Intuition behind the reason why gas balloons rise

In this post I will try to explain the intuition that allows us to understand why gas balloons rise into the atmosphere. This post will prepare the readers to better understand the physics and mathematics of gas balloons that I will be presenting in my next post.

Water tank analogy
Imagine that you have a tank full of water. Imagine also that you have a one litter plastic bottle filled-in with air and with a string attached to the neck of the bottle. Now imagine that there is a magical hand (occupying no volume) that picks up the wire and dives the bottle in the water tank. The bottle will only stay suspended under the water if the magical hand keeps holding the string. If the hand let the string go, the plastic bottle will emerge. This is common knowledge, but lets try to gain an intuition on what is happening. When the one litter bottle is submerged in the tank, it has to displace one litter of water so that there is room for the bottle to occupy its space in the tank. This means also that the water level in the tank will increase in one litter (if the magical hand had volume we would have to consider it also to know the increase of the water level in the tank). Now lets think the following. The air is lighter than water, this means that the one litter air bottle is lighter than one litter of water. This means that, when the magical hand releases the bottle string, the one litter of water that was displaced to make room for the bottle, as it is heavier than the bottle, will want to claim its place back, and when it does, the bottle will have to occupy the space where that water was, and this process will be repeated as long has the bottle is surrounded by water. When the bottle emerges it will have water bellow and air above. As the bottle is lighter than the water it will not submerge again. And as it is heavier than the air (because of the weight of the bottle itself) it will not raise up into the atmosphere. This is the law of buoyancy. 

Fig. firstly the water tank is empty; then a magical hand dives an air plastic bottle; afterwards the magical hand releases the air bottle string and the water surrounding the bottle, which is heavier, claims it space where the bottle is located; finally the bottle is pushed up into the surface because whenever the bottle is surrounded by the heavier water it will claim the space it occupies and as the contents of the tank can only expand to above, it is to the above that the bottle moves until is not surrounded any more by water but also by the lighter air (@Paulo Carmo)

So, to summarise:

- Lets have two things. Thing 'A' and thing 'B'; A and B can be objects, gas, etc.;
- 'A' is lighter than 'B';
- If 'A' is surrounded by 'B', the space occupied by A will claimed by B;
-  Now we have 'A' occupying the space of 'B' and vice versa;
- If 'B' continues being surrounded by heavier things, then the space it occupies shall be claimed again;

- 'B' moves in the direction of where it can claim space

With the water tank analogy clear on our mind let's now mode deep into the atmosphere.

Gas balloons in the atmosphere

While the behaviour of objects in the water is very well know since we practice with it on a daily basis, the interaction of objects with the atmosphere is a bit more esoteric to most people. So, equipped with the previous water tank analogy, lets now imagine that all this air above our heads, the atmosphere, is in contained inside a air tank (only open on the top at the end of the atmosphere, 10000Km above the earth's sea level). Now imagine that we fill in a balloon with an element lighter than air like hydrogen (the lightest existing element) or with helium (the second lightest existing element) and string it to an heavy rock. If we do that we get an object which is lighter than the air that surrounds it (when released from the rock), and if we make it big enough we can even attach objects to the balloon that its total weight is still less than the air surrounding it. Now the mechanics of the raise of the balloon is exactly the same of when we had a water tank. When the gas balloon string is released, the air that was displaced to make room for the gas balloon, as it is heavier than the balloon, will want to claim its place back, and when it does, the balloon will have to occupy the space where that air was, and this process will be repeated as long has the balloon is surrounded by air and no other event (explained bellow) occur, which will make the balloon continuously rise deep into the atmosphere. But now the question is, and does it stop rising? Well, theoretically, as long as the gas balloon is surrounded by elements lighter that the balloon itself, it could keep rising until the "end of days", but here a different phenomena occurs - the atmospheric pressure - that does not allow the balloon to rise forever. So in the atmosphere the same law of buoyancy applies.

Atmospheric pressure and its impact in the gas balloons

The atmospheric pressure, or air pressure, is the force applied into a surface by the weight of air existing in the atmosphere above that surface. This is applicable to the Planet Earth or any other planet with an atmosphere. The referred force is applied to any object in the planet including yourself, or in what concerns this post, to a balloon.
The air in the atmosphere is constituted by small particles called air molecules, and depending on the altitude, the concentration of air molecules in the atmosphere is bigger (at sea level) or smaller (in the higher altitudes of the atmosphere). The figure bellow, which was built on data taken from the following site, shows the force which is applied by the atmospheric pressure to a man at different altitudes.  

Fig. Atmospheric pressure at different altitudes - not in scale (@Paulo Carmo)

For instance, at sea level, 0 meters of altitude, the atmospheric pressure applies a force of 1.03 Kg/cm2 to a man. This is due to the weight of all air existing in the atmosphere above the man. If the man were at an altitude of 12192 meters, the atmospheric pressure would be almost 9 times less, particularly of 0.191 Kg/cm2. This is due to the fact that the man at 12192 meters has much less air (molecules) above is head. Other related phenomena is that the air density (in the right side of the figure) decreases with the altitude (as does the air pressure). The air density is the mass per unit volume of Earth's atmosphere, and it changes also with changes in temperature or humidity. The air density is related with the fact that the air can be compressed to fit in a smaller volume (volume is the space that contains an object. In the case of a gas, or air, the volume of the container will tell you the volume of the gas. Volume is typically measured in litres or milliliters. For example, an open 1-litre bottle at sea-level contains 1-liter of air).When the air is compressed it is said to be under high pressure, which is what happens, for instance, at sea level.
A question that can be formulated is the following: why don't we fell, at sea level, the force of 1.03 Kg/cm2, that is applied to us by the air existing in the atmosphere above us? Well the reason is that we have air inside our body too, and that air balances out the pressure outside so that we do not feel the air pressure.

But all this information is to explain what is the impact of the air pressure in a gas balloon as it rises into the atmosphere. Well, as the balloon rises, and as we have seen, the atmospheric pressure drops, and so the force which is applied to the exterior of the balloon decreases. Since the gas inside the balloon stays the same (the balloon is sealed), also the force that it exerts outward stays the same while the balloon rises.
So, when you are at the Earth surface and fill-in the balloon with gas, the equilibrium of internal and external forces tin the balloon will result in a given size of the balloon. As the balloon rises, the air pressure external to the balloon will drop and since the internal gas pressure is maintained, what happens is that the balloon will increase in size - because the pressure that the gas inside the balloon applies encounter less resistance from the external air pressure. When the internal gas pressure is bigger than the sum of the external air pressure and the pressure (resistance) of the balloon fabric, the balloon bursts. And this is the event that prevents the balloon to rise forever - at a given altitude the balloon bursts.