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If the earth's gravity exerts a net downward gravitational force on all air molecules, how come the molecules don't eventually lose their momentum and all settle down? How is the atmosphere is still miles thick after billions of years?

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First of all, gravity does continually accelerate the air molecules. I don't see how this could make them lose their momentum.

What is the net effect of gravity on the atmosphere? Simple, gravity prevents the atmosphere from flying off in space, and instead it keeps it comfortably wrapped around our planet!

The reason why the atmosphere is still thick after billions of years is because you have two net effects on the air molecules, gravity, which keeps it as close as possible to the ground, and inertia, who has the opposite net effect. So as long as the molecules do not slow down they "orbit" our planet.

The reason is the same as why is the moon orbiting the Earth after billions of years. There's a balance between the kinetic energy of the moon and the gravitational potential energy - or a balance between gravity and inertia.

The other answers give you a summary explanation of what determines the temperature of the atmosphere and hence its molecules' average velocity. The reality is way more complicated as the temperature of the atmosphere is not constant with height and you have to take into account many more factors like varying pressure, convection and so on. Modelling the Earth's atmosphere accurately is very complicated.

In conclusion the basic mechanisms are outlined above. I hope they answer your question.

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I don't understand the talk about inertia and comparison to moon and Earth. This problem has nothing to do with inertia. There is only energy (molecules want to get as close to Earth as possible) vs. entropy (molecules want to occupy as big phase space as possible and so they want to go higher) battle. Of course, in reality the picture is complicated by many processes, but this is the main point. – Marek Dec 18 '10 at 16:17
@Marek, why would entropy decrease if the molecules fell down the gravity well? Gravity increases entropy (see, or maybe I am misunderstanding your comment? – Sklivvz Dec 18 '10 at 16:35
Huh? I never said anything like that. Systems in equilibrium want to minimize their energy and maximize their entropy. But this can't be done at the same time so there is a battle between the two phenomena (which can be made precise by Boltzmann statistics). But your answer doesn't talk about temperature/entropy part of the problem at all and therefore it is wrong. – Marek Dec 18 '10 at 16:37
@Marek, the talk about inertia is the following: gravitational attraction keeps the atmosphere wrapped around us. Without gravitation the atmosphere would simply float away because molecules going outwards would keep their constant motion due to inertia. – Sklivvz Dec 18 '10 at 16:38
@Marek: "you don't agree with me, and are therefore wrong" is a poor argument. The phenomenon can be explained without using entropy, in my opinion (although you can use it if you like). – Sklivvz Dec 18 '10 at 16:39

The key ingredient is temperature.

If it were zero then all the air would indeed just fall down to the ground (actually, this is a simplification I'll address later). As you increase the temperature the atoms of the ground will start to wiggle more and they'll start to kick the air molecules giving them non-zero average height. So the atmosphere would move a little off the ground. The bigger the temperature is the higher the atmosphere will reach.

Note: there are number of assumptions above that simplify the picture. They are not that important but I want to provide a complete picture:

  1. Even at the zero temperature the molecules would wiggle a little because of quantum mechanics
  2. the atmosphere would freeze at some point (like 50K) so under that temperature it would just lie on the ground
  3. I assumed that the ground and the atmosphere have the same temperature because they are in the thermal equilibrium; in reality their temperatures can differ a little because of additional slow heat-transfer processes

To leave you with something a little bit more concrete (although quite simplified), statistical mechanics tells us that probability $p$ that some portion of the system will occupy a given energy $E$ level is $p \propto \exp({-E \over kT})$ where $k$ is a Boltzmann constant (note that you have to normalize this probability to 1 by summing over all energy levels; that's why it's only proportional, $\propto$).

For the atmosphere of ideal gas where molecules have only potential energy (kinetic energy is not important here because the molecules don't interact with each other in the ideal gas) we have $E = mgh$. This gives us $p \propto \exp({-mgh \over kT})$. You can see that atmosphere "wants" to get lower ($p$ is bigger for $h$ lower) but the temperature "forces" it to go up. Play with the equation for different values of $T$ and $g$ (gravity strength) to see what happens.

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+1 for Boltzmann distribution. – gigacyan Dec 18 '10 at 8:42
I won't downvote this because everything is perfectly true, but I don't think it adresses the question. From what I read, the question is asking Why doesn't the earth's gravity reduce temperature by pulling down the molecules?. The answer would be conservation of energy, lack of friction, and the elastic nature of colisions between molecules. – Malabarba Dec 18 '10 at 20:10
@Bruce: I'll have to think about it but my first impression is that you totally misinterpreted the question. Of course, it's possible that I did the very same. Anyway, I don't get your talk about lack of friction and elastic nature of collisions. First is just not true (there is friction) and second is unimportant. – Marek Dec 19 '10 at 2:04
@Marek: Sure, it might just be me who's misinterpreting stuff, the only way to be sure would be if the OP made a comment. Even I turned out to be right I wouldn't delete this answer anyway, there's never any harm in teaching people some statistical mechanics. =) – Malabarba Dec 19 '10 at 22:59
@Marek: As for the friction. Friction as it is macroscopically (a force that opposes relative motion between bodies in contact) does not exist on the molecular level. A gas molecule moving inside a box doesn't suffer the friction we experience on a macroscopic level. It does, of course, suffer numerous collisions which we can call resistance, but I wouldn't call friction. Is that what you meant? – Malabarba Dec 19 '10 at 23:01

The other answers are correct but to understand them you have to get an idea of how much thermal energy does an average molecule have.

