Why does hot air rise in a column instead of cold air pressing down?

With the same argument, I could deduce (and I know that it's wrong) that the cold air above is denser, so it will go down, pressing the hot air away sideways.

Replace your hot air with a helium balloon. You can see there's no force on the balloon to push it sideways. The buoyancy forces it to accelerate upward (and some cool air around it to accelerate downward). If you don't stop at one, but keep creating balloons (similar to you continuing to heat the air from the pan), then you'll get a trail that forms a column.

The asymmetry in the situation is that you're creating a small amount of heated air in a large amount of cooler air.

If you reversed the situation by placing a block of ice near the ceiling, then you would get a column of cooler air falling through the relatively warmer air.

in my mind I envision a picture of (red) hot air molecules separated more than the (blue) cold molecules which slip down between the red ones.

Molecules in a gas have a distribution of speeds. So the cooler gas has almost as many fast molecules as the warmer one does.

But the problem here is that at such a scale, the size of your heated parcel is huge. A few molecules will do that at the edge (diffusion), but not quickly. The mean free path of an air molecule in your room is less than 100 nanometers, while the size of your heated parcel is probably several centimeters. Most will hit and remain close to their neighbors. It's much faster for the entire parcel to lift, so that process dominates.

Welcome to the magic of convection cells =)

The first thing to remember is that you're working with a large number of gas molecules. The effect theoretically occurs no matter how many particles you have, but the effects are much easier to describe using bulk terms that handle many molecules at once, rather than trying to track each molecule. As BowlOfRed mentioned, the free path length in open air is about 100nm, which means almost all effects you see are going to be macroscopic statistical effects, like density and mass transfer.

Consider a volume of hot air above the hot plate. I am going to claim its shape is roughly cylindrical. This is easy to prove early on, when the hot air is all concentrated in a disc shape just above the hot plate. We'll use induction to show that it remains cylindrical as the system evolves.

Now its easy to see that the system is in an energized state. Its ground state would have all of the high energy (low density) particles up high, and all of the low energy (and thus high density) particles down low, because that minimizes the potential energy of the air column. However, we have to figure out how it accomplishes this.

Consider, just for a moment, the radial movement of air. Cold air, trying to work its way down to its lower potentials is willing to displace hot air. Thus the hot air tries to move up, in all directions including outward, and the cold air tries to move down, in all directions including inward. However, we can't have two crossing streams of molecules, because they collide. This collision keeps the inward/outward velocity of most molecules of air very low. But this isn't true everywhere.

Near the bottom, right on the hot plate, there is no hot air clashing with cold air. Once you're down on the surface of the hot plate, there no more hot air trying to go up, but there's still cold air trying to go down. This is where we start to see movement. The cold air sweeps in horizontally, until the pressure equalizes.

Now we can start to see the cycle that forms. Cold air, trying to minimize its potential energy, goes as straight down as gasses ever do, blowing in horizontally along the hot plate. When it does, the slightest of low pressure areas appears above the hot air, as some of the cold air joins this slight breeze around the cylinder.

Remember, the sides of the cylinder don't permit much movement because the radial velocity of the gasses is basically zero. There's only diffusion mixing along that boundary. However, we now have the cold air sneaking in along the hot plate sideways, and the hot air pushing upwards. This is the basis of a convection cell.

To finish the iterative cycle, the hot plate warms some of the cold air that just came in, turning it to hot air. Now we have the same situation we had before, only with two changes:

• The cylinder is taller now, because the hot air has moved upwards
• There is now a slight upward current of air in the hot region, and a slight downward current of air in the cold region.

If you repeat the process, the same effect occurs, except now the low pressure area above cylinder is even lower pressure because there's a mass flow of cold air leading away from it.

Eventually material limits do limit the process, but hopefully that explains why the hot air rises straight up. There's a convection cell with a countercurrent of cold air right next to the hot air. Inbetween, the radial velocity is very low, so we see very little mixing. At the bottom, we see cold air moving in sideways, and at the top we see hot air being pulled upwards by lower pressures.

You ask why a column of hot air (as in a chimney) rises, given that denser cold air is above it, pushing down. It rises, because denser cold air around the BOTTOM of the chimney is under higher pressure than the cold air at the top of the chimney. The extra pressure due to a chimney-height of cold air is pushing it down. The lesser density of hot air means that the chimney-height of hot air causes less pressure than the surrounding cold air.

In formulae, the situation is: outside the chimney,

$$ P_\textrm{(chimney-top)} + \rho_\textrm{(cold-air)}\cdot g \cdot h_\textrm{chimney} = P_\textrm{(cold-chimney-bottom)}$$

for the cold air, and inside the chimney

$$P_\textrm{(chimney-top)} + \rho_\textrm{(hot-air)} \cdot g \cdot h_\textrm{chimney} = P_\textrm{(hot-chimney-bottom)}$$

where '$\rho$' is the density of the air.

$$\rho_\textrm{(cold-air)} \gt \rho_\textrm{(hot-air)}$$

Thus, the cold air at the chimney bottom is higher pressure than the hot, it pushes its way in, and the hot air is displaced (it rises).