A sample of 1 mole gas at temperature T is adiabatically expanded… - Vidyadip

MCQ

A sample of $1$ mole gas at temperature $T$ is adiabatically expanded to double its volume. If adiabatic constant for the gas is $\gamma=\frac{3}{2}$, then the work done by the gas in the process is:

A
$RT [2-\sqrt{2}]$
B
$\frac{ R }{ T }[2-\sqrt{2}]$
C
$RT [2+\sqrt{2}]$
D
$\frac{ T }{ R }[2+\sqrt{2}]$

✓

Answer

$ TV ^{\gamma-1}=\text { constant }$
$\Rightarrow T ( V )^{\frac{3}{2}-1}= T _{ f }(2 V)^{\frac{3}{2}-1}$
$\Rightarrow TV ^{\frac{1}{2}}= T _{ f }(2)^{\frac{1}{2}}(V)^{\frac{1}{2}}$
$\Rightarrow T_{ f }=\left(\frac{ T }{\sqrt{2}}\right)$
$\text { Now, W.D. }=\frac{ nR \Delta T}{1-\gamma}=\frac{1 \cdot R \left[\frac{ T }{\sqrt{2}}- T \right]}{1-\frac{3}{2}}$
$\Rightarrow \text { W.D. }=2 RT \left[1-\frac{1}{\sqrt{2}}\right]$
$\Rightarrow \text { W.D. }= RT [2-\sqrt{2}]$

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