A thermally insulated container is divided into two compartments by a movable piston. One side contains $n$ moles of an ideal gas at temperature $T$, and the other side is a vacuum. If the piston is released and the gas expands to fill the entire volume, what is the final temperature of the gas?

Solution

1. The gas undergoes free expansion into a vacuum, so the heat exchanged $Q=0$ and the work done $W=0$.
2. According to the first law of thermodynamics, $$\begin{array}{rl}\Delta U& =Q-W\\ & =0\end{array}$$.
3. For an ideal gas, the internal energy depends only on temperature, so $$\begin{array}{rl}\Delta U& =n{C}_v\Delta T\\ & =0\end{array}$$.
4. Since $n$ and $C_v$ are non-zero, it follows that $\Delta T=0$, meaning the final temperature remains $T$.
5. Hence the answer is (A).