Formula
The efficiency of a heat engine is the work divided by the heat taken from the source: η = (Q₁ − Q₂) / Q₁.
ENT-level problem
A gas absorbs a quantity of heat $Q = 500$ J and at the same time does work $A = 200$ J. Find the change in internal energy of the gas.
The first law of thermodynamics: $Q = \Delta U + A$, hence
$$\Delta U = Q - A = 500 - 200 = 300 \text{ J}$$
The internal energy of the gas increased by 300 J.
A heat engine takes heat Q₁ from the source, converts part of it into work A = Q₁ − Q₂ and rejects the rest Q₂ to the cold reservoir. The efficiency shows what fraction of the heat became work.
By the second law of thermodynamics the efficiency is always below one: heat cannot be converted into work completely. Internal combustion engines have η ≈ 20–40 %, power plants — up to 40–45 %.
By the second law some heat must be rejected to the cold reservoir (Q₂ > 0), so η = A / Q₁ < 1.
A = Q₁ − Q₂. With the efficiency known: A = η · Q₁. For Q₁ = 1000 J and η = 0.4 the work is 400 J.
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