Flat-earthers have this common argument and objection that according to the "globe model" Earth's atmosphere is a big bunch of gas adjacent to a vacuum, which according to them must mean that the atmosphere should quickly expand to this vacuum of space until there's nothing left on the Earth.
Ok, let me explain why the Earth has an atmosphere and why it doesn't escape to space.
You agree that objects that have mass fall down towards the surface of the Earth, don't you? This is, of course, trivially observable. You might not want to call that phenomenon "gravity" (for a reason that nobody, not even you, can understand), and instead you want to call it "buoyancy". But whatever you want to call it, it's undeniable that the phenomenon exists: Objects that have mass fall towards the ground. If the substance is fluid enough, it might form layers with the least dense parts on the top and most dense at the bottom, but at its own density level the direction of falling is down. It will not go up to a less dense part above it.
(On a side note: Yes, buoyancy is a thing, and it exists, and you can read the Wikipedia articles about it, and all kinds of academic material about it. The thing is: Buoyancy works in the direction it does because of gravity. In fact, buoyancy works because of gravity. No gravity = no buoyancy. Anyway, this is just a side note. You don't need to accept this in order to read this explanation.)
Your main problem is that you seem to think that gas is somehow different from other forms of matter, and that the same physical laws don't apply to it, that it behaves completely different from anything else. You seem to think that there are two sets of natural laws: One that applies to gases, and another different one that applies to all other substances.
That's not so: The exact same laws apply to all substances in the same way. Gravity and buoyancy both apply to gases in the same way as it applies to liquids, in the same way as it applies to very fine material (like dust), in the same way as it applies to low-density particles. And yes, even the same way it applies to big solid objects. (The only difference here is that when it comes to big solid objects, their physical size and geometry start playing a significant role in how they behave when they come in contact with other objects. But this is not because different laws of nature somehow applied to them: It's simply a consequence of their size, shape and geometry.)
So yes: Gas also falls towards the ground in the exact same way as anything else. This is because, like all substances, also gas has mass.
If the surface of the Earth would be a complete vacuum, and you released a bottle of gas near the surface, it would indeed fall towards the ground. Obviously it would spread very thin over a very large area of the surface because of its liquid nature, but it would still be on the ground. Gas is not somehow some kind of magical element that ignores gravity/buoyancy and would fly off to space. If the gas is visible (eg. some kind of smoke) you could literally see it falling towards the ground, and spreading out as it hits it, to form a very thin layer over a large area.
You might be asking: "So why isn't the atmosphere just a very thin layer on the ground?"
Because there's a huge amount of gas in the atmosphere. There's so much that it covers the entire surface of the Earth, forming a very thick layer of gas, miles and miles high. It's the sheer amount of it.
This can be seen by the fact that the higher you go in the atmosphere, the lower the atmospheric pressure. In other words, the thinner the air is. That's because lower down there's a lot more air on top, pressing it down. Higher up there's less air above pressing it down.
Keep going up, and up, and up, and what do you think eventually happens? That's right: Eventually the gas is so thin that it's almost undetectable. Keep going up even further, and there will be nothing. It's not a sharp surface like the surface of water: It gradually gets thinner and thinner until there's nothing. Everything below is pressing against the air below it, all the way down, until we get to the air that's pressing against the ground itself.
Think of it like this: Suppose that instead of air we have a big bunch of small pieces of very lightweight foam. Let's assume we are in a vacuum. If you throw these pieces of foam, they will fall to the ground. Keep throwing more and more small pieces of foam. What starts happening? The foam starts piling up on top of itself. Keep piling up more and more foam, and you'll start noticing that the foam on the bottom will start getting more and more compressed. Keep doing this and you'll get a clear gradient: Very compressed foam at the bottom, and the foam gets gradually less and less compressed as you go up, until you reach the top, where it's not compressed at all.
Air is no different from this. It works the same way. The only difference is that instead of pieces of foam you have molecules, which are much, much smaller. However, these molecules behave essentially the same way. They may spread out a lot easier than foam because they are so small, and they may have much stronger fluid dynamics than the pieces of foam, but in the grand scale of things they essentially behave the same way: Very compressed at the bottom, and less and less compressed as you go up, until you effectively reach the surface, where there are no air molecules anymore.
In other words, the atmosphere stays on the surface of the Earth because it has mass and gravity, or buoyancy if you prefer that term, keeps it down. Gas has no magical properties that would allow it to escape Earth's gravity any more than other objects.
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