Entropy in free expansion

Problem

An ideal gas occupies half a container, while the other half is empty. The partition is removed and the gas undergoes free expansion. State whether its entropy increases, decreases or stays the same.

Solution

This is an irreversible adiabatic free expansion: the container is isolated, so $Q=0$, and the gas does no work against the vacuum, $L=0$. For an ideal gas the internal energy depends only on $T$, so $\Delta U=0\Rightarrow T$ stays constant. The volume doubles, so we compute $\Delta S$ using the initial and final states: $\Delta S_\text{gas}=nR\ln\frac{V_B}{V_A}=nR\ln 2>0$ Entropy increases: more available volume means more positional microstates accessible to the molecules. Since the system is isolated, there is no exchange with the surroundings, so $\Delta S_\text{tot}=\Delta S_\text{gas}>0$: an irreversible process. $\ev{\Delta S>0}$

Links

Topics: Entropy and the second law Concepts: Entropy · Thermodynamic processes Skills: Micro-macro interpretation Objects: Ideal gas