I realize that I didn't address one of the questions (or one of the possible questions): "Why don't all the air molecules just fall to the ground and stay there"? In case anyone was wondering about that question, the answer is that the air molecules DO fall toward the ground, but they continually run into other air molecules (or molecules in the ground if they get that far down), all of which share a nonzero absolute temperature and therefore are in random motion, and in collisions sometimes a molecule will be knocked upward. When you work out the statistical mechanics of all this, you get an exponential falloff of density (in the approximation of a constant-temperature non-convective atmosphere). This falloff is a bit faster for the lower-mass nitrogen molecules than for the oxygen molecules, but as I explained in a previous note, both of these molecular species have mean heights of around 8000 meters, so you shouldn't expect much difference in oxygen vs nitrogen between your cellar and your attic.
A picturesque way of looking at this is the following. Imagine there is no atmosphere, and you're sitting at a table out in the open (in your spacesuit). Place a cup on the table. The atoms in the bottom of the cup are in contact with atoms in the top of the table, and all of these atoms are moving with random thermal motion related to the absolute temperature. At any given moment, there is a finite (but exquisitely small) probability that all of the atoms in the table underneath the cup happen to all be heading upward. In that case the cup will leap up off the table, knocked upward by the upward-moving atoms in the table. This would not violate conservation of energy or conservation of momentum (the Earth would recoil), but it would violate the Second Law of Thermodynamics, because given the gigantic number of atoms lying underneath the cup, the probability of all those atoms simultaneously heading upward is vanishingly small. You might have to wait for billions of billions of billions of years to observe the leap. Suppose instead of placing a cup on the table you place a single molecule of oxygen. Now it's not so improbable that an atom in the table might impart a significant upward speed to this single molecule of oxygen. Statistical mechanics provides the tools for calculating quantitatively the probabilities of various upward speeds. What you find is that the average speed imparted to an oxygen molecule by an atom in a table at room temperature is a speed sufficient for the oxygen molecule to go up 7920 meters before falling back down! In other words, statistical mechanics gives the answer (the same answer) to two different questions: 1) What is the average height attained by one oxygen molecule in contact with a table at room temperature? (Ans. 7920 m) 2) What is the average height of all the oxygen molecules in a constant-temperature atmosphere? (Ans. 7920 m) (I'm deliberately playing rather loose with the word "average" here, but the basic idea is correct.) There's yet another source of amusement in this statistical picture. Suppose you have a box whose sides have an accurately known mass. Suppose you weigh the box in an airless room (to avoid buoyancy effects) with and without the box being filled with atmospheric-density air. You're not astonished that the extra mass with the air is equal to the mass of the air added to the box. But maybe you should be astonished, because at any given instant almost none of the air molecules are touching the inside of the box! The reason why the scale measures an increase is because of the e to the (-mgy/kT) density gradient. The air density and pressure are just a tiny bit higher at the bottom of the air (in contact with the bottom of the box) than at the top of the air (in contact with the top of the box). Momentum transfer per second from the bottom of the air to the bottom of the box is very slightly greater than the momentum transfer per second from the top of the air to the top of the box. When you work out the details, you find that this difference provides the conspiracy that let's you think you're measuring the mass of the air. The difference is small, but so is the mass of the air. Sometimes one describes air pressure at sea level as "the weight per area of the column of air above that area". But almost none of those air molecules are in contact with your measuring device! However, the number of molecules per cubic meter, and their average y component of velocity, is such (conspiratorially) as to hit your area with the same force as though an object with the mass of the total column of air sat on this area. Related amusement: Consider a steel ball bearing dropped from a height h onto a scale, and rebounding to nearly the same height every time. If the scale can respond very quickly, you will see sudden sharp spikes when the ball bearing hits, and zero at other times. Now suppose that the scale is sluggish, and/or h is small enough that the ball bearing hits the scale at a high rate (though small speed). What you can calculate is that the average reading of the scale is exactly the same as if you simply place the ball bearing at rest on the scale! Bruce ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
