The Boltzmann factor

We can learn a lot about how particles in a gas share energy by using this model below. Imagine a 6 by 6 grid of squares with a counter on each square. We are going to take counters off squares and put them on other squares.

So what has this got to do with gases?

Our gas particles are bashing into each other exchanging energy. Each time they collide they may gain or lose energy. If we imagine that this energy comes in packets, or quanta, then the model above is a good representation.

Most of the particles in the gas have energy k T. Some have more energy. Knowing that the relationship is exponential means we can calculate the fraction of particles which have a certain amount of additional energy. This is very useful as we shall see.

 

    

example: A gas is at a temperature of 300K.

k T = 6.6 x 10-34  x  300 = 1.98 x 10-31J

additional energy = 10kT - kT = 9kT

e -(e/kT) = e -9  = 1.23 x 10-4

 Atmospheric pressure decreases exponentially with height. If we know the temperature at a certain height we can calculate the air pressure at that height. Climbers at the top of Everest can still breathe, but only just!