A monatomic 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

{(\frac{L_{1}}{L_{2}})}^{2 / 3}

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\frac{L_{1}}{L_{2}}

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\frac{L_{2}}{L_{1}}

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{(\frac{L_{2}}{L_{1}})}^{2 / 3}

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Solution

The correct option is A ${(\frac{L_{2}}{L_{1}})}^{2 / 3}$
Here $T V^{γ - 1} = c o n s t a n t$
As $γ = 5 / 3$ , hence $T V^{2 / 3} = c o n s t a n t$
Now $T_{1} L_{1}^{2 / 3} = T_{2} L_{2}^{2 / 3} (∵ V \propto L)$
Hence $\frac{T_{1}}{T_{2}} = {(\frac{L_{2}}{L_{1}})}^{2 / 3}$ .

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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 $\frac{T_{1}}{T_{2}}$ is given by