According to Maxwell-Boltzmann distribution, the most probable speed of an air (say, nitrogen) molecule at room temperature is $v_p = \sqrt { \frac{2kT}{m} } = 422 m/s$. Without collisions with other molecules it can travel upwards $h=\frac{v_{0}^{2}}{2g}$ = 9 kilometers before the gravity stops it and pulls back to Earth. Basically, potential energy of molecules in gravitational field is too small compared to their kinetic energy to keep them low.

Update: Still, gravity is the reason why we have atmosphere after billions of years.

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incorrect, the atmosphere is hundreds of kilometers thick. – Sklivvz Dec 23 '10 at 10:07
At this point I'd like to express big thanks to @Sklivvz because he made me look at what the real atmosphere looks like. I have to say I was stunned by e.g. the fact that menopause is about $170K$ because of $CO_2$ cooling while slightly above it thermosphere is about $2000K$ hot; that exosphere reaches as far as half the way to the moon (where solar wind finally overcomes gravity) and that there exists turbopause after which gases no longer mix with each other. I never imagined atmosphere to be such a lively place :-) – Marek Dec 23 '10 at 13:03
Of course, the above is just a little excerpt of all the processes that go on in the atmosphere that I found interesting. By no means is it an exhaustive list. – Marek Dec 23 '10 at 13:05
@Sklivvz: I know that and I didn't say that all molecules stop at 9 km. I just tried to give feeling of how strong thermal motion is compared to Earth's gravity. – gigacyan Dec 23 '10 at 17:28

This is because the sun is supplying them with kinetic energy all the time, which you know is temperature. Then the ideal gas law states p*V=nRT, so the volume keeps pretty much constant.

The pressure is causes by the electrostatic repulsion between the molecules pushing them apart, and the gravity pulling them towarsd earth.

Another important reason they dont 'settle down' is that temperature is the lowest form of energy, this means that what you actually ask for cannot happen in any closed system, the temperature energy can only go towards heating something else, the core of the atmosphere is in this sense not a perfect analogy, because some of its energy will be "lost" to heating the earth and the oceans, and most will be radiated into the universe.

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But what pressure is keeping them up? It seems like there's plenty of free space in air. – wrongusername Dec 18 '10 at 4:43
Well, it only seems so. – mbq Dec 18 '10 at 13:58
@wrongusername: imagine a gas in a box. The only pressure there is, is exerted on the walls of the box. The molecules bounce off the wall and go back. But our atmosphere doesn't really have walls. There is only wall on the bottom which we call ground. So molecules are in fact falling down and then jumping up all the time. But this wouldn't work if the temperature of the ground were zero because bouncing off the ground dissipates energy and after a little while atmosphere would fall down (same way the ball does if you let it fall). You need "hot" ground to give some energy back to the air. – Marek Dec 18 '10 at 16:27
@wrongusername: of course, I was talking just about ideal gas. In that case you don't really have any pressure in the atmosphere because molecules don't interact with each other. But if you'd put some object there (like balloon) it would get hit by the molecules all the time and it would experience the pressure. So it's important to distinguish between the pressure the air is able to exert and the actual pressure that happens by bouncing off the walls. By the way, the pressure would obey the same exponential law I stated above for probability. So that it would be pretty low up high (cont.) – Marek Dec 18 '10 at 16:30
@wrongusername: in accordance with gigacyan's answer. When the molecules fly 9 km up they have to overcome gravitational attraction and lose almost all they kinetic energy. So on average the molecule 9km up is very slow and therefore exerts almost no pressure (on a balloon that would happen to be there, say). – Marek Dec 18 '10 at 16:32

The sun keeps the earth warm. Without it, the atmosphere would freeze solid.

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If the atmosphere were made out of ideal gas, it wouldn't freeze (and there are actually some non-ideal real gases that don't freeze). So this answer doesn't address the main point of why doesn't gas fall down. – Marek Dec 18 '10 at 16:35
@Marek: actually the gas "falls down" all the time. – Sklivvz Dec 23 '10 at 10:11
@Sklivvz: no. Only individual particles. But average height (which is the only thing that is important here) doesn't change in first order. – Marek Dec 23 '10 at 10:13
@Marek: it was a point about terminology. I think we both understand the physics. :-) – Sklivvz Dec 23 '10 at 10:15
The gas would "compress" or "sublimate" :-) – Sklivvz Dec 23 '10 at 10:15

Hydrostatic balance...the vacuum of space wants to suck all the molecules of air out but it's perfectly counter balanced with the gravity of Earth...all the while we keep spinning and rotating thousands of miles per hour in space around a giant nuclear reactor just waiting to destroy us all :)

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