Physics - Heat & Thermo Question with Solution | TestHub

Work done by gas in adiabatic process is:

$W=-\frac{\left(P_2{V}_2-P_1{V}_1\right)}{\gamma -1}$

  $=-\frac{nR\left(T_2-T_1\right)}{\gamma -1}$

 For an adiabatic process, $T_1{V_1}^{\gamma -1}=T_2{V_2}^{\gamma -1}$

$$\begin{gathered} \Rightarrow T{V}^{\frac{3}{2}-1}=T_2{\left(2V\right)}^{\frac{3}{2}-1} \\ \Rightarrow T{V}^{\frac{1}{2}}=T_2{\left(2V\right)}^{\frac{1}{2}} \\ \Rightarrow T_2=\frac{T}{\sqrt{2}} \end{gathered}$$

Now, work done is $W=\frac{R\left(T_1-T_2\right)}{\gamma -1}=\frac{R\left(T-{\displaystyle \frac{T}{\sqrt{2}}}\right)}{{\displaystyle \frac{1}{2}}}=RT\left(2-\sqrt{2}\right)$