Question

The rms velocity (\( u_{\text{rms}} \)), mean velocity (\( u_{\text{av}} \)), and most probable velocity (\( u_{\text{mp}} \)) of a gas differ from each other at a given temperature. Which of the following ratios regarding them is correct?

• A. \( \frac{u_{\text{rms}}}{u_{\text{av}}} = 1.20 \)
• B. \( \frac{u_{\text{av}}}{u_{\text{mp}}} = 1.12 \)
• C. \( \frac{u_{\text{rms}}}{u_{\text{mp}}} = 1.15 \)
• D. \( \frac{u_{\text{av}}}{u_{\text{rms}}} = 0.98 \)

Hint:

The different velocities in gas laws represent different statistical speeds of gas molecules at a given temperature.

Answer 1

The Correct Option is B

Step 1: Define Velocity Relations The relationships between the different velocities are: \[ u_{\text{rms}} = \sqrt{\frac{3RT}{M}} \] \[ u_{\text{av}} = \sqrt{\frac{8RT}{\pi M}} \] \[ u_{\text{mp}} = \sqrt{\frac{2RT}{M}} \]
Step 2: Compute Ratios
- \( \frac{u_{\text{rms}}}{u_{\text{av}}} = 1.08 \)
- \( \frac{u_{\text{av}}}{u_{\text{mp}}} = 1.12 \)
- \( \frac{u_{\text{rms}}}{u_{\text{mp}}} = 1.22 \) Since the second ratio is correct, the answer is \( \frac{u_{\text{av}}}{u_{\text{mp}}} = 1.12 \).