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A monoatomic ideal gas, initially at temperature $T_1$, is enclosed in a cylinder fitted with a mmminmin frictionless piston. The gas is allowed to expand adiabatically to a temperature. ${ }_2$ by releasing the piston suddenly. If $L_1$ and $L_2$ are the lengths of the gas column before and after expansion respectively, then $T_1 / T_2$ is given by
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The correct answer is:
$\left(\frac{L_2}{L_1}\right)^{2 / 3}$
$T_1 V_1^{\gamma-1}=T_2 V_2^{\gamma-1} \Rightarrow \frac{T_1}{T_2}=\left(\frac{V_2}{V_1}\right)^{\gamma-1}=\left(\frac{L_2 A}{L_1 A}\right)^{\frac{5}{3}-1}=\left(\frac{L_2}{L_1}\right)^{\frac{2}{3}}$
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