# A heat engine operates between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. The engine performs one complete cycle. Explain why the thermal efficiency of this engine must always be less than 1, even if all mechanical components are frictionless.

> IB DP Physics Higher Level (2023 syllabus) — B.4 Thermodynamics (HL only) · Explain · 4 marks

> A heat engine absorbs heat Q_H from a hot reservoir, does work W on the surroundings, and expels heat Q_C to a cold reservoir in each complete cycle.

## Mark scheme (4 marks)

1. The second law of thermodynamics requires that the total entropy of an isolated system cannot decrease, so some heat must always be transferred to the cold reservoir.
2. Because Q_C > 0 (heat must be expelled to the cold reservoir), not all of Q_H can be converted to work, so W < Q_H.
3. Efficiency η = W/Q_H = 1 − Q_C/Q_H, and since Q_C/Q_H > 0, it follows that η < 1.
4. The maximum possible efficiency is the Carnot efficiency η_C = 1 − T_C/T_H, which is less than 1 for any finite T_C > 0, confirming that a perfect (100% efficient) heat engine is thermodynamically impossible.

## Key terms

- [thermal efficiency](https://www.gradenine.co.uk/glossary/thermal-efficiency)

## Related

- [Revision notes for IB DP Physics Higher Level (2023 syllabus)](https://www.gradenine.co.uk/learn)
- [How to answer "Explain" questions](https://www.gradenine.co.uk/tools/command-word-cheatsheet)
- [Practice this with AI marking (free)](https://www.gradenine.co.uk/start)

---
Source: [GradeNine](https://www.gradenine.co.uk/q/a-heat-engine-operates-between-a-d3ba89cc) · Published by Druglandscape Ltd.