You may have heard the maxim that “warm air rises.” False: no matter its temperature, air is drawn downward by Earth’s gravity. There’s no force sending it toward the sky—unless something heavier pushes in with a force greater than its weight, displacing it upward.
And what’s heavier than warm air? Well, all the liquids and solids that we observe on the ground (e.g. you and me), but also… cold air.
Why? The ideal gas law we saw last week states that cold air is denser than warm air. That is, the same volume contains more mass and therefore more weight.
Warm air rises if (and only if) it’s displaced by cold air. A better maxim would be “cold air sinks.”
Hot air balloons take advantage of that fact to achieve lift. The hotter the air in the balloon’s envelope, the more kinetic energy its molecules have; the more kinetic energy they have, the more they collide and drive each other apart. In other words, the hot air spreads itself thin, becoming less dense. Since the envelope’s volume remains constant, a lower density means less mass and a lower weight.
Take a balloon whose envelope holds about 75,000 cubic feet (ft³) of air. 75,000 ft³ at 60°F weighs about 6,000 lb, whereas 75,000 ft³ at 200°F weighs just 4,700 lb. Heating the envelope from 60°F to 200°F decreases the balloon’s weight by 1,300 lb!
As soon as the air within the balloon becomes lighter than the atmosphere, the surrounding air sinks around its form and starts pushing it upward. The greater the difference in densities between the envelope’s contents and the air outside, the greater the force buoying it toward the sky and the faster it ascends.
At this point it’s clear that temperature is only relevant because it affects air’s density. Cold air sinks relative to warm air, true—but more broadly speaking, dense air sinks relative to thin air. The best maxim? “Dense air sinks.”
In other words, you’ll get the same lift if you inflate the balloon with a gas that’s inherently less dense than air regardless of temperature. Our atmosphere is mostly nitrogen and oxygen, 7 and 8 in the periodic table respectively, so our contenders for less-dense gas must come above them. Do we have anything that fits the bill?
Our gaseous options are elements 1 and 2, i.e. hydrogen and helium. Since H and He have fewer elementary particles per atom than N and O, the same number of molecules have less mass. Hydrogen and helium don’t need to be heated and forced to spread thin because they’re already lighter than the atmosphere at ambient temperatures.
Sure enough, elements 1 and 2 have proven effective at floating skyward when trapped in light envelopes. (Bring on the room-temperature party balloons!)
So why is the Albuquerque sky speckled with hot air balloons, but not airships inflated with lighter gases? Alas, helium is hard to procure in large quantities and hydrogen has the undesirable side effect of being highly flammable. Heating plain old air is thus the safest, easiest way to fill your flying machine with a less-dense gas. Happy liftoff!