The question Why does the air pressure at the surface of the earth exactly equal the weight of the entire air column above it asks why the air pressure at any elevation is equal to the weight of the column of air above it in the atmosphere. I'm not convinced by any of the answers. Also, I'm going to slightly change the topic from a tall column of air in the atmosphere to the much simpler problem of a tall column of gas in a laboratory; the many processes that take place in the earth's atmosphere muddy the waters too much.
All of the answers boil down to considering a small parcel of gas; gravity pushes that parcel downwards, the gas-pressure differential pushes the parcel upwards, static equilibrium requires that the two forces be equal, and this results in the desired conclusion. Indeed, Feynmann (Chapter 40.1) makes this same argument in his Lectures, so I suppose it must be correct :-). Many physics textbooks make essentially the same argument for equilibrium water pressure, as well.
My problem is that a gas does not exist in small packets surrounded by thin weightless membranes, as those explanations all seem to imply. Gas molecules flow freely throughout space. I don't understand what it means for a "packet" of gas to be in equilibrium when molecules are entering and leaving the packet all of the time. Presumably the same number of molecules enter it as leave it, but the conditions for that to occur seem like they must be substantially more complex than the above explanations imply, especially when collisions are taken into account.
Is this another case where, because the mean free path of gas molecules is so small, they drift very slowly and hence behave as if they were really in small fixed packets? If so, would the above claims not apply in a less dense atmosphere (e.g., a high vacuum)?
Here are two explanations that seem plausible to me. Is either of them correct? Note that:
Again, I'm neglecting large-scale atmospheric effects such as solar radiation, so that the answers will be relevant to a column of gas in a laboratory.
Both explanations disregard collisions. However, a collision between two molecules obviously occurs only if the two molecules are at the same height. Since collisions are elastic and thus conserve total kinetic energy, the average kinetic energy of all molecules at any height is thus not affected by collisions. Since both proposals below are based largely on the concept of gravitational potential energy changing into kinetic energy as molecules fall, they should thus not be affected by collisions.
Here are the two theories:
- Gravity makes molecules gradually accelerate downwards. Neglecting collisions, the molecules closer to the earth would thus be (on average) moving faster. They are still moving in random directions, but are moving faster than the molecules higher up due to their lower gravitational potential energy. Similarly, because molecules tend to fall, there are more molecules lower down than higher up. Thus, lower heights will have a higher concentration of molecules, moving at higher speeds. This will result in a diffusion gradient of molecules moving upwards. This diffusion gradient will eventually balance the gravitational pull downwards, resulting in equilibrium. Result: close to the bottom we will have more molecules, and they will be moving faster, resulting in more kinetic energy per unit volume, which means more pressure. But while the argument seems to make sense qualitatively, that's a long way from a quantitative proof.
- Here's a completely different (and quantitative) argument. Consider dropping a single molecule from some height h. It will accelerate downwards until it bounces elastically into the earth and then slow down as it rises up to the original height h, and then do it over and over again. A bit of simple Newtonian mechanics quickly predicts the kinetic energy as a function of height, and then the velocity as a function of height, and then the amount of time spent in any small height interval, and then the fraction of the total time that the molecule spends in any small height interval. Now do this with many molecules that never collide. The number density at any height is then proportional to the fraction of time a molecule spends at that height. And the pressure at any height must be the number density times the kinetic energy per molecule. In fact, a few simple calculations (that I would be happy to include if requested) quickly result in precisely the conclusion that we wanted --- that the pressure change in any parcel is exactly equal to the weight of that parcel. It also predicts that temperature increases closer to the ground, due to the gravity-induced higher kinetic energy per molecule. Wonderful! But is it correct? Note that since molecules move more slowly at higher altitude, it also predicts that the number density increases at high altitude, which seems wrong.
I like theory #2 because it's quantitative and it exactly predicts that pressure at any height will equal the weight of the gas above it. Its prediction that number density increases at higher altitudes seems hard to stomach, but seems unavoidable. If molecules move more slowly at high altitudes (which they almost certainly do), then they spend more time traversing the same height delta at high altitudes than low, and they thus are statistically more likely to be found at high altitudes.