Statistical Mechanics: Maxwell-Bolztmann and playing games
It's much nicer to have a proper plotted Maxwell-Boltzmann distribution to use when teaching, rather than continually sketching them yourself. Here's an Excel file which has:
- Under the 'Change the variables and see' tab, a graph of the distribution which will re-plot whenever you change the variables in the yellow cells so you can show how the distribution changes at different temperatures.
- Under the 'Graph for various temperatures' tab, multiple plots of the distribution at different temperatures from 100K to 1000K which can be easily printed out for pupils.
THE MAXWELL-BOLTZMANN DISTRIBUTION: A GAME
At school level, the M-B distribution can perhaps sometimes seem like a bit of a mystery in terms of where it comes from and why it must be true. This is an activity - adapted from the Advancing Physics course from a few years ago - which uses dice throws to randomly shuffle counters (which represent particles in a material) on an imaginary ladder (the rungs of which represent energy levels). Eventually, a distribution very similar to Maxwell-Boltzmann emerges naturally. This file contains the resources you need, apart from the actual dice:
ANOTHER GAME TO SHOW HOW RANDOM WALKS WORK
Another interesting result that comes out of statistical thermodynamics is that a particle engaged in a random walk involving collisions - the obvious example being a smoke particle undergoing Brownian motion - is statistically most likely to achieve a non-zero displacement over an extended period of time. Given that each collision is random, this can be anti-intuitive, yet is more due to statistics than physics. This activity - also adapted from Advancing Physics - involves tossing a coin, moving a counter and seeing how far from the starting point the counter gets after a set number of goes. From experience, you need quite a large sample size before you start to get convincingly close to the 'right' result, so there's also an Excel file with some previous actual results:
[From experience, I've found that having each participant rolling four six-sided dice and subtracting four from the result works better than using a single 20-sided dice. Of course, this is exactly the opposite of what 'should' happen, since four six-sided dice rolled together are much more likely to give numbers in the teens than numbers like 4 or 24. Within the limitations of the game, though - specifically, with a relatively small number of throws - it actually pays to rig the statistics a little in favour of some particles being slightly more likely to get significantly more energy than their partners. It might be interesting to see which of your pupils spot this 'cheat'.]
HOW FAR DOES AN AIR MOLECULES TRAVEL IN A LESSON?
Once you've done enough of the above, you can combine it with other results from thermodynamics for a rather intellectually satisfying finale. This sheet takes you through the calculations necessary to show that, over a 1-hour period, a single air molecule in a classroom will collide with its neighbours ten thousand billion times, travel the equivalent distance of going from London to Rome yet only achieve a displacement of about 30cm.