Which of the following statement(s) is (are) are correct regarding the root mean square speed `(u_("rms"))` and average translational kinetic energy `(E_("av"))` of a molecule in a gas at equilibrium?

Options

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

Explanation

To determine which statements regarding the root mean square speed `(u_rms)` and average translational kinetic energy `(E_av)` of a molecule in a gas at equilibrium are correct, we can analyze each statement based on the formulas and principles of kinetic theory.

### 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 universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.
- The average translational kinetic energy `(E_av)` of a molecule is given by:
\[ E_{av} = \frac{3}{2}RT \]
This shows that the average kinetic energy is directly proportional to the temperature.

2. **Analyzing Statement A**:
- Statement A claims that \( u_{rms} \) is universally proportional to the square root of its molecular mass.
- From the formula, we see that \( u_{rms} \) is inversely proportional to the square root of the molar mass \( M \). Thus, this statement is **incorrect**.

3. **Analyzing Statement B**:
- Statement B states that \( E_{av} \) increases four times when the temperature increases four times.
- Since \( E_{av} \) is directly proportional to \( T \), if \( T \) increases four times, \( E_{av} \) will also increase four times. Therefore, this statement is **correct**.

4. **Analyzing Statement C**:
- Statement C claims that \( u_{rms} \) doubles when the temperature increases four times.
- Since \( u_{rms} \) is proportional to the square root of the temperature, if the temperature increases four times, \( u_{rms} \) will increase by a factor of \( \sqrt{4} = 2 \). Thus, this statement is **correct**.

5. **Analyzing Statement D**:
- Statement D states that \( E_{av} \) at a given temperature does not depend on molecular mass.
- From the formula for \( E_{av} \), it is clear that it only depends on \( T \) and not on \( M \). Therefore, this statement is **correct**.

### Conclusion:
The correct statements are:
- Statement B: Correct
- Statement C: Correct
- Statement D: Correct
- Statement A: Incorrect

### Final Answer:
The correct statements regarding the root mean square speed and average translational kinetic energy are B, C, and D.

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