Question

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

• A. $(L_1 /L_2)^{2/3}$
• B. $(L_1 /L_2)$
• C. $L_2/L_1$
• D. $(L_2 /L_1)^{2/3}$

Answer 1

Answer: (D)

$(L_2 /L_1)^{2/3}$

Answer 2

During adiabatic expansion, we know
$\, \, \, \, \, TV^{\gamma -1}=constant \, \, \, \, or \, \, T_1V_1^{\gamma-1}=T_2V_2^{\gamma-1}$
For a monoatomic gas $\gamma=\frac{5}{3}$
$\therefore \, \, \, \, \, \, \, \frac{T_1}{T_2}=\bigg(\frac{V_2}{V_1}\bigg)^{\gamma-1} =\bigg(\frac{AL_2}{AL_1}\bigg)^{(5/3)-1}$
\hspace20mm (A = Area of cross-section of piston)
\hspace20mm $=\bigg(\frac{L_2}{L_1}\bigg)^{2/3}$