Which of the following statements is/are true :

Options

A. The ratio of the mean speed of the rms speed is independent of temperature of gas
B. The square of the mean speed of the molecules is equal to the mean squared speed at a certain temperature
C. Mean translational kinetic energy of the gas molecules at any given temperature is independent of the molecular mass of gas
D. The difference between rms speed and mean speed at any temperature for different gases diminishes as larger and yet larger molar masses are considered.

Explanation

To solve the question regarding which statements are true, we will analyze each statement one by one based on the principles of kinetic theory of gases.

### Step-by-Step Solution:

1. **Statement A**: "The ratio of mean speed to the RMS speed is independent of temperature of the gas."
- **Analysis**:
- The root mean square (RMS) speed (\(v_{rms}\)) is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- The mean speed (\(v_{mean}\)) is given by:
\[ v_{mean} = \sqrt{\frac{8RT}{\pi M}} \]
- The ratio of mean speed to RMS speed is:
\[ \frac{v_{mean}}{v_{rms}} = \frac{\sqrt{\frac{8RT}{\pi M}}}{\sqrt{\frac{3RT}{M}}} = \frac{\sqrt{8}}{\sqrt{3} \cdot \sqrt{\frac{\pi}{1}}} = \frac{8}{3\pi} \]
- This ratio does not depend on temperature (T) or molar mass (M).
- **Conclusion**: Statement A is **true**.

2. **Statement B**: "The square of mean speed of the molecule is equal to mean square speed at a certain temperature."
- **Analysis**:
- The mean speed squared (\(v_{mean}^2\)) is:
\[ v_{mean}^2 = \left(\sqrt{\frac{8RT}{\pi M}}\right)^2 = \frac{8RT}{\pi M} \]
- The mean square speed (\(v_{rms}^2\)) is:
\[ v_{rms}^2 = \left(\sqrt{\frac{3RT}{M}}\right)^2 = \frac{3RT}{M} \]
- Clearly, \(v_{mean}^2 \neq v_{rms}^2\) since \(\frac{8}{\pi} \neq 3\).
- **Conclusion**: Statement B is **false**.

3. **Statement C**: "Mean translational kinetic energy of a gas molecule at any temperature is independent of molecular mass of gas."
- **Analysis**:
- The mean translational kinetic energy (KE) is given by:
\[ KE = \frac{3}{2}RT \]
- This expression shows that kinetic energy depends only on temperature (T) and is independent of the molecular mass (M).
- **Conclusion**: Statement C is **true**.

4. **Statement D**: "The difference between RMS speed and mean speed at any temperature for different gases diminishes as larger and larger molar mass are considered."
- **Analysis**:
- As molar mass increases, both \(v_{rms}\) and \(v_{mean}\) decrease. Since both speeds are inversely proportional to the square root of molar mass, the difference between them becomes negligible as molar mass approaches infinity.
- **Conclusion**: Statement D is **true**.

### Final Answer:
The true statements are **A, C, and D**.

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