The ratio of number of collisions per second at the walls of containers by He and `O_(2)` gas molecules kept at same volume and temperature is (assume normal incidence on walls)

Options

A. `2:1`
B. `1:2`
C. `2 sqrt2 :1`
D. `1:2 sqrt2`

Answer

To solve the problem of finding the ratio of the number of collisions per second at the walls of containers by helium (He) and oxygen (O₂) gas molecules kept at the same volume and temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Collision Frequency**:
The collision frequency (z) is given by the formula:
\[ z = 2 \sigma^2 n v \]
where:
- \( \sigma \) is the diameter of the gas molecules,
- \( n \) is the number density of the gas (number of molecules per unit volume),
- \( v \) is the average speed of the gas molecules.

2. **Average Speed of Gas Molecules**:
The average speed \( v \) of gas molecules can be expressed as:
\[ v = \sqrt{\frac{8RT}{\pi M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass (molecular weight) of the gas.

3. **Setting Up the Ratio**:
Since the volume and temperature are the same for both gases, we can compare the collision frequencies for helium and oxygen:
\[ \frac{z_{He}}{z_{O_2}} = \frac{2 \sigma^2 n v_{He}}{2 \sigma^2 n v_{O_2}} = \frac{v_{He}}{v_{O_2}} \]

4. **Substituting Average Speeds**:
Now substituting the expression for average speeds:
\[ \frac{v_{He}}{v_{O_2}} = \frac{\sqrt{\frac{8RT}{\pi M_{He}}}}{\sqrt{\frac{8RT}{\pi M_{O_2}}}} = \sqrt{\frac{M_{O_2}}{M_{He}}} \]

5. **Using Molecular Weights**:
The molecular weights are:
- \( M_{He} = 4 \, \text{g/mol} \)
- \( M_{O_2} = 32 \, \text{g/mol} \)

Therefore:
\[ \frac{z_{He}}{z_{O_2}} = \sqrt{\frac{32}{4}} = \sqrt{8} = 2\sqrt{2} \]

6. **Conclusion**:
The ratio of the number of collisions per second at the walls of containers by helium and oxygen gas molecules is:
\[ \frac{z_{He}}{z_{O_2}} = 2\sqrt{2} \]

### Final Answer:
The ratio of the number of collisions per second at the walls of containers by He and O₂ gas molecules is \( 2\sqrt{2} \).