The ratio of most probable velocity to that of average velocity is

Options

A. `(pi)/(2)`
B. `(2)/(pi)`
C. `(sqrtpi)/(2)`
D. `(2)/(sqrtpi)`

Explanation

To find the ratio of the most probable velocity to the average velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formulas for most probable velocity and average velocity:**
- The formula for the most probable velocity (v_mp) is given by:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
- The formula for the average velocity (v_avg) is given by:
\[ v_{avg} = \frac{8RT}{\pi M} \]

2. **Set up the ratio of most probable velocity to average velocity:**
- We need to find the ratio:
\[ \text{Ratio} = \frac{v_{mp}}{v_{avg}} = \frac{\sqrt{\frac{2RT}{M}}}{\frac{8RT}{\pi M}} \]

3. **Simplify the ratio:**
- First, we can rewrite the ratio:
\[ \text{Ratio} = \frac{\sqrt{2RT}}{M} \times \frac{\pi M}{8RT} \]
- Cancel \(M\) and \(RT\) from the numerator and denominator:
\[ \text{Ratio} = \frac{\sqrt{2}}{8} \times \pi \]

4. **Further simplify the expression:**
- This can be simplified to:
\[ \text{Ratio} = \frac{\sqrt{2\pi}}{8} \]
- To express this in a more standard form, we can write:
\[ \text{Ratio} = \frac{\sqrt{\pi}}{4} \]

5. **Final Result:**
- Therefore, the ratio of the most probable velocity to the average velocity is:
\[ \text{Ratio} = \frac{\sqrt{\pi}}{2} \]