A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with a frictionless…
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
$\left(\frac{L_1}{L_2}\right)$
$\left(\frac{L_2}{L_1}\right)^{2 / 3}$
$\left(\frac{L_1}{L_2}\right)^{2 / 3}$
$\left(\frac{L_2}{L_1}\right)$
Solution
For adiabatic process:
$\begin{aligned} & T V Y-1=\text { constant } \\ & \Rightarrow T_1 V_2-1=T_2 V_2^{\gamma-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{A}{A} \frac{L_2}{L_1}\right)^{\frac{5}{3}-1} \\ & \Rightarrow \frac{T_1}{T_2}=\left(\frac{L_2}{L_1}\right)^{\frac{2}{3}}\end{aligned}$