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(A) This process is known as free expansion of an ideal gas into a vacuum.Since the walls are insulating,the heat exchange with the surroundings is ze

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$P/2, T$

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(A) This process is known as free expansion of an ideal gas into a vacuum.Since the walls are insulating,the heat exchange with the surroundings is zero $(Q = 0)$.Since the gas expands into a vacuum,the work done by the gas is zero $(W = 0)$.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. Since $\Delta U = 0$,the temperature remains constant,so $T_2 = T$.Using the ideal gas law for the initial and final states: $P_1 V_1 = P_2 V_2$.Initially,the gas occupies volume $V_1 = V$ at pressure $P_1 = P$.After opening the valve,the gas occupies the total volume $V_2 = 2V$.Substituting these values: $P \cdot V = P_2 \cdot (2V)$.Therefore,$P_2 = P/2$.

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