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 ${L_1}$ and ${L_2}$ are the lengths of the gas column before and after expansion respectively, then ${T_1}/{T_2}$ is given by
IIT 2000,JEE MAIN 2021, Diffcult
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(d) ${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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