Mean Free Path of a Water Molecule in Water Vapor

Given:
- Temperature (T) = 373 K
- Diameter of the water molecule (d) = 2 × 10^-10 m
- Number of molecules per unit volume at STP (n) = 2.7 × 10^25 m^-3

Mean Free Path (λ):
The mean free path is the average distance traveled by a molecule between successive collisions with other molecules. It can be calculated using the formula:

λ = 1 / (√2 * π * d^2 * n)

where,
λ is the mean free path,
d is the diameter of the molecule, and
n is the number of molecules per unit volume.

Calculations:

1. Calculate the value of λ using the given formula:

λ = 1 / (√2 * π * (2 × 10^-10 m)^2 * (2.7 × 10^25 m^-3))

2. Simplify the equation:

λ = 1 / (√2 * π * 4 × 10^-20 m^2 * 2.7 × 10^25 m^-3)

λ = 1 / (2.7 × √2 * π * 10^5 m^2 * m^-3)

λ = 1 / (2.7 × √2 * π * 10^5 m)

λ = 1 / (2.7 × 1.414 * 3.14159 * 10^5 m)

λ ≈ 1 / (11.327 * 3.14159 * 10^5 m)

λ ≈ 1 / (35.5892 * 10^5 m)

λ ≈ 2.81 × 10^-7 m

Mean Free Path:
The mean free path for a water molecule in water vapor at a temperature of 373 K is approximately 2.81 × 10^-7 meters.

Explanation:
- The mean free path of a molecule is the average distance it travels before colliding with another molecule.
- In this case, we are considering water molecules in water vapor at a temperature of 373 K.
- The diameter of a water molecule is given as 2 × 10^-10 meters.
- The number of water molecules per unit volume at standard temperature and pressure (STP) is given as 2.7 × 10^25 m^-3.
- By using the formula for mean free path, we can calculate the average distance traveled by a water molecule between successive collisions.
- The calculated mean free path is approximately 2.81 × 10^-7 meters.
- This means that on average, a water molecule in water vapor at 373 K travels a distance of 2.81 × 10^-7 meters before colliding with another molecule.