Verified 30 May 2026.
For an ideal gas undergoing an adiabatic process where $PV^{\gamma} = \text{const}$ and $\gamma = \frac{5}{3}$, the work done when volume changes from $V_1$ to $2V_1$ is:
$\frac{P_1 V_1 \left(1 - 2^{-2/3}\right)}{\frac{2}{3}}$
$\frac{3P_1 V_1}{2}\left(1 - 2^{-2/3}\right)$
$P_1 V_1 \ln 2$
$\frac{2P_1 V_1}{\gamma - 1}$
For an adiabatic process: $PV^{\gamma} = P_1 V_1^{\gamma} = \text{const}$
$$W = \int_{V_1}^{2V_1} P \, dV = P_1 V_1^{\gamma} \int_{V_1}^{2V_1} V^{-\gamma} \, dV$$
$$= P_1 V_1^{\gamma} \cdot \frac{V^{1-\gamma}}{1-\gamma}\Bigg|_{V_1}^{2V_1}$$
$$= \frac{P_1 V_1}{\gamma - 1}\left(1 - 2^{1-\gamma}\right) = \frac{3P_1 V_1}{2}\left(1 - 2^{-2/3}\right)$$
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