A sample of an ideal gas ( C_p / C_V = γ ) has an initial pressure...

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

Substituting the given values $p_1 = p_0$, $V_1 = V_0$, and $p_2 = \dfrac{p_0}{2}$:

$p_0 V_0^{\gamma} = \dfrac{p_0}{2} V_2^{\gamma}$

$V_2^{\gamma} = 2 V_0^{\gamma}$

$V_2 = 2^{1/\gamma} V_0$

For the subsequent isothermal process, the temperature remains constant, so $p_2 V_2 = p_3 V_3$.

Substituting $p_2 = \dfrac{p_0}{2}$ and $p_3 = \dfrac{p_0}{4}$:

$\dfrac{p_0}{2} V_2 = \dfrac{p_0}{4} V_3$

$V_3 = 2 V_2$

Substituting the value of $V_2$ obtained from the adiabatic process:

$V_3 = 2 \times 2^{1/\gamma} V_0$

$V_3 = 2^{(\gamma + 1)/\gamma} V_0$