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 (VRMS), average speed (Vaverage), and most probable speed (Vmps) of a gas at a given temperature, we will follow these steps:

### Step 1: Understand the Definitions
- **Most Probable Speed (Vmps)**: The speed at which the maximum number of molecules are moving. It is given by the formula:
\[ V_{mps} = \sqrt{\frac{2RT}{M}} \]
- **Average Speed (Vaverage)**: The mean speed of all molecules in the gas. It is calculated using the formula:
\[ V_{average} = \sqrt{\frac{8RT}{\pi M}} \]
- **Root Mean Square Speed (VRMS)**: The square root of the average of the squares of the speeds of the gas molecules. It is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

### Step 2: Compare the Speeds
From the formulas, we can derive the relationships:
- The value of \( V_{rms} \) is greater than \( V_{average} \), which is greater than \( V_{mps} \).
- In numerical terms:
- \( V_{rms} = \sqrt{3} \cdot \sqrt{\frac{RT}{M}} \approx 1.732 \cdot \sqrt{\frac{RT}{M}} \)
- \( V_{average} = \sqrt{\frac{8}{\pi}} \cdot \sqrt{\frac{RT}{M}} \approx 1.595 \cdot \sqrt{\frac{RT}{M}} \)
- \( V_{mps} = \sqrt{2} \cdot \sqrt{\frac{RT}{M}} \approx 1.414 \cdot \sqrt{\frac{RT}{M}} \)

### Step 3: Establish the Correct Relationship
From the above comparisons, we can conclude:
\[ V_{rms} > V_{average} > V_{mps} \]
Thus, the correct relationship at a given temperature for a gas is:
\[ V_{rms} > V_{average} > V_{mps} \]

### Conclusion
The correct answer to the question is that the root mean square speed is greater than the average speed, which is in turn greater than the most probable speed.