For a given gas, which of the following relationships is correct at a given temperature ?

Options

A. `u_(rm s) gt u_(av) gt u_(m p)`
B. `u_(rm s) lt u_(av) lt u_(m p)`
C. `u_(rm s) gt u_(av) lt u_(m p)`
D. `u_(rm s) lt u_(av) gt u_(m p)`

Explanation

To solve the question regarding the relationships between the root mean square speed (u_rms), average speed (u_avg), and most probable speed (u_mp) of a gas at a given temperature, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Formulas**:
- The formulas for the speeds of a gas are:
- **Root Mean Square Speed (u_rms)**:
\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]
- **Average Speed (u_avg)**:
\[ u_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
- **Most Probable Speed (u_mp)**:
\[ u_{mp} = \sqrt{\frac{2RT}{M}} \]

2. **Compare the Values**:
- To compare these speeds, we will express them in a common format:
- \( u_{rms} = \sqrt{\frac{3RT}{M}} \)
- \( u_{avg} = \sqrt{\frac{8RT}{\pi M}} \)
- \( u_{mp} = \sqrt{\frac{2RT}{M}} \)

3. **Calculate the Ratios**:
- We can calculate the ratios of these speeds to understand their relationships:
- \( \frac{u_{rms}}{u_{mp}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{2RT}{M}}} = \sqrt{\frac{3}{2}} \)
- \( \frac{u_{avg}}{u_{mp}} = \frac{\sqrt{\frac{8RT}{\pi M}}}{\sqrt{\frac{2RT}{M}}} = \sqrt{\frac{8}{2\pi}} = \sqrt{\frac{4}{\pi}} \)
- \( \frac{u_{rms}}{u_{avg}} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{8RT}{\pi M}}} = \sqrt{\frac{3\pi}{8}} \)

4. **Evaluate the Numerical Values**:
- Now, we can evaluate the numerical values of these ratios:
- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{\frac{8}{\pi}} \approx 1.595 \) (since \( \pi \approx 3.14 \))
- From these calculations, we can conclude:
- \( u_{rms} > u_{avg} > u_{mp} \)

5. **Conclusion**:
- Therefore, the correct relationship at a given temperature for a gas is:
\[ u_{rms} > u_{avg} > u_{mp} \]

### Final Answer:
The correct relationship for the speeds of a gas at a given temperature is:
\[ u_{rms} > u_{avg} > u_{mp} \]