A sample of one mole of gas at temperature T is adiabatically compressed to half of its volume. The work done on the gas in the process is (given $\gamma =\frac{3}{2}$ )

For one mole of ideal gas, the work done in adiabatic process
$W=\frac{R}{r-1}[T_1-T_2]$ but $T_1{V}_1^{{ }^{\gamma -1}}=T_2{V}_2^{\gamma -1}$ 
$T_2=T_1{\left(\frac{V_1}{V_2}\right)}^{\gamma -1}={\left(\frac{V}{V/2}\right)}^{3/2-1}T=T{\left(2\right)}^{1/2}=\sqrt{2}T$ 
$W=\frac{R}{\frac{3}{2}-1}[T-\sqrt{2}T]$ 
$W=2RT[1-\sqrt{2}]\Rightarrow W=RT(2-2\sqrt{2})$