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

For the isothermal process, the temperature remains constant, so $p V = \text{constant}$.

Let the volume after the isothermal process be $V_1$.

$p_0 V_0 = \left(\dfrac{p_0}{2}\right) V_1$

$V_1 = 2 V_0$

For the subsequent adiabatic process, $p V^{\gamma} = \text{constant}$.

Let the final volume be $V_2$.

$p_1 V_1^{\gamma} = p_2 V_2^{\gamma}$

\left(\dfrac{p_0}{2}\right) (2 V_0)^{\gamma} = \left(\dfrac{p_0}{4}\right) V_2^{\gamma}$<br><br>$\dfrac{1}{2} \times 2^{\gamma} V_0^{\gamma} = \dfrac{1}{4} V_2^{\gamma}$<br><br>$V_2^{\gamma} = 4 \times \dfrac{1}{2} \times 2^{\gamma} V_0^{\gamma}$<br><br>$V_2^{\gamma} = 2 \times 2^{\gamma} V_0^{\gamma}$<br><br>$V_2^{\gamma} = 2^{\gamma + 1} V_0^{\gamma}$<br><br>$V_2 = 2^{(\gamma + 1)/\gamma} V_0