For an ideal gas undergoing an adiabatic process, if PV^γ = constant and γ = 5/3, the work done when volume changes from V₁ to 2V₁ is:

For an adiabatic process: $PV^{\gamma} = P_1 V_1^{\gamma} = \text{const}$

$$W = \int_{V_1}^{2V_1} P \, dV = P_1 V_1^{\gamma} \int_{V_1}^{2V_1} V^{-\gamma} \, dV$$

$$= P_1 V_1^{\gamma} \cdot \frac{V^{1-\gamma}}{1-\gamma}\Bigg|_{V_1}^{2V_1}$$

$$= \frac{P_1 V_1}{\gamma - 1}\left(1 - 2^{1-\gamma}\right) = \frac{3P_1 V_1}{2}\left(1 - 2^{-2/3}\right)$$