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 $L_{1},\ L_{2}$ are the lengths of the gas column before and after expansion respectively, then $T_{1}/T_{2}$ is given by
A) $\left( \frac{L_{1}}{L}_{2} \right)^{2/3}$
B) $\left( \frac{L_{1}}{L}_{2} \right)$
C) $\frac{L_{1}}{L}_{2}$
D) $\left( \frac{L_{2}}{L}_{1} \right)^{2/3}$
Explanation
(d) During adiabatic expansion
$TV^{\gamma - 1} =$ constant of $T_{2}V_{2}^{\gamma - 1} = T_{1}V_{1}^{\gamma - 1}$
$\frac{T_{1}}{T_{2}} = \left( \frac{V_{2}}{V_{1}} \right)^{\gamma - 1}$
For monoatomic gas, $\gamma = 5/3$
$\frac{T_{1}}{T_{2}} = \left( \frac{AL_{2}}{AL_{1}} \right)^{5/3 - 1} = \left( \frac{L_{2}}{L_{1}} \right)^{2/3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.