1. Maxwell-Boltzmann velocity distribution

2. Maxwell-Boltzmann speed distribution

3. Maxwell-Boltzmann speed distributionvm

4. Gas pressure and the ideal gas law
Kinetic theory provides a natural interpretation of the absolute temperature of a dilute gas. Namely, the temperature is proportional to the mean kinetic energy (e ) of the gas molecules.
The mean kinetic energy is independent of pressure, volume, and the molecular species, i.e. it is the same for all molecules.

6. The probability density function
The random motions of the molecules can be characterized by a probability distribution function.
Since the velocity directions are uniformly distributed, we can reduce the problem to a speed distribution function which is isotropic.
Let f(v)dv be the fractional number of molecules in the speed range from v to v + dv.
A probability distribution function has to satisfy the condition
We can then use the distribution function to compute the average behavior of the molecules: