A monoatomic ideal gas, initially at temperature $T_{1}$ is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $T_{2}$ by releasing the piston suddenly. If $L_{1}$ , $L_{2}$ are the lengths of the gas column before and after expansion respectively, then $T_{1} {/T}_{2}$ is given by :

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${(L_{1} {/L}_{2})}^{2/3}$

$L_{2} {/L}_{1}$

$L_{1} {/L}_{2}$

${(L_{2} {/L}_{1})}^{2/3}$

answer is D.

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Detailed Solution

$I n a d i a b a t i c p r o c e s s, r e l a t i o n b e t w e e n t e m p e r a t u r e T a n d V o l u m e V i s$

$T V^{γ - 1} = c o n s \tan t, h e r e γ i s s p e c i f i c h e a t r a t i o$

$\Rightarrow T_{1} {V_{1}}^{γ - 1} = T_{2} {V_{2}}^{γ - 1}$

$\Rightarrow \frac{T_{1}}{T_{2}} = {(\frac{V_{2}}{V_{1}})}^{γ - 1}$

$v o l u m e V = A r e a o f b a s e x h e i g h t g i v e n V_{1} = A L_{1}$

$V_{2} = A L_{2}, f o r m o n o a t o m i c g a s γ = \frac{5}{3}$

$\Rightarrow \frac{T_{1}}{T_{2}} = {(\frac{{AL}_{2}}{{AL}_{1}})}^{5 / 3 - 1}$

$\Rightarrow \frac{T_{1}}{T_{2}} = {(\frac{L_{2}}{L_{1}})}^{2 / 3}$

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