Everyday Thermodynamics I: Why does warm air rise?

A new Fall semester started at Florida State University. In other parts of the world, a riot of colors slowly descends with
beauty, grace and briskness. The air has a newfound cold bite, and is filled with the anticipation of Saturdays and college football. Not so, in Tallahassee.

In Tallahassee, we sweat.

And for the first 10 days in the Engineering Building we steamed as the AC broke yet again. And I noticed something interesting. It is funny, how what we notice depends on what we think. I am teaching thermodynamics for the third straight year (which is great in terms of the time such a luxury affords me), and I noticed right away that my office which is on the third floor was much warmer than the ground floor although the air-conditioning for the entire building had failed.

What had happened was obvious. "Warm air rises, cold air sinks!". Unfortunately for the kids taking ECH 3101, I have this section called "bonus surprise tests" (whose popularity is variable, but usually negative) and I decided that this was, what people who sound like experts call a "teaching moment". Pronto, I put this on their first surprise test, and asked them to explain why warm air rises, assuming that air is an ideal gas (which in this case is not a bad approximation)!

The answer is fairly trivial. To aid thinking it is useful to ask why water sinks and oil rises. It is immediately clear that the quantity to go after is density. The rest, however important, is ironing out the details.

The ideal gas law says PV=NRT, where symbols have their usual meaning. Therefore, N/V=P/RT. N/V is the molar density. The pressure doesn't change too much over 20 m (air pressure drops with altitude, by about 0.01 bar per 100 m.). Therefore density is inversely proportional to temperature. Thus cold air is more dense. QED.

The same effect, of-course, underlies the rising of hot air balloons like the one pictured left.

Just to dispel any lingering suspicion that you may have figured everything out here is an interesting question as an epilogue. If warm air rises, why is it cold in the mountains? Think about it for a while and then click here (pdf) for the answer.