Question

A sample of gas at temperature 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 : \((\mu=1\) mole)

(1) \(\mathrm{RT}[\sqrt{2}-2]\)

(2) \(\mathrm{RT}[1-2 \sqrt{2}]\)

(3) \(\mathrm{RT}[2 \sqrt{2}-1]\)

(4) \(\mathrm{RT}[2-\sqrt{2}]\)

Answer 1

Correct option is :  (4) \(\mathrm{RT}[2-\sqrt{2}]\)   

\(\mathrm{W}=\frac{\mathrm{nR} \Delta \mathrm{T}}{1-\gamma}\)

\(\mathrm{TV}^{\gamma-1}=\) cons tan \( \mathrm{t}=\mathrm{T}_{\mathrm{f}}(2 \mathrm{V})^{\gamma-1}\)

\(\mathrm{T}_{\mathrm{f}}=\mathrm{T}\left(\frac{1}{2}\right)^{1 / 2}=\frac{\mathrm{T}}{\sqrt{2}}\)

\(\mathrm{W}=\frac{\mathrm{R}\left(\frac{\mathrm{T}}{\sqrt{2}}-\mathrm{T}\right)}{1-\frac{3}{2}}=2 \mathrm{RT} \frac{(\sqrt{2}-1)}{\sqrt{2}}\)

\(=\mathrm{RT}(2-\sqrt{2}) \)