`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)ltV_(av)ltV_(mp)`
B. `V_(rmr)gtV_(av)gtV_(mp)`
C. `V_(mp)ltV_(rms)ltV_(av)`
D. `V_(mp)gtV_(rms)gtV_(av)`

Explanation

To solve the problem regarding the relationship between the root mean square speed (v_rms), average speed (v_av), and most probable speed (v_mp) of gas molecules obeying Maxwellian velocity distribution, we can follow these steps:

### Step 1: Understand the definitions and formulas
- **Root Mean Square Speed (v_rms)**: This is defined as:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a gas molecule.

- **Most Probable Speed (v_mp)**: This is defined as:
\[ v_{mp} = \sqrt{\frac{2kT}{m}} \]

- **Average Speed (v_av)**: This is defined as:
\[ v_{av} = \sqrt{\frac{8kT}{5m}} \]

### Step 2: Calculate the ratios of the speeds
To compare these speeds, we can express them in terms of a common variable. Let's denote \( x = \sqrt{\frac{kT}{m}} \). Then we can rewrite the speeds as:
- \( v_{rms} = \sqrt{3} x \)
- \( v_{mp} = \sqrt{2} x \)
- \( v_{av} = \sqrt{\frac{8}{5}} x \)

### Step 3: Compare the values
Now, we can compare the values of \( v_{rms} \), \( v_{mp} \), and \( v_{av} \):
- \( v_{rms} = \sqrt{3} x \)
- \( v_{mp} = \sqrt{2} x \)
- \( v_{av} = \sqrt{\frac{8}{5}} x \)

To determine the order of these speeds, we can calculate their approximate numerical values:
- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{\frac{8}{5}} \approx 1.264 \)

### Step 4: Establish the order
From the numerical values:
- \( v_{rms} \) (1.732) > \( v_{av} \) (1.264) > \( v_{mp} \) (1.414)

Thus, we conclude:
\[ v_{rms} > v_{av} > v_{mp} \]

### Step 5: Conclusion
The correct statement regarding the relationship between the speeds is:
- The root mean square speed is the greatest, followed by the average speed, and the most probable speed is the least.

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

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