|
Here's a different way to get to the same result, without knowing various densities, just with some rules of thumb. The first rule of thumb is that for a given substance that has both liquid and gas phases, the density of the liquid is about 1000 times higher than the density of the gas. It's not exact; for water the density of the liquid is 1000 g/l and of the gas is 0.8 g/l, so the actual ratio is 1250, but 1000 is easier to remember. Second rule of thumb is that our density is about the same as the density of water (it's slightly lower, since we can float, but barely). Third rule of thumb is that the density of a gas is proportional with the (average) molecular mass of the substance that forms it. Air is made of 20% O2 and 80% N2, which have masses of 32 and 28, so it's average molecular mass is about 29. Helium is just He, with a mass of 4, so air has a density about 29/4 higher than Helium, which is 7.25. Or Helium has about 13% the density of air. The same rule of thumb is to get the ratio of water vapor and air. Water has a molecular mass of 18, and air about 29, the ratio is 29/18 = 1.6. (By the way, this approximation is surprisingly correct, the actual ratio is about 1.605; the ratio for air/helium is also very close to 7.25; using the numbers above it comes as 7.22). So, 1kg of human is about the same as 1kg of water, which is the same as 1 liter of liquid water, which is (rule thumb 1) about the same as 1000 liters of water vapor. 1000 liters of air will be about 1.6 times as heavy (rule thumb 3), so a balloon filled with vacuum will be able to lift about 1.6 kg (of water, or human, it doesn't matter, a kilogram is a kilogram). But the balloon is not filled with vacuum, it's filled with helium, which has about 13% the density (rule thumb 3), or about 0.2 kg. So the buoyancy of the helium-filled balloon is about 1.4 kg for each 1000 liters. (If for some reason we remember that water is actually 1250 times more dense in liquid form than in gas form, rather than 1000 times, we can divide the 1.4 by 1.25 and get the 1.1 calculated above). If we throw in some adjustment for the mass of the balloon itself plus the gear around it, and a reasonable rounding for our imprecision, we get to 1000 liters of helium for each 1 kg of mass. |