Which of the following statements are true?

Options

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

Answer

To determine which statements are true regarding the properties of gases, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Analyzing Option 1:**
- The statement claims that the ratio of mean speed to root mean square (RMS) speed is independent of temperature.
- The formula for mean speed (μ_mean) is given by:
\[ \mu_{\text{mean}} = \sqrt{\frac{8RT}{\pi m}} \]
- The formula for RMS speed (μ_rms) is:
\[ \mu_{\text{rms}} = \sqrt{\frac{3RT}{m}} \]
- The ratio of mean speed to RMS speed is:
\[ \frac{\mu_{\text{mean}}}{\mu_{\text{rms}}} = \frac{\sqrt{\frac{8RT}{\pi m}}}{\sqrt{\frac{3RT}{m}}} = \frac{\sqrt{8}}{\sqrt{3} \sqrt{\frac{1}{\pi}}} = \frac{8}{3\pi} \]
- Since this ratio does not contain temperature (T), the statement is **true**.

2. **Analyzing Option 2:**
- The statement claims that the square of mean speed is equal to the mean squared speed at a certain temperature.
- We calculate the square of mean speed:
\[ \mu_{\text{mean}}^2 = \left(\sqrt{\frac{8RT}{\pi m}}\right)^2 = \frac{8RT}{\pi m} \]
- The mean squared speed (which is the square of RMS speed) is:
\[ \mu_{\text{rms}}^2 = \left(\sqrt{\frac{3RT}{m}}\right)^2 = \frac{3RT}{m} \]
- Setting these equal gives:
\[ \frac{8RT}{\pi m} = \frac{3RT}{m} \]
- Canceling RT and m leads to:
\[ \frac{8}{\pi} = 3 \]
- This is not true since \( \frac{8}{\pi} \neq 3 \). Thus, this statement is **false**.

3. **Analyzing Option 3:**
- The statement claims that the mean kinetic energy of a fixed number of gas molecules at a given temperature is independent of the molecular weight of the gas.
- The formula for mean kinetic energy (KE) is:
\[ KE = \frac{3}{2}RT \]
- This expression does not include molecular weight (m). Therefore, the statement is **true**.

4. **Analyzing Option 4:**
- The statement claims that the difference between RMS speed and mean speed diminishes at larger molar masses.
- As the molar mass (m) increases, both mean speed and RMS speed decrease:
\[ \mu_{\text{mean}} = \sqrt{\frac{8RT}{\pi m}}, \quad \mu_{\text{rms}} = \sqrt{\frac{3RT}{m}} \]
- As m increases, both speeds approach zero, and their difference diminishes. Thus, this statement is **true**.

### Conclusion:
The true statements are:
- **Option 1**: True
- **Option 2**: False
- **Option 3**: True
- **Option 4**: True

### Final Answer:
The correct options are **1, 3, and 4**.