Mathematical Diversions: Producing Helium from Hydrogen

posted Sep 20, 2013
in Bloviation

In response to this story about a possible impending Helium shortage, someone suggested on Hacker News that perhaps someday we can use nuclear fusion to produce helium.

As it happens I'd idly chatted with my wife about that a few weeks ago, but that wasn't enough motivation to run the numbers. This was. Could we produce enough helium to satisfy our commercial production of it through fusion, if we just assume we have fusion?

A fusion reaction that starts from 4 hydrogen atoms (with no neutrons) produces one atom of helium. Let's just ignore the fact that this is a very hard reaction to do, even by fusion standards, and let's just ignore how the energy comes out and deal with raw amounts for now. Let's produce 1 mole of helium, which according to Wolfram Alpha, is .025 cubic meters of helium at standard temperature and pressure.

To produce that mole of helium, we need 4 moles of hydrogen. According to Wikipedia, 4 atoms of hydrