In a mixture of nitrogen and helium kept at room temperarture. As compared to a helium molecule nitrogen molecule hits the wall

Options

A. With greater average speed
B. with smaller average speed
C. with greater average kinetic energy
D. with smaller average kinetic energy

Answer

To solve the problem, we need to compare the average speed and average kinetic energy of nitrogen (N₂) and helium (He) molecules in a mixture at room temperature.

### Step-by-Step Solution:

1. **Understanding Average Speed**:
The average speed of gas molecules can be derived from the kinetic theory of gases. The formula for average speed (v) is given by:
\[ v = \sqrt{\frac{8kT}{\pi m}} \]
where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the molar mass of the gas.

2. **Comparing Molar Masses**:
- The molar mass of nitrogen (N₂) is 28 g/mol.
- The molar mass of helium (He) is 4 g/mol.
Since the molar mass of nitrogen is greater than that of helium, we can conclude:
\[ m_{N_2} > m_{He} \]

3. **Average Speed Relation**:
From the formula, we see that average speed is inversely proportional to the square root of molar mass:
\[ v \propto \frac{1}{\sqrt{m}} \]
Therefore, since \( m_{N_2} > m_{He} \), it follows that:
\[ v_{N_2} < v_{He} \]
This means nitrogen molecules hit the wall with a smaller average speed compared to helium molecules.

4. **Understanding Average Kinetic Energy**:
The average kinetic energy (KE) of gas molecules is given by:
\[ KE = \frac{f}{2} nRT \]
where \( f \) is the degrees of freedom, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

5. **Degrees of Freedom**:
- For nitrogen (N₂), which is a diatomic gas, the degrees of freedom \( f \) is 5 (3 translational + 2 rotational).
- For helium (He), which is a monatomic gas, the degrees of freedom \( f \) is 3 (3 translational).

6. **Calculating Average Kinetic Energies**:
- Average kinetic energy for nitrogen:
\[ KE_{N_2} = \frac{5}{2} nRT \]
- Average kinetic energy for helium:
\[ KE_{He} = \frac{3}{2} nRT \]

7. **Comparison of Kinetic Energies**:
Since \( \frac{5}{2} nRT > \frac{3}{2} nRT \), we conclude:
\[ KE_{N_2} > KE_{He} \]
This means nitrogen molecules have greater average kinetic energy compared to helium molecules.

### Final Conclusion:
- **Average Speed**: Nitrogen molecules hit the wall with a smaller average speed.
- **Average Kinetic Energy**: Nitrogen molecules hit the wall with greater average kinetic energy.