# Explain why the entropy of an ideal gas increases when it undergoes a free expansion into a vacuum, even though no heat is exchanged with the surroundings and no work is done.

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

> An ideal gas is initially confined to one half of an insulated rigid container. A partition separating the gas from the evacuated half is suddenly removed, and the gas expands to fill the entire container.

## Mark scheme (4 marks)

1. In a free expansion, no work is done (W = 0) and no heat is exchanged (Q = 0), so the internal energy and temperature of an ideal gas remain unchanged.
2. The process is irreversible because the gas will not spontaneously return to the original half of the container; the reverse process is never observed.
3. After expansion the gas molecules occupy a greater volume, so there are significantly more possible microstates (ways to arrange positions and momenta of the molecules) available to the system.
4. Since entropy is related to the number of microstates by S = k_B ln Ω, an increase in the number of microstates means entropy increases; the second law states that for an isolated system entropy cannot decrease, and here it increases.

## Key terms

- [free expansion](https://www.gradenine.co.uk/glossary/free-expansion)
- [entropy](https://www.gradenine.co.uk/glossary/entropy)

## 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)
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