Let `barv,v_(rms) and v_p` respectively denote the mean speed. Root mean square speed, and most probable speed of the molecules in an ideal monatomic gas at absolute temperature T. The mass of a molecule is m. Then

Options

A. no molecules can have a speed greater than `sqrt2v_(rms)`
B. no molecule can have a speed less than `v_p//sqrt2`
C. `v_p lt barv lt v_(rms)`
D. the average kinetic energy of a molecules is `3/4mv_p^2`.

Explanation

To solve the problem, we need to analyze the relationships between the mean speed (\( \bar{v} \)), root mean square speed (\( v_{rms} \)), and most probable speed (\( v_p \)) of molecules in an ideal monatomic gas at absolute temperature \( T \). The mass of a molecule is denoted by \( m \).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- **Most Probable Speed (\( v_p \))**: This is the speed at which the maximum number of molecules are moving. For an ideal monatomic gas, it is given by:
\[ v_p = \sqrt{\frac{2RT}{m}} \]
- **Mean Speed (\( \bar{v} \))**: This is the average speed of all the molecules in the gas. It is given by:
\[ \bar{v} = \sqrt{\frac{8RT}{\pi m}} \]
- **Root Mean Square Speed (\( v_{rms} \))**: This is the square root of the average of the squares of the speeds of the molecules. It is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{m}} \]

2. **Comparing the Speeds**:
- We need to compare \( v_p \), \( \bar{v} \), and \( v_{rms} \):
- From the formulas:
- \( v_p = \sqrt{\frac{2RT}{m}} \)
- \( \bar{v} = \sqrt{\frac{8RT}{\pi m}} \)
- \( v_{rms} = \sqrt{\frac{3RT}{m}} \)

3. **Establishing the Inequalities**:
- To establish the relationship between these speeds, we can compare their squares:
- Comparing \( v_p \) and \( \bar{v} \):
\[ v_p^2 = \frac{2RT}{m} \quad \text{and} \quad \bar{v}^2 = \frac{8RT}{\pi m} \]
Since \( \pi \) is approximately 3.14, we find:
\[ \frac{8}{\pi} \approx 2.55 \quad \Rightarrow \quad v_p^2 < \bar{v}^2 \quad \Rightarrow \quad v_p < \bar{v} \]

- Comparing \( \bar{v} \) and \( v_{rms} \):
\[ v_{rms}^2 = \frac{3RT}{m} \quad \text{and} \quad \bar{v}^2 = \frac{8RT}{\pi m} \]
Since \( \frac{8}{\pi} > 3 \):
\[ v_{rms}^2 < \bar{v}^2 \quad \Rightarrow \quad v_{rms} > \bar{v} \]

- Comparing \( v_p \) and \( v_{rms} \):
\[ v_{rms}^2 = \frac{3RT}{m} \quad \text{and} \quad v_p^2 = \frac{2RT}{m} \]
Thus:
\[ v_p^2 < v_{rms}^2 \quad \Rightarrow \quad v_p < v_{rms} \]

4. **Final Conclusion**:
- From the comparisons, we conclude:
\[ v_p < \bar{v} < v_{rms} \]

### Summary of Results:
- The relationships established are:
- Most Probable Speed \( v_p < \) Mean Speed \( \bar{v} < \) Root Mean Square Speed \( v_{rms} \).