Which of the following combinations is correct ?

Options

A. `U_(rms) prop sqrtT and U_(rms) prop sqrtM`
B. `U_(rm s) prop sqrtT and U_(rm s) prop (1)/(sqrtM)`
C. `U_(rm s) prop (1)/(sqrtT) and U_(rm s) prop (1)/(sqrtM)`
D. `U_(rm s) prop (1)/(sqrtT) and U_(rm s) prop sqrtM`

Explanation

To solve the question regarding which combination is correct concerning the Root Mean Square (RMS) velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand RMS Velocity**:
- The Root Mean Square (RMS) velocity is defined as the square root of the average of the squares of the velocities of gas molecules. It is represented by the formula:
\[ v_{RMS} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Analyze the Relationship**:
- From the formula \( v_{RMS} = \sqrt{\frac{3RT}{M}} \), we can derive the relationships:
- **Directly Proportional to Temperature (T)**: \( v_{RMS} \) is directly proportional to the square root of the temperature, \( \sqrt{T} \).
- **Inversely Proportional to Molar Mass (M)**: \( v_{RMS} \) is inversely proportional to the square root of the molar mass, \( \sqrt{M} \).

3. **Evaluate the Options**:
- Now, we need to evaluate the given options based on the derived relationships:
- **Option 1**: Both are directly proportional (Incorrect).
- **Option 2**: Directly proportional to \( \sqrt{T} \) and inversely proportional to \( \sqrt{M} \) (Correct).
- **Option 3**: Both are inversely proportional (Incorrect).
- **Option 4**: Opposite of the correct relationship (Incorrect).

4. **Conclusion**:
- Based on the analysis, the correct combination is **Option 2**: \( v_{RMS} \) is directly proportional to \( \sqrt{T} \) and inversely proportional to \( \sqrt{M} \).

### Final Answer:
The correct combination is **Option 2**.