceiling	0.25	ratio
Total reservoir entropy change	-0.1	J/K
Two-reservoir second-law screen passes	0	flag
Reversible refrigerator COP	3	ratio
Reversible heat-pump COP	4	ratio

The proposal closes energy but violates the stated two-reservoir second-law limit and is rejected.

For a cyclic working substance, Delta U and Delta S return to zero. Reservoir entropy is −Qhot/Thot+Qcold/Tcold; nonnegative total is necessary. Entropy is a state function even for an irreversible route. Reversible equality is a limiting idealization, not evidence of zero heat transfers. Refrigerator COP=Tc/(Th−Tc), heat-pump COP=Th/(Th−Tc) for the reversible bounds, and neither is an engine efficiency.

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

The proposal closes energy but violates the stated two-reservoir second-law limit and is rejected.

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 state, system and observation

Build the algebraic model

This original proposal exchanges heat only with reservoirs at 400 and 300 K. The reversible efficiency ceiling is 1−Tc/Th=0.25. For Qh=600 J and proposed work fraction 0.30, work would be 180 J and rejected heat 420 J. Energy closes, but total reservoir entropy change is −600/400+420/300=−0.1 J/K, violating the second law for this closed cycle plus reservoirs.

Check the original evidence packet

At efficiency 0.20, work is 120 J, rejected heat 480 J and total reservoir entropy increases by 0.1 J/K. At 0.25 it is zero in the reversible limit. Entropy is a state function: an actual irreversible route has the same system entropy change as any reversible comparison route between its endpoints. Individual subsystems can lose entropy while the total isolated account increases. Equilibrium maximizes entropy only under the relevant fixed isolated constraints; spreading energy can reduce its availability for work without destroying energy.

Challenge the domain and transfer

A refrigerator uses work to move heat from cold to hot. For reversible operation between these reservoirs, COPref=Tc/(Th−Tc)=3 and COPheat=Th/(Th−Tc)=4. With 20 J work, the reversible comparison removes 60 J from the cold reservoir and delivers 80 J to the hot one. A COP above one is not an efficiency above 100%: delivered heat includes energy extracted from the other reservoir. None of these bounds proves a practical machine can be built or operated safely.

Worked original case

A proposed 800 J heat input with 0.4 work fraction would reject 480 J and yield entropy change −2+1.6=−0.4 J/K. Reject it despite its correct first-law arithmetic.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

A reversible engine works between 500 K and 300 K. Find its maximum efficiency as a decimal fraction.

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 limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. Packets are supplied or generated teaching evidence, not measured apparatus. No heating, pressure vessels, physical oscillators, sound exposure or construction is requested. Public checks and saved reports do not establish supervised practical skill, equipment safety or independent mastery.

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