Domain: Kinetic theory

In a kinetic-theory analysis of atmospheric gases, one wishes to estimate the intrinsic molecular collision activity occurring within a microscopic volume of air. Consider dry air at 20 °C and a total pressure of 1.00 bar. Assume that nitrogen constitutes 80% of the molecules by number. Treat nitrogen molecules as identical hard spheres with a collision cross section of N₂ is 4.50 × 10⁻¹⁹ m².

Using ideal-gas behavior and the Maxwell–Boltzmann distribution for molecular speeds, determine the total frequency of nitrogen–nitrogen collisions occurring per second within a volume of exactly one cubic centimeter of air under these conditions.

Express your final answer in units of s⁻¹·cm⁻³.

Grading rubric:
criterion: The solution correctly uses the molar ideal gas formulation rho = N_A P / (R T) in L·bar units before converting to m^-3.
weight: 20
criterion: The solution correctly converts the ideal-gas constant from molar form to a number-density expression using Avogadro’s constant.
weight: 10
criterion: The solution explicitly reports the intermediate number density as 2.0 × 10^22 L^-1 before converting to 2.0 × 10^25 m^-3.
weight: 20
criterion: The solution computes the mean molecular speed using sqrt(8RT / (pi M)) and does NOT use the relative-speed expression sqrt(16kT / (pi m)).
weight: 10
criterion: The collision expression is written in the exact form Z = sigma * <u> * rho^2 / sqrt(2).
weight: 20
criterion: The solution explicitly includes the 1 divided by sqrt(2) factor in the collision frequency expression for identical particles.
weight: 10
criterion: The final result is approximately 6.0 × 10²⁸ s⁻¹ cm⁻³.
weight: 10

Answer:
6.0 × 10^28 s^-1 cm^-3