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 the correct combination of relationships involving the root mean square velocity (URMS), temperature (T), and molar mass (M), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for URMS**:
The root mean square velocity (URMS) of gas molecules is given by the formula:
\[ U_{RMS} = \sqrt{\frac{P \cdot R \cdot T}{M}} \]
where:
- \(P\) = pressure of the gas,
- \(R\) = gas constant,
- \(T\) = absolute temperature,
- \(M\) = molar mass of the gas.

2. **Identify Relationships**:
From the formula, we can derive the relationships:
- \(U_{RMS}\) is directly proportional to the square root of temperature (\(T\)):
\[ U_{RMS} \propto \sqrt{T} \]
- \(U_{RMS}\) is inversely proportional to the square root of molar mass (\(M\)):
\[ U_{RMS} \propto \frac{1}{\sqrt{M}} \]

3. **Evaluate the Options**:
Now we will evaluate the given options based on the derived relationships:
- **Option 1**: \(U_{RMS} \propto \sqrt{T}\) and \(U_{RMS} \propto \sqrt{M}\) - **Incorrect** (it should be inversely proportional to \(\sqrt{M}\)).
- **Option 2**: \(U_{RMS} \propto \sqrt{T}\) and \(U_{RMS} \propto \frac{1}{\sqrt{M}}\) - **Correct**.
- **Option 3**: \(U_{RMS} \propto \frac{1}{\sqrt{T}}\) - **Incorrect**.
- **Option 4**: \(U_{RMS} \propto \frac{1}{M}\) - **Incorrect** (it should be inversely proportional to \(\sqrt{M}\)).

4. **Conclusion**:
Based on the evaluations, the correct option is **Option 2**.