From the Kinetic theory of gases the number of bimolecular collisions per second per unit volume among molecules of one species is given by:

Question

Accepted Answer

The correct answer to the question regarding the number of bimolecular collisions per second per unit volume among molecules of one species, according to the kinetic theory of gases, is given by Option 1:

$$ Z = 2n^{2}\sigma^{2}(8\pi kT/\mu)^{1/2} $$

### Explanation for Option 1:

1. **Bimolecular Collisions**: In the kinetic theory of gases, bimolecular collisions refer to interactions between pairs of molecules. The rate of these collisions is influenced by factors such as the number density of molecules, the cross-sectional area for collisions, and the relative velocity of the molecules.

2. **Number Density (n)**: The term $ n $ represents the number density of molecules, which is the number of molecules per unit volume. For bimolecular collisions, the rate is proportional to the square of the number density, $ n^2 $, because collisions involve pairs of molecules.

3. **Collision Cross-Section ($\sigma$)**: The collision cross-section, $ \sigma $, is a measure of the effective area that a molecule presents for collision. The rate of collisions is proportional to the square of the collision cross-section, $ \sigma^2 $, because it involves interactions between two molecules.

4. **Relative Velocity**: The term $ (8\pi kT/\mu)^{1/2} $ represents the average relative velocity of the molecules, where $ k $ is the Boltzmann constant, $ T $ is the temperature, and $ \mu $ is the reduced mass of the colliding molecules. The relative velocity is crucial in determining the frequency of collisions.

5. **Combining Factors**: The expression $ 2n^{2}\sigma^{2}(8\pi kT/\mu)^{1/2} $ combines these factors to accurately describe the rate of bimolecular collisions per unit volume, making Option 1 the correct choice.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $ Z = 2n~\sigma(8\pi kT/\mu) $**
  - This option incorrectly uses $ n $ and $ \sigma $ without squaring them. Since bimolecular collisions involve pairs of molecules, the rate should be proportional to $ n^2 $ and $ \sigma^2 $, not $ n $ and $ \sigma $.

- **Option 3: $ Z = 2n~\sigma(8\pi kT/\mu)^{1/2} $**
  - Similar to Option 2, this option fails to square $ n $ and $ \sigma $. The correct expression for bimolecular collisions should include $ n^2 $ and $ \sigma^2 $.

- **Option 4: $ Z = 2n^{2}\sigma^{2}(8\pi kT/\mu) $**
  - Although this option correctly includes $ n^2 $ and $ \sigma^2 $, it incorrectly uses $ (8\pi kT/\mu) $ instead of $ (8\pi kT/\mu)^{1/2} $. The latter represents the average relative velocity, which is essential for calculating the collision rate.

In conclusion, Option 1 is the correct answer because it accurately incorporates all the necessary factors—number density squared, collision cross-section squared, and the square root of the average relative velocity—to describe the rate of bimolecular collisions per unit volume according to the kinetic theory of gases.