The ratio of root mean square speed, average speed and most probable speed is

Options

A. `3pi : 8 :2pi`
B. `pi : 2pi : 3`
C. `3pi :2pi: 8`
D. `3pi:2pi:6`

Explanation

To find the ratio of root mean square speed (Vrms), average speed (Vavg), and most probable speed (Vmp), we will use the formulas for each type of speed and then derive the ratio.

### Step-by-Step Solution:

1. **Identify the Formulas**:
- The formula for root mean square speed (Vrms) is:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
- The formula for average speed (Vavg) is:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
- The formula for most probable speed (Vmp) is:
\[ V_{mp} = \sqrt{\frac{2RT}{M}} \]

2. **Set Up the Ratio**:
We want to find the ratio:
\[ V_{rms} : V_{avg} : V_{mp} \]

3. **Substitute the Formulas into the Ratio**:
Substituting the formulas into the ratio gives:
\[ \sqrt{\frac{3RT}{M}} : \sqrt{\frac{8RT}{\pi M}} : \sqrt{\frac{2RT}{M}} \]

4. **Factor Out Common Terms**:
Since \(RT\) and \(M\) are common in all terms, we can factor them out:
\[ \sqrt{RT} : \sqrt{\frac{8RT}{\pi}} : \sqrt{2RT} \]

5. **Simplify the Ratio**:
This simplifies to:
\[ \sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \]

6. **Convert to a Common Form**:
To express this ratio in a more manageable form, we can square each term (since squaring does not change the ratio):
\[ 3 : \frac{8}{\pi} : 2 \]

7. **Multiply to Eliminate Fractions**:
To eliminate the fraction, multiply through by \(\pi\):
\[ 3\pi : 8 : 2\pi \]

8. **Final Ratio**:
Thus, the final ratio of root mean square speed, average speed, and most probable speed is:
\[ 3\pi : 8 : 2\pi \]