Question

At 256 K, the rms speed of ${SO2}$ molecules is 3.16$\times 10^2$ m/s. What is the most probable velocity (in m/s) of the same gas molecules at the same temperature?

• A. 2.911$\times 10^2$
• B. 2.58$\times 10^2$
• C. 3.16$\times 10^2$
• D. 1.29$\times 10^2$

Hint:

Memorize the speed ratios for an ideal gas: $v_{\text{mp}} : v_{\text{avg}} : v_{\text{rms}} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}$.

Answer 1

The Correct Option is B

For an ideal gas, the root mean square speed is $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$, and the most probable speed is $v_{\text{mp}} = \sqrt{\frac{2RT}{M}}$.
Given $v_{\text{rms}} = 3.16 \times 10^2$ m/s:
\[ \frac{v_{\text{mp}}}{v_{\text{rms}}} = \sqrt{\frac{2RT/M}{3RT/M}} = \sqrt{\frac{2}{3}} \] \[ v_{\text{mp}} = v_{\text{rms}} \times \sqrt{\frac{2}{3}} = (3.16 \times 10^2) \times \sqrt{\frac{2}{3}} \] \[ \sqrt{\frac{2}{3}} \approx 0.8165, \quad v_{\text{mp}} = (3.16 \times 10^2) \times 0.8165 \approx 2.58 \times 10^2 \text{ m/s}. \] This matches option (2).