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(A) The process described is free expansion of an ideal gas into a vacuum.Work done by a gas is given by the formula:$W = \int P_{ext} dV$Since the ga

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No work is done by $X$.

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(A) The process described is free expansion of an ideal gas into a vacuum.Work done by a gas is given by the formula:$W = \int P_{ext} dV$Since the gas expands into a vacuum,the external pressure $P_{ext} = 0$.Therefore,the work done is:$W = \int 0 \cdot dV = 0$Since the container is fully insulated,there is no heat exchange with the surroundings,so $Q = 0$.According to the first law of thermodynamics,$\Delta U = Q - W = 0 - 0 = 0$.For an ideal gas,internal energy $U$ depends only on temperature $(U \propto T)$. Since $\Delta U = 0$,the temperature remains constant $(\Delta T = 0)$.Thus,no work is done by the gas,and its internal energy and temperature remain unchanged.

$A$ closed container is fully insulated from the outside. One half of it is filled with an ideal gas $X$ separated by a plate $P$ from the other half $Y$,which contains a vacuum as shown in the figure. When $P$ is removed,$X$ expands into $Y$. Which of the following statements is correct?

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