Question

An ideal gas (γ = 1.5) is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of molecules 2 times

• A. 4 times
• B. 16 times
• C. 8 times
• D. 2 times

Answer 1

Answer: (B)

16 times

Answer 2

To reduce the rms velocity two times, temperature should be reduced by four times (As $v_{rms} \propto \sqrt{T}$)

∴ $T_{1} = T$ $T_{2} = \frac{T}{4}$, $V_{1} = V$

From adiabatic law $TV^{\gamma - 1} =$constant we get

$\left( \frac{V_{2}}{V_{1}} \right)^{\gamma - 1} = \frac{T_{1}}{T_{2}} = 4$

⇒ $\frac{V_{2}}{V_{1}} = (4)^{\frac{1}{\gamma - 1}}$ [γ = 3/2 given]

⇒ $V_{2} = V_{1}(4)^{\frac{1}{3/2 - 1}} = V_{1}(4)^{2} = 16V_{1}$ ∴ $\frac{V_{2}}{V_{1}} = 16$