Which of following statements are correct

Options

A. Average velocity of molecules of a gas in a container is zero
B. All molecules in a gas are moving with the same speed
C. If an open container is heated from 300 K to 400 K the fraction of air which goes out with respect to originally present is `1/4`
D. If compressibility factor of a gas at STP is less than unity then its molar volume is less than 22.4 L at STP

Explanation

To determine which of the given statements are correct, we will analyze each statement one by one.

### Statement 1: Average velocity of molecules of a gas in a container is zero.
- **Analysis**: The average velocity of gas molecules is indeed zero because the molecules are in constant random motion in all directions. When you average the velocities (considering both magnitude and direction), the positive and negative components cancel out, resulting in an average velocity of zero.
- **Conclusion**: This statement is **true**.

### Statement 2: All molecules of gas are moving with the same speed.
- **Analysis**: According to Maxwell's distribution of molecular speeds, gas molecules do not all move at the same speed. Instead, they have a range of speeds, with some moving faster and some moving slower. Therefore, this statement is incorrect.
- **Conclusion**: This statement is **false**.

### Statement 3: If an open container is heated from 300 to 400 Kelvin, then the fraction of air that goes out or escapes is 1/4 of the original air.
- **Analysis**: We can use the relation \( n_1/T_1 = n_2/T_2 \). Let the initial number of moles be \( n_1 = n \) at \( T_1 = 300 \) K and the final temperature \( T_2 = 400 \) K. Rearranging gives:
\[ n_2 = n \cdot \frac{T_2}{T_1} = n \cdot \frac{400}{300} = \frac{4n}{3} \]
The number of moles that escaped is:
\[ n - n_2 = n - \frac{4n}{3} = n \cdot \left(1 - \frac{4}{3}\right) = -\frac{n}{3} \]
This indicates a mistake in the calculation. The correct interpretation should be:
\[ n_2 = n \cdot \frac{300}{400} = \frac{3n}{4} \]
Thus, the number of moles that escaped is:
\[ n - n_2 = n - \frac{3n}{4} = \frac{n}{4} \]
Therefore, the fraction of air that escapes is:
\[ \frac{n/4}{n} = \frac{1}{4} \]
- **Conclusion**: This statement is **true**.

### Statement 4: If the compressibility factor of a gas at STP is less than unity, then its molar mass is less than 22.4 liters at STP.
- **Analysis**: The compressibility factor \( Z \) is defined as \( Z = \frac{PV}{RT} \). If \( Z < 1 \), it indicates that the gas is more compressible than an ideal gas, which implies that the molar volume is less than the ideal molar volume of 22.4 L at STP. Thus, if \( Z < 1 \), the molar volume is indeed less than 22.4 L.
- **Conclusion**: This statement is **true**.

### Final Conclusion:
- Statements 1, 3, and 4 are correct, while statement 2 is incorrect.

### Summary of Correct Statements:
- **Correct Statements**: 1, 3, and 4.

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