The Kinetic Theory of Gases I

The Kinetic Theory of Gases is the third lecture within the Properties of Matter section of PH1011. It covers the evidence for the constituents of gases, basic assumptions made in kinetic theory, equations for pressure and the average molecular kinetic energy.

Previous: Thermal Physics and Temperature

Next: The Kinetic Theory of Gases II

Evidence and Assumptions
Evidence:
 * Gases will always expand to fit a given volume
 * Gases can and will diffuse into one another
 * Evaporation causes expansion from 1ml liquid to 1000ml vapour
 * Atoms are in constant random motion (Brownian motion)

Assumptions:
 * Gases consist of identical molecules
 * Particles have 0 size and exert no force
 * All collisions are elastic.

Pressure
Pressure is assumped to be caused by gas molecules striking the walls of a container. An equation for this can be found by finding the impulse => the rate of impulse => force of molecle on wall => total force from summation => Pressure from dividing by area. Like so:


 * Change in momentum of a single molecule on one direction's surface is given by Δp=2mvx/sub>. This occurs in Δt = 2L/v x (L being the container's length).
 * As force is defined as the change in pressure by time, Δp/Δt, it can be given as F = m[vx2]/L.
 * Given that there are nNA molecules in total, overall force is F = nNAm[vx2]/L.
 * As pressure is F/A = F/L2 = ((nNAm)/L3x[vx2] = nM/V x [vx2]
 * For a single molecule, v2 is the sum of each direction's velocity squared. For a number of molecules, this is [v2]; the average of all particles' summed velocities.

If molecules are in random motion, then [vx2] = [vy2] = [vz2]; 1/3 of [v2]. This gives P = 1/3nM/V [v2] = 1/3 ρ [v2]. This in turn gives -

vrms = √[(3PV)/nM] = √[(3RT)/M] as PV = nRT.

Kinetic Energy
The kinetic energy of a molecule is given by Ek = 1/2mv2. In relation to the above equation, this gives an average energy of 1/2m√[(3RT)/M] = 3/2(R/NA)T = 3/2kT. k is Boltzmann's constant, R/NA = 1.38x10-23.

This proves that the kinetic energy of a molecule is only dependent on its temperature. Molecules have an energy of 1/2kT per degree of freedom - in this example the atom is approximated as a point, and therefore has translational degrees of freedom but no rotational degrees of freedom.

Summary
Gases are made up of very small individual molecules at low density, as shown by expansion, diffusion, evaporation and Brownian motion. The pressure of a gas within a container can be found if the container's measurements and the properties of the gas are known, and from that the root mean square velocity can be calculated. The kinetic energy of a gas can use vrms to prove that it is only dependent upon temperature, and has 1/2kT energy per degree of freedom.