`v_(rms), v_(av)` and `v_(mp)` are root mean square average and most probable speeds of molecules of a gas obeying Maxwellian velocity distribution. Which of the following statements is correct ?

Options

A. `v_(rms) lt v_(av) lt v_(mp)`
B. `v_(rms) gt v_(av) gt v_(mp)`
C. `v_(mp) lt v_(rms) lt v_(av)`
D. `v_(mp) gt v_(rms) gt v_(av)`

Explanation

To solve the problem, we need to compare the three velocities: root mean square velocity (\(v_{rms}\)), average velocity (\(v_{av}\)), and most probable velocity (\(v_{mp}\)) of gas molecules that follow the Maxwellian velocity distribution.

### Step-by-Step Solution:

1. **Identify the formulas for each velocity:**
- The formula for root mean square velocity (\(v_{rms}\)) is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- The formula for average velocity (\(v_{av}\)) is:
\[ v_{av} = \sqrt{\frac{8RT}{\pi M}} \]
- The formula for most probable velocity (\(v_{mp}\)) is:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]

2. **Calculate the numerical values of each velocity:**
- For \(v_{rms}\):
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \approx 1.73 \sqrt{\frac{RT}{M}} \]
- For \(v_{mp}\):
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \approx 1.41 \sqrt{\frac{RT}{M}} \]
- For \(v_{av}\):
\[ v_{av} = \sqrt{\frac{8RT}{\pi M}} \approx 1.6 \sqrt{\frac{RT}{M}} \]

3. **Compare the values:**
- Now we have:
- \(v_{rms} \approx 1.73 \sqrt{\frac{RT}{M}}\)
- \(v_{av} \approx 1.6 \sqrt{\frac{RT}{M}}\)
- \(v_{mp} \approx 1.41 \sqrt{\frac{RT}{M}}\)

4. **Determine the order of the velocities:**
- From the calculated values, we can see:
\[ v_{rms} > v_{av} > v_{mp} \]

5. **Conclusion:**
- The correct statement is that the root mean square velocity is the highest, followed by the average velocity, and the most probable velocity is the lowest.

### Final Answer:
The correct statement is that \(v_{rms} > v_{av} > v_{mp}\).

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