Is it equal to atmospheric pressure? If so why? Surely the amount of air particles inside the bottle is much lower than the amount of the air particles above the bottle (i.e. the column of air above it) which should make it collapse as soon as the container gets sealed?
[Physics] the pressure inside a closed bottle of air
ideal-gaspressure
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It took me quite some time to clearly understand the experiment you're describing.
Actually, pouring a full bottle in a container is a quite intriguing thing.
Consider the following starting configuration :
This of course is an unstable situation, as the pressure $P_0$ cannot be at the same time the pressure of the air in the bottle, and the atmospheric pressure since the height of water in the bottle is higher than the level in the container.
So we should quickly get to this configuration instead :
You'll agree that along the red line, the pressure is $P_0$, so what is the pressure $P_1$ ?
Using simple hydrostatics, $ P_1 = P_0 - \rho \, g \, H$
Notice that in the picture as well as in this calculation, we consider the height $H$ to not have changed, i.e. very little water has moved out of the bottle into the container. We'll see why now.
What is now the volume of air in the bottle ?
Using the law of perfect gases $P_0 * V_0 = P_1 * V_1$, hence $$\frac{V_1-V_0}{V_0} = \frac{1}{\frac{P_0}{\rho \, g \, H} - 1} = \frac{1}{\frac{10^5}{10^3 \, 10 \, 10^{-1}} - 1} \approx 1 \% $$
For this numerical estimation I took a water height in the bottle of $10 \, cm$. The variation in volume is so small, it will be hardly noticeable !
The reason why pouring the bottle is intriguing is that it empties itself in bursts. A bubble of air gets in, and water gets out at once. But if you do it in a controlled way, you will end up in the initial configuration I described, and from that point onwards, no air can get in. The variation of volume of the air in the bottle we just obtained obviously corresponds to a volume of water that gets out of the bottle, but again, it is small, and hardly noticeable.
What if you take a longer bottle ?
gigacyan is right, something will happen after a while. Recall that I did the calculation assuming the amount of water exiting the bottle is very small, this assumption is now false. If you have a significant height of water, the pressure will be enough to push out quite a bit of water out of the bottle, in which case the pressure of the air in the bottle will go down, and the level of water in the container go up.
If you consider a very wide container, its level will stay roughly the same, but the level of water in your bottle will go down. A simple calculation leads to: $$P_0-\rho \, g \, h_{final}+\rho \, g \, (H-h_{final})=P_0$$ Hence $h_{final} = H/2$, which is the point when the low pressure in the air is able to lift the weight of the water underneath, down to the free surface.
Several interesting remarks can now be made.
To begin with, the pressure in the air keeps on dropping, $P_1 = P_0 - \rho \, g \, h_{final}$. Nothing prevents it from going to negative values, which happens when $h_{final} = \frac{P_0}{\rho \, g} = 10 \, m$. That's where this famous value of 10 meters comes from.
Now, if you think about trees, at first you may imagine they rely on capillary action to carry sap to their leaves, but that can't be the case, as the pressure drops too much after 10 vertical meters against gravity. Any presence of air would make the wood crumple under its own applied pressure.
Which means there is absolutely no air whatsoever in the sap canals of a tree (a.k.a. xylem).
The trees rely principally on another mechanism to pump up sap, known as evaporation. This easily produces (highly) negative pressures in the sap, and the actual limit to the size of a tree is the point when this pressure is small enough that a cavity of water vapor spontaneously appears in its canals, through cavitation. Pull hard enough on water, and you will create two interfaces and evaporate some of the liquid ! This cavitation pressure is around $-120 \, MPa$.
This catastrophic failure is know as embolism, and is also a bad health condition for humans (a gas bubble in a blood vessel).
Shaking the soda bottle makes the dissolved gas escape the liquid. As long as the lid is on, this can't happen (or happens for a little bit and then stops): the air in the bottle is already saturated with gas.
Then you poke a hole in the bottle, below the surface. The liquid may come out, reducing its volume in the bottle, thus expanding the volume of the air inside the bottle. In a non-fizzy beverage, the expanding air reduces its pressure and "sucks the liquids back in", as equilibrium is established. No liquid flows out of the hole.
In a fizzy beverage, the expansion of the air volume makes it so that it isn't saturated anymore; the gas solved in the liquid can move out of the solution; the air in the bottle thus is both expanding and gaining particles. As long as these two factors balance each other, the air can expand without dropping in pressure, allowing the liquid to flow out. When the separation of the gas from the solution can't supply a number of molecules large enough to keep the pressure constant, we would fall back into the "non-fizzy beverage" case, stopping the flow.
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Taking your argument about the column of air above the bottle and how it should crush it:
You are not getting crushed right now. Whilst you are correct that this will mean that lower down in the atmosphere there will be slightly higher pressure, it is essentially negligable compared to a few metres above the bottle.
Consider unbalanced forces, if the pressure in the bottle was substantially lower than the pressure outside, there would be an unbalanced force and the material of the bottle will accelerate inwards (you can see this in a cheap plastic bottle if you suck some of the air out).