Simple Models of Matter and Boltzmann Factor

Gas Laws

When describing an ideal gas, the following assumptions are made:

Gas particles take up negligible volume
All collisions between particles and the walls of the container and each other are completely elastic
There are negligible forces between particles except during collisions
All particles of a particular gas are identical
The internal energy of the gas is entirely kinetic
Newton's laws of motion apply
Gravitational, electrostatic and Van de Waals forces can be ignored
The motion of all molecules is random
All molecules travel in straight lines
The top three assumptions in bold are given explicitly in the specification.

Boyle's Law states that for a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure:

Avogadro's Law states that for a constant pressure and temperature, the volume of a gas is directly proportional to the number of moles:

Temperature is a measure of the average kinetic energy of the particles in a substance. Due to the random nature of collisions between particles in a gas, there is a distribution of speeds, and therefore kinetic energies of particles. The range of speeds are represented by the Maxwell Boltzmann distribution.

Charles' Law states that for a constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature:

For in Kelvin (K).

By combining these laws, we get:
Where is the ideal gas constant, with a value of .

The Boltzmann Constant is similar to the gas constant, , but only for one particle. It is equal to the gas constant divided by , the Avogadro constant:

By combining with , we get: , so we can rewrite the ideal gas equation as:

At a constant temperature, the distribution of speeds will remain the same, as well as the average speed. The root mean square speed is:

Temperature of a gas is proportional to the average kinetic energy of its particles:

For monatomic particles (and one dimension), a simple estimation is often used:

Gas Law Derivation

Consider a particle of mass moving at velocity in a cubic box with side length .

Assume that the particle is moving only perpendicular to a given wall. In general, one third of the total kinetic energy is in each perpendicular direction^[1].
When the particle collides with a wall, it applies an impulse of .
For a given wall, this happens every seconds, so there are collisions with a given wall per second.
The force exerted on this wall is equal to the rate of change of impulse, so
The pressure is equal to force per unit area, so
Rearranging, we find
is just the volume, so for a single particle
Considering particles each with mean squared speed , we get .

Equating , we find:

(cancelling from both sides)
(pulling out a factor of for KE)

Specific Heat Capacity

Specific heat capacity is a measure of the energy needed to raise the temperature of of a substance by . The equation for SHC is:

Brownian Motion

Particles in a liquid or gas undergo random motion because fast free-moving molecules move around and collide with each other. The mean free path is the average distance a particle travels before colliding (the size of a 'step'). If a particles takes steps, on average its end point is steps away from its starting point.

Boltzmann Factor

For many physical processes, a certain amount of energy is needed for the process to take place. This is the activation energy E; for example, in chemistry, reagents are often heated so they have enough kinetic energy to react. Other physical processes requiring an activation energy are:

Changes of state
Thermionic emission, where electrons are emitted from a heated metal surface
Ionisation, where electrons are removed from an atom
- Conduction in semiconductors
Viscous flow
Nuclear fusion
The first five of these processes are given explicitly in the specification.

Hope you like this diagram because we used up our entire yearly Tikz diagram budget on it.

Many physical processes require energies between and , for example water evaporation requires around at freezing.

Footnote on kinetic energies in perpendicular directions
Kinetic energies in perpendicular directions add (by the Pythagorean theorem), so on average a third of the total kinetic energy is in any given direction.↩︎