Thermodynamics · Carnot Engine Chain

Follow rejected heat through five engines and recover η

Each identical Carnot engine absorbs the heat rejected by the previous one. If one engine has efficiency η, the heat passed forward is multiplied by 1 − η at every stage.

2026 Paper 1 Q10 · five identical Carnot engines
Engine efficiency η
Current value 0.333
Heat passed forward
Engine 1 absorbs Q₀ and rejects Q₁ = (1 − η)Q₀. Each following engine repeats the same multiplication.
Relationηnet = 1 − (1 − η)⁵
Resultη = 1/3
Heat flowremaining heat to next engine
Work outputη times absorbed heat
Final rejectionQ₅ = (1 − η)⁵Q₀

Derivation path

01 / ONE ENGINE

Work and rejected heat

If an engine absorbs Q, it produces ηQ work and rejects (1 − η)Q.

Wᵢ = ηQᵢ₋₁, Qᵢ = (1 − η)Qᵢ₋₁
02 / FIVE ENGINES

Rejected heat forms a geometric chain

After five identical engines, the heat finally rejected is Q₅ = (1 − η)⁵Q₀.

W/Q₀ = 1 − Q₅/Q₀ = 1 − (1 − η)⁵
03 / GIVEN NET EFFICIENCY

Solve the fifth power

The given value leaves Q₅/Q₀ = 32/243 = (2/3)⁵.

1 − η = 2/3, so η = 1/3
Final value
ηnet = 1 − (1 − η)5 = 211/243.
Therefore (1 − η)5 = 32/243 = (2/3)5, giving η = 1/3.
Answer: η = 1/3 ≈ 0.333