Question

An ideal gas ($\gamma$ = 1.5) is expanded adiabatically. To reduce root mean square velocity of molecules two times, the gas should be expanded

• A. 20 times
• B. 16 times
• C. 12 times
• D. 8 times

Answer 1

Answer: (B)

16 times

Answer 2

To reduce the root mean square (rms) velocity two times, the temperature must be reduced by a factor of 4, because $v_{\text{rms}} \propto \sqrt{T}$.

Using the adiabatic law for an ideal gas, $T \cdot V^{\gamma - 1} = \text{constant}$, where $\gamma = 1.5$, we have:

$$\frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma - 1}$$

Given that $\frac{T_2}{T_1} = \frac{1}{4}$, we substitute:

$$\frac{1}{4} = \left(\frac{V_1}{V_2}\right)^{0.5}$$

Squaring both sides:

$$\frac{1}{16} = \frac{V_1}{V_2} \implies \frac{V_2}{V_1} = 16$$

Therefore, the gas should be expanded 16 times its initial volume.