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 \(L_{2}\) be the lengths of the gas column before and after expansion respectively, then \(\frac{T_{1}}{T_{2}}\) is

• A. \(\left(\frac{L_{1}}{L_{2}}\right)^{\frac{2}{3}}\)
• B. \(\left(\frac{L_{1}}{L_{2}}\right)\)
• C. \(\left(\frac{L_{2}}{L_{1}}\right)\)
• D. \(\left(\frac{L_{2}}{L_{1}}\right)^{\frac{2}{3}}\)

Hint:

Use TV^(gamma-1) = constant with gamma = 5/3 and volume proportional to the gas column length.

Answer 1

The Correct Option is D

Step 1: Formula First
$\dfrac{T_1}{T_2}=\left(\dfrac{V_2}{V_1}\right)^{\gamma-1}$.

Step 2: Substitution
$\dfrac{V_2}{V_1}=\dfrac{L_2}{L_1}$ and $\gamma-1=\dfrac{2}{3}$.

Step 3: Compare the options
This matches option (D).

Final Answer:
The temperature ratio is L2 over L1 to the power two thirds. \[ \boxed{\text{(D) }\left(\dfrac{L_{2}}{L_{1}}\right)^{\frac{2}{3}}} \]