Which of the following is/are correct regarding root mean square speed `(U_(rms))` & average translation K.E. `(E_(av))` of molecules in a gas at equilibrium

Options

A. `U_(rms)` is inversely proportional to square root of its molecular mass
B. `U_(avg)` at a given temperature does not depend on its molecular mass
C. `U_(avg)` is doubled when its temperature is increased four times
D. `U_(rms)` is doubled when its temperature is increased four times

Explanation

To solve the question regarding the root mean square speed `(U_(rms))` and average translational kinetic energy `(E_(av))` of molecules in a gas at equilibrium, we will analyze each statement provided in the question based on the relevant formulas.

### Step-by-Step Solution:

1. **Understanding the Formulas**:
- The root mean square speed `(U_(rms))` is given by the formula:
\[ U_{rms} = \sqrt{\frac{3RT}{m}} \]
where \( R \) is the gas constant, \( T \) is the temperature, and \( m \) is the molecular mass of the gas.
- The average translational kinetic energy `(E_(av))` is given by the formula:
\[ E_{av} = \frac{3}{2} RT \]

2. **Analyzing Statement 1**:
- **Statement**: "Root mean square speed is inversely proportional to the square root of its molecular mass."
- **Analysis**: From the formula for \( U_{rms} \), we see that \( U_{rms} \) is inversely proportional to the square root of \( m \) (molecular mass). Thus, this statement is **correct**.

3. **Analyzing Statement 2**:
- **Statement**: "Average translational kinetic energy at a given temperature does not depend on its molecular mass."
- **Analysis**: In the formula for \( E_{av} \), there is no \( m \) (molecular mass) present. Therefore, the average translational kinetic energy is independent of molecular mass at a given temperature. This statement is also **correct**.

4. **Analyzing Statement 3**:
- **Statement**: "Average translational kinetic energy is doubled when its temperature is increased four times."
- **Analysis**: Since \( E_{av} \) is directly proportional to \( T \), if \( T \) is increased four times, \( E_{av} \) will also increase four times, not double. Thus, this statement is **incorrect**.

5. **Analyzing Statement 4**:
- **Statement**: "Root mean square speed is doubled when its temperature is increased four times."
- **Analysis**: Since \( U_{rms} \) is proportional to \( \sqrt{T} \), if \( T \) is increased four times, \( \sqrt{T} \) will increase by a factor of 2 (since \( \sqrt{4} = 2 \)). Therefore, \( U_{rms} \) will indeed be doubled. This statement is **correct**.

### Conclusion:
The correct statements regarding root mean square speed and average translational kinetic energy are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Incorrect
- Statement 4: Correct

Thus, the correct options are 1, 2, and 4.