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

The gas is slowly compressed while permitted to exchange heat with its surroundings, which indicates an isothermal process.

For an isothermal process, the equation of state is $p_1 V_1 = p_0 V_0$.

Given $V_1 = \dfrac{V_0}{2}$, the pressure after the first compression is:

$p_1 = p_0 \left( \dfrac{V_0}{V_0 / 2} \right) = 2 p_0$

The gas is then suddenly compressed, which indicates an adiabatic process.

For an adiabatic process, the equation of state is $p_2 V_2^{\gamma} = p_1 V_1^{\gamma}$.

Given $V_2 = \dfrac{V_0}{4}$, the final pressure is:

$p_2 = p_1 \left( \dfrac{V_1}{V_2} \right)^{\gamma}$

Substituting the values of $p_1$, $V_1$, and $V_2$:

$p_2 = 2 p_0 \left( \dfrac{V_0 / 2}{V_0 / 4} \right)^{\gamma}$

$p_2 = 2 p_0 (2)^{\gamma} = 2^{\gamma + 1} p_0$