The Kinetic Theory of Gases II

The Kinetic Theory of Gases II is the fourth lecture within the Properties of Matter section of PH1011. It covers the collisions of molecules in a gas, the distance moved on average by random walk, and estimations of molecular size by density of gases and liquids.

Previous: The Kinetic Theory of Gases I

Next: Statistical Mechanics

Mean Free Path
Typical values for vrms are very fast. However, diffusion of gases is very slow; this is due to collisions against other gas molecules which cause random changes in direction and therefore hinder the overall motion of a gas. The average distance between these collisions is the mean free path, λ.

The assumptions made in approximating λ are
 * that molecules are spheres, speed v and diameter d; collisions occur when molecules come within d of one another.
 * In considering a single molecule, it can be useful to approximate the radius as d rather than the diameter, and consider all other molecules as points. This Big MoleculeTM moves a distance given by vΔt, sweeping out a path of volume V = πd2vΔt.
 * The number of resulting collisions in this cylinder is given by n=(N/V)πd2vΔt; (N/V) being the number of molecules in a unit volume.
 * Mean free path is the length of the path/average number of collisions; vΔt/(N/V)πd2vΔt; V/d2N. In practice this becomes vΔt/(N/V)πd2vΔt; V/(√2)d2N.

Number of collisions is given by s = vt/λ.

Random Walk
Random walk refers to the random motion caused by these collisions. Consider the triangle - the cosine rule states that [xs2] = [xs-12] + λ2 - [2xs-1]λcosθ. Often in random walk cosθ will be 0 (where θ=π/2 and 3/2π). The previous step would be given by [xs-12] = [xs-22] + λ2, and the first step would have been [x12] = 0 + λ2.

This gives [xs2] = sλ2. The mean distance of this is the square root of [xs2]; λ√s. xrms=λ√s.

Substituting s = vt/λ gives xrms = λ√(vt/λ) => xrms = √(λvrmst). The diffusion distance is therefore proportional to the square root of time.


 * Example: A bromine diffusion front moves 0.1m in 500s. The vrms is ~200ms-1. Find λ.
 * λ = xrms2/vrmst
 * = (0.1)2/200x500 = 10-7 = 100nm

Molecular Size from Density
Kinetic theory allows estimation of the size of molecules. In a gas, molecules occupy a volume of πd2λ. In a liquid, this volume is just d3. The volume varies against 1/density, giving ρliquid/ρgas = πd2λ/d3. d = πλ(ρliquid/ρgas). This proves that diffusion rate and density can be used to find accurate statistics for the microscopic properties of gases. This further allows deduction of the number of atoms in a mole, further proving Avogadro right.

Summary
The mean free path is the average distance that a molecule can travel before it has a collision with another. This can be calculated using the molecular size, assuming one molecule to have radius d and the others to be points. The number of collisions allows calculation of the random walk distance - how far molecules actually manage to move in a given time. Approximated by triangles this distance is proportional to the square root of time. If the density and diffusion rates of a gas is known, the size of its atoms can be calculated. Nobody should have doubted Avogadro.