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

## Content

### 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.