Energies

The picture below represents the density of the earth's atmosphere from sea level upwards.  The triangle represents Mount Everest to give a scale.

Why does the atmosphere thin out as we go up?  Everyone knows that the pressure gets less with height, so it seems that the lower parts have to support the upper parts. But the air is mostly empty space between the molecules, so why doesn't the upper air fall down and fill the gaps?  After all, the pressure at the bottom of the atmosphere is equal to the pressure caused by only about nine metres of water, so teh air is obviously thin compared with water.

When a molecule hits the ground, it isn't hitting a dead surface - the molecules there all vibrating with an energy related to the temperature.  So air molecules and ground molecules share their energy.  Sharing also happens when air molecules collide, so they all end up with a distribution of velocities of which the mean represents the temperature.

This velocity is high enough to shoot a molecule miles into the air.  Of course it will hit others on the way, so the ones that reach miles high are not the ones that bounce off the ground.  The molecules also have energy because of their height, of the kind that is used in a hydro-electric power station when the height energy is converted into turbine motion.  From the decreasing density of the air we know that the higher the energy, the fewer the molecules.

Why is this so?  Suppose the air had a sharp boundary, with air on one side and empty space on the other.  What one of the highest molecules were hit from below by a lower one?  It would be knocked upwards from its original position.  Molecules all the way through the air are being knocked in all directions.  But only those at or near the top can on average go away from where they already are.  What actually happens is that the air automatically forms a distribution in which every molecule on average has an equal chance of going up or down.  We saw that this cannot be achieved with a sharp edge, and calculation shows that an exponential is the right answer.

That this must be the case is shown by the following argument.  Suppose that we dig a hole one kilometre deep . The same distribution must exist in the hole, as there is nothing magic about the exact radius of the earth.  In fact if the earth were one km smaller all round, and we added enough air to get the same densities in all the original places, we would continue the distribution 1 km deeper, as in the hole.  There is only one distribution that remaine the same shape for all heights, and that is an exponential.

This a universal rule - all other things being equal, systems will share out energy so that the energy density is less at higher energies.  We have to work hard to achieve any other arrangement.