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The Maxwell-Boltzmann Distribution

posted Dec 15, 2016, 7:29 AM by Grace Ong   [ updated Dec 15, 2016, 7:32 AM ]
The following post was first posted on Blogger on Friday, 5 April 2013.

Molecules in a gas sample do not all have the same energy. Rather, there is a distribution of energies. The Maxwell-Boltzmann distribution is a graph of the number of molecules with a given amount of energy against that amount of energy. It shows the distribution of molecular energies in a gas sample, and this is what it looks like at different temperatures:

maxwell boltzmann distribution curves at different temperatures

Although it looks like an easy graph to draw, I often find students making silly mistakes. I suspect these came about because they were learning the graph by rote, without fully understanding what it represents.

Common mistake #1: The graph does not begin at the origin.

maxwell boltzmann distribution - common student error - origin (1)
maxwell boltzmann distribution - common student error - origin (2)

Common mistake #2: The axis labels are switched.

maxwell boltzmann distribution - common student error - axes

To help my students better understand the Maxwell-Boltzmann distribution, I decided to take them down memory lane, back to their time in primary school when they were learning about bar graphs.

I began with a very, very, very tiny gas sample comprising of just 29 molecules, and assigned a certain amount of energy to each of them such that
  • 0 molecules have 0 J of energy (since temperature of the gas is above the absolute temperature of 0 K)
  • 3 molecules have 1 × 10−21 J of energy
  • 5 molecules have 2 × 10−21 J of energy
  • 6 molecules have 3 × 10−21 J of energy
  • 5 molecules have 4 × 10−21 J of energy
  • 4 molecules have 5 × 10−21 J of energy
  • 3 molecules have 6 × 10−21 J of energy
  • 2 molecules have 7 × 10−21 J of energy
  • 1 molecule has 8 × 10−21 J of energy
The above information was then presented in a bar graph as follows:

maxwell boltzmann distribution made simple
Total number of molecules = 3 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 29

At this point, it became very clear that if this system were to be scaled up to one that contained a very huge number of molecules, one would arrive at the smooth Maxwell-Boltzmann distribution curve. And the area under the curve is the total number of molecules in the system.

(There was this look of realisation dawning on my students when I related the simple bar graph to the Maxwell-Boltzmann distribution. Let's just hope that I don't see them making the same mistakes again.)