A sample of gas at temperature $\mathrm{T}$ is adiabatically expanded to double its volume. Adiabatic constant for the gas is $\gamma=3 / 2$. The work done by the gas in the process is : $(\mu=1 \mathrm{~mole})$
JEE MAIN 2024, Diffcult
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$\mathrm{W}=\frac{\mathrm{nR} \Delta \mathrm{T}}{1-\gamma}$
$\mathrm{TV}^{\gamma-1}=\text { constant }=\mathrm{T}_{\mathrm{f}}(2 \mathrm{~V})^{\gamma-1}$
$\mathrm{~T}_{\mathrm{f}}=\mathrm{T}\left(\frac{1}{2}\right)^{1 / 2}=\frac{\mathrm{T}}{\sqrt{2}}$
$\mathrm{~W}=\frac{\mathrm{R}\left(\frac{\mathrm{T}}{\sqrt{2}}-\mathrm{T}\right)}{1-\frac{3}{2}}=2 \mathrm{RT} \frac{(\sqrt{2}-1)}{\sqrt{2}}$
$=\mathrm{RT}(2-\sqrt{2})$
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