Everyone has seen a few balloons in his lifetime, whether they are helium or air filled. If this is true for you, than you probably have noticed that balloons filled with Helium float high into the sky, while the ones filled with air drop to the ground. You may also have wondered why this is the case, and what about hot air balloons?

To understand this, we have to look at the physics behind it. The most important thing we have to understand is that gases can be very closely described by the ideal gas law (pV = cT). The ideal gas law is an equation that allows us to relate pressure to the volume and the temperature. In this case if the pressure is kept constant, the volume and the temperature will change proportionally.
In more basic terms; the hotter it gets, the more it expands.
What happens is that the density of the gas, the amount of gas particles per volume, goes down when a gas expands. (The amount of particles stays the same while the volume in which they are becomes larger.)

Secondly, we need to understand the law of Archimedes, which states that the amount of displaced volume of a liquid or gas by an object, is proportional to the upward buoyancy force that this liquid or gas exerts back up on the object. Hence an object floats if the weight of the displaced material is higher than the weight of the material which displaced it.
If an object with a lower density pushes into an object with higher density, the higher density material will push it back up.
In more basic terms: If the amount of water which has been displaced is heavier than the amount of air which displaced it, it floats. (See second picture.)
This law explains how boats keep afloat in the water, namely by displacing enough water to make the upward force cancel the downward pull of gravity.

Now when we combine these two principles, we can understand why hot air balloons fly.
This is because the air in the balloon is warmer, means that the density of the hot air is lower than that of colder air. When compared to the example, this causes the cold air to function as water and the warm air as the boat.

The cold air pushes the pocket of warm air up, exactly as water does with a boat. We call this process convection. However, unlike water the balloon does not go to the top of the atmosphere but stops at the height that the operator wants it to. This is because the mass of the balloon also comes into the equation, counting as extra downward pull by gravity that has to be countered by the upwards lift of the cold air. If these forces cancel out completely, then the balloon is stationary. Make the lift force larger and it goes up, make it smaller and it goes down.

Now keeping this balance in mind we can look at a few kinds of balloons. The first is the soap bubble, a bubble filled with ordinary air, maybe slightly warmer than its surroundings. These bubbles float because the thin soap film weighs almost nothing compared to the air inside, and the balance is not disturbed.
Secondly, there are latex balloons filled with the same air, which don’t float at all. This is because the latex surrounding the air has a significant mass compared to the air, causing the gravitational pull to win from the upwards lift. However if we take the same balloon, but now filled with water and thrown into a pool, then we will see that it would float somewhere halfway. This is because compared with the water, the mass of the latex balloon is again negligible, like with the soap film in soap bubbles.

Finally Helium balloons work similarly, but because Helium has about a tenth of the density of air (This means that one kg of Helium takes up about ten times more space than a kg of air.) we can ignore the ideal gas law and focus solely on the Archimedes principle.
If the pressures of the gases are the same, the helium in the balloon will always be a lot less dense than the air around it, so it will undergo a large lift force. It will even lift with some additional ballast.

Knowing this, it is not hard to imagine that the buoyancy forces exerted by the air are easily strong enough to lift the balloon out of the atmosphere. However, this never happens because as the balloon will break at a certain altitude due to the pressure difference. (The air pressure decreases significantly on high altitudes, while the pressure in the balloon stays the same, making it pop.) The Helium itself is perfectly capable of leaving our planet. Infact, helium escapes our atmosphere constantly, making it an expensive non-reusable gas.

Convection can be found in many other processes, for instance; it is the cause of the bubbles in boiling water, the hot winds being pushed above the cold winds and lava lamps.