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  4. A monoatomic ideal gas, initially at temperature T1, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature T2 by releasing the piston suddenly. If L1 and L2 are the lengths of the gas column before and after expansion respectively, then T1 / T2 is given by

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Q. 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

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Solution:

During adiabatic expansion, we known
$TV^{\gamma -1}=$ constant or $T_1\,V_1^{\gamma-1}=T_2V_2^{\gamma-1}$
For a monoatomic gas $\gamma=\frac{5}{3}$
$\therefore \frac{T_1}{T_2}=\left(\frac{V_2}{V_1}\right)^{\gamma-1} =\left(\frac{AL_2}{AL_1}\right)^{(5/3)-1}$
$$(A = Area of cross-section of piston)
$=\left(\frac{L_2}{L_1}\right)^{2/3}$