A given mass of a gas, with a specific heat ratio C_p / C_V = γ ,...

The initial state of the gas has pressure $p_0$, volume $V_0$, and temperature $T_0$. Since the gas is permitted to exchange heat with its surroundings, the surroundings are at the initial temperature $T_0$.

The gas is first suddenly compressed from $V_0$ to $\dfrac{V_0}{2}$. This sudden process is adiabatic, causing the temperature of the gas to increase above $T_0$.

Subsequently, the gas is slowly compressed to a final volume of $\dfrac{V_0}{4}$. Because this process is slow and the gas can exchange heat with its surroundings, it will reach thermal equilibrium with the surroundings. Thus, the gas cools down and its final temperature becomes equal to the surrounding temperature $T_0$.

Since the initial and final temperatures of the gas are both $T_0$, the overall change from the initial to the final state can be treated using Boyle's law ($T = \text{constant}$):

$p_0 V_0 = p_f V_f$

Substituting the final volume $V_f = \dfrac{V_0}{4}$:

$p_0 V_0 = p_f \left(\dfrac{V_0}{4}\right)<br><br>$p_f = 4 p_0$<br><br>Answer:$4 p_0