Question

A mono-atomic 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. \(L_{1}\) and \(L_{2}\) are the lengths of gas columns before and after the expansion, respectively. The value of \(\frac{T_{1}}{T_{2}}\) is \(\left[\gamma =\frac{5}{3}\right]\)

• A. \(\left(\frac{L_{1}}{L_{2}}\right)^{\frac{2}{3}}\)
• B. \(\left(\frac{L_{2}}{L_{1}}\right)^{\frac{2}{3}}\)
• C. \(\left(\frac{L_{1}}{L_{2}}\right)^{\frac{5}{3}}\)
• D. \(\left(\frac{L_{2}}{L_{1}}\right)^{\frac{5}{3}}\)

Hint:

For an adiabatic process the product of temperature and volume raised to the power (gamma minus 1) stays constant, and volume is proportional to the gas column length.

Answer 1

The Correct Option is B

Step 1: Check with expansion
In expansion $L_{2}>L_{1}$, so $T_{1}>T_{2}$, as in adiabatic cooling.

Step 2: Compare the options
Options A and C give a ratio below 1 for expansion. Option D uses $\gamma$ instead of $\gamma-1$. Only option B fits.

Step 3: Final answer
Only $\left(\frac{L_{2}}{L_{1}}\right)^{2/3}$ agrees with adiabatic cooling.

Final Answer:
Only $\left(\frac{L_{2}}{L_{1}}\right)^{2/3}$ agrees with adiabatic cooling. \[ \boxed{\text{(B) }\left(\dfrac{L_{2}}{L_{1}}\right)^{\frac{2}{3}}} \]