Warm air has more energy than cold air. This means that according to the Einstein equation $E = mc^2$ the warmer air has a greater mass than the cold one. Why is the warm air rising, if it has a greater mass, which means that the attraction of gravity between the Earth and the warm air is greater?
[Physics] Why does the warm air rises up
atmospheric sciencemass-energyrelativitytemperaturethermodynamics
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I think the expansion explains only part of it -- a small part I'd say.
I think the more likely explanation is that blowing with your mouth open results in an airflow that moves much slower than an airflow blown through a small opening in the lips. The faster moving air will cause a quicker rate of evaporation on your hand, which makes it feel colder.
But proper measurements are always more convincing than theories :)
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.
Best Answer
As other answers point out, the reason is buoyancy. This post is to show just how small the opposing (relativistic) effect of increased gravitational force is.
The increase in the gravitational attraction associated with kinetic energy is proportional to the Lorentz factor, $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$.
We could look up molecular masses, and get into the Maxwell-Boltzmann distribution which describes the range of speeds of gas molecules, but for guesstimation purposes, let's use the rule of thumb that typical molecular speeds in a gas are on the same order as the speed of sound through the gas--for air near the surface of the Earth, roughly 300 m/s.
Let's say we increase the temperature of our parcel of air by 10%, from 300 K (a warm day) to 330 K (roughly the hottest day in the hottest desert). That means we increase the average molecular velocity by 5%, from our fudged 300 to 315 m/s. This takes $\gamma$ from $1+5 \times 10^{-13}$ to $1+6 \times 10^{-13}$, an increase of 1 part in $10^{13}$.
That is how much the gravitational force increases. You could cancel that increase by moving half a part in $10^{13}$ further from the center of Earth. That's a third of a micron.