The Elements of Flight
When Larry Walters attached 42 helium-filled weather balloons to his lawn chair in 1982, he became briefly famous, shortly after his arrest for violating federal airspace in a non-airworthy craft.
Landing in power lines and causing a 20 minute blackout in Long Beach, California, he survived rising 16,000 feet in the air (he had planned only 100), and ended his 45 minute flight alive, but was awarded an honorable mention in the 1982 Darwin Awards nonetheless.
This example of knowing just enough to be dangerous came to mind when one of my readers at Scitoys.com asked how to make a solar hot air balloon out of garbage bags that was big enough to lift one person. I answered the question anyway, since the scale of the task was big enough that this kid was unlikely to get his project off the ground.
A little hot air...
A typical hot air balloon holds 90,000 cubic feet of air and can lift 1,600 pounds.
We can assume then that to lift a 200 pound person we would need about 12,000 cubic feet of hot air.
But that air is very hot, as the typical hot air balloon is heated with a lot of propane.
In a solar balloon, where thin black plastic is heated only by the sun, we can assume there is a much lower temperature -- hopefully low enough as to keep the plastic from melting. I picked 120 degrees Fahrenheit for the temperature of the air in the solar balloon, because I am an optimist.
The law that tells us just how hot air rises is called Charles' Law. It states that the volume of a gas is proportional to the number of degrees above absolute zero the temperature is. If the outside air is 65 degrees, and the air in the balloon is 120 degrees, the air in the balloon is only 10% hotter in absolute terms.
Air at 65 degrees weighs about 0.075 pounds per cubic foot. Inside the balloon it weighs only 0.067 pounds. The difference is 0.008 pounds per cubic foot. So to lift a 200 pound person, we will need 25,000 cubic feet of hot air.
That's a cube 30 feet on a side, or a sphere 37 feet in diameter. That's a lot of garbage bag.
On a lighter note...
In 1814, Amedeo Avogadro published a scientific paper where he explained that the number of molecules in a volume of gas is a constant. So if we want to make a volume of gas lighter, we should choose a lighter molecule.
Hydrogen is the lightest element. An atom of hydrogen weighs about 1 atomic mass unit, since it is little more than a single proton plus one tiny electron. There are two atoms of hydrogen in a hydrogen molecule, so the weight is about 2 amu.
Air is made up mainly of three parts nitrogen molecules and one part oxygen molecules. Nitrogen weighs about 28 amu, and oxygen about 32. So air is about 29. That's about 14 times heavier than hydrogen.
To lift a 200 pound person, you would need a spherical bag of hydrogen about 18 feet across.
At this point you might have pictures in your head of flaming dirigibles crashing into New Jersey,
and be thinking that helium might be a better gas to use. Helium is pretty good. An atom of helium weighs about 4 amu (it has two protons and two neutrons). And, helium doesn't combine well with other elements, or even itself, so each "molecule" of helium is just one atom. So even though a cubic foot of helium weighs twice as much as one of hydrogen, it still weighs 7 times less than air. And, since the volume of the sphere goes up as the cube of the diameter, lifting a 200 pound person still takes less than a 19 foot radius bag.
There are other gases we could use. Pure nitrogen is lighter than air, by a little bit. Methane only weighs 16 amu, so it is about half the weight of air. Ammonia is 17 amu, and neon is 20.
I sometimes ask kids to guess which is heavier -- humid air, or dry air. The natural guess is humid air. After all, water is heavy, right? Water vapor weighs 18 amu -- only a little heavier than methane. Displacing heavy air with light water vapor makes a lighter mixture, not a heavier one. Pilots of airplanes sometimes need to take the humidity into account when they calculate how much the plane can lift.
Of course, to keep the water from condensing on the inside of the balloon, we would need to keep the temperature above boiling, which would add even more to our lift.