Is pressure only due to weight?

There is no “adjustment” which occurs when or after the bottle is closed. The absolute pressure, and the distribution of velocities of the air molecules inside the bottle, will be the same as outside the bottle — as if the bottle did not exist around that region of gas.

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Suppose the bottle is open (with the opening facing up) and things are at equilibrium. Then we know the pressure inside the bottle is the same as the pressure outside the bottle (at equal heights), because if it wasn't there would be a net flow of air molecules to correct it.

Now put the cap on the bottle. This means we have placed a barrier between the air inside the bottle and outside the bottle. But we have not actually changed anything by doing so — we did not compress the air inside the bottle (except possibly by an insignificant amount depending on the design of the cap).

Therefore, the absolute pressure is again the same. If the pressure were not the same on the two sides of the cap, then we would feel a force resisting putting the cap on. If you look at kinetic energies and individual gas molecules, “there is no net flow” (when open) translates into “the molecules outside and inside strike the top and bottom of the cap, respectively, with equal numbers and velocity” (when closed).

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There isn't really a ‘shorter air column’ inside the bottle. The air column model is useful for understanding how atmospheric pressure comes about, but for a small container the gravitational effects are totally insignificant and you can just think about pressure by itself.

But if you want, you can say that, yes, there is a shorter column inside the bottle — but still at the original atmospheric pressure. Therefore, if it were free to move like the atmosphere is, it would rapidly expand to lower pressure, but it isn't, in two individually sufficient ways:

• If the bottle is open, then the pressure of the “short column” is opposed by the pressure of the remaining tall column of atmospheric air above it.

• If the bottle is closed, then the cap of the bottle is holding in the pressure.

  • If the bottle remains where it is, then the pressure on each side of the cap is equal (at least until the weather changes) and so it is under no net force and merely preventing mixing of otherwise-identical air.

  • If we transport our bottle into a vacuum chamber or up a handy space elevator, then the surrounding pressure is now zero, but the pressure inside the bottle is still the same — but because the balance of pressures and therefore forces has changed, the bottle is now at risk of breaking if it is not strong enough.

Does your bottle contain anything other than air? If it is a closed, unbreakable bottle, why do you assume that its pressure will be equal to the atmospheric pressure? (or are you defining that in your thought experiment?)

I think pressure is fundamentally related to molecular motion, and only indirectly to weight. Pressure can be exerted in zero gravity when warming up a balloon; the gas molecule travel more quickly, and impact the inner surface of the balloon at higher velocity, this with more kinetic energy, this with more force, thus with more pressure. I see no weight involved in that scenario.

Let's consider a scenario where weight appears to be involved, such as a column of a fluid in a container (I suppose this is you scenario). At the bottom of the column there is a certain pressure, given by the height of the column and the density of the fluid (and yes, the gravitational constant). But what is really going on? The molecules are moving, impacting that bottom surface...

Let's assume our container contains water and air. Why do the water molecules impact the bottom more than the top? They are also acted in by the gravitational force I suppose, which draws them downward. Gravity draws the water downward more strongly than it draws the air, because the water is "heavier". Perhaps there is a situation where one molecule of the liquid is lighter (less mass) than one molecule of the "air" gas. What we observe as humans is the bulk effect, so it is not the masses of single molecules that really matters but the mass of the bulk substance, which relates to its molecule by its density.

Either way, gravity directs the water downward, so the molecules tend to have more collisions with the bottom of the container than with the air in the container or the top of the container.

The pressure at the bottom surface is not due to gravity directly, but due to the collisions of the molecules with the surface; the molecules are (on average) directed toward the "bottom" by gravity, so more collisions occur there than at a higher surface. This higher pressure. Gravity plays a role in these scenarios, but it is not required for pressure to exist. Particle motion is required to produce pressure.

Oh yes, and pressure can produced by light particles, "massless" photons!