The process consists of two steps:

Step 1: Isothermal Expansion Initial state: $P_1 = P, V_1 = V$ Final state of step 1: $V_2 = 2V$ For an isothermal process, $PV = \text{constant}$ (Boyle's Law). $P_1 V_1 = P_2 V_2$ $P \cdot V = P_2 \cdot (2V)$ $P_2 = \frac{P}{2}$

Step 2: Adiabatic Expansion Initial state of step 2: $P_2 = \frac{P}{2}, V_2 = 2V$ Final state: $V_3 = 16V$ For an adiabatic process, $PV^{\gamma} = \text{constant}$. $P_2 V_2^{\gamma} = P_3 V_3^{\gamma}$ Substituting the values: $(\frac{P}{2}) (2V)^{\gamma} = P_3 (16V)^{\gamma}$ $P_3 = \frac{P}{2} \left( \frac{2V}{16V} \right)^{\gamma}$ $P_3 = \frac{P}{2} \left( \frac{1}{8} \right)^{\frac{5}{3}}$ $P_3 = \frac{P}{2} \left( (\frac{1}{2})^3 \right)^{\frac{5}{3}}$ $P_3 = \frac{P}{2} \left( \frac{1}{2} \right)^{5}$ $P_3 = \frac{P}{2} \cdot \frac{1}{32} = \frac{P}{64}$