Calculate the mean free path of air molecules at 3.50 * 10^-13 atm and 300 K. (This pressure is readily attainable in the laboratory; see Exercise 18.23.) As in Example 18.8, model the air molecules as spheres of radius 2.0 * 10^-10 m.

Verified step by step guidance

1. Understand the concept of mean free path: It is the average distance traveled by a molecule between collisions. It depends on the density of the gas and the size of the molecules.
2. Use the formula for mean free path \( \lambda \): \( \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P} \), where \( k_B \) is the Boltzmann constant \( 1.38 \times 10^{-23} \text{ J/K} \), \( T \) is the temperature, \( d \) is the diameter of the molecules, and \( P \) is the pressure.
3. Convert the radius of the air molecules to diameter: Since the radius is given as \( 2.0 \times 10^{-10} \text{ m} \), the diameter \( d \) is \( 4.0 \times 10^{-10} \text{ m} \).
4. Substitute the given values into the mean free path formula: \( T = 300 \text{ K} \), \( P = 3.50 \times 10^{-13} \text{ atm} \). Convert pressure from atm to pascals using \( 1 \text{ atm} = 1.013 \times 10^5 \text{ Pa} \).
5. Calculate \( \lambda \) using the substituted values: Ensure all units are consistent, particularly converting pressure to pascals, and perform the calculation to find the mean free path.

Key Concepts

Mean Free Path

The mean free path is the average distance a molecule travels between collisions in a gas. It is influenced by factors such as the size of the molecules, the density of the gas, and the temperature. In this context, it helps determine how far air molecules move before interacting with each other under specified conditions.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure (P), volume (V), and temperature (T) of a gas to the number of moles (n) and the ideal gas constant (R). It is crucial for calculating the density of the gas, which is needed to determine the mean free path of the molecules at a given pressure and temperature.

Molecular Diameter

The molecular diameter is the effective size of a molecule, often modeled as a sphere for simplicity. In this problem, the air molecules are considered spheres with a given radius, which is essential for calculating the mean free path. The diameter directly affects the frequency of collisions between molecules in the gas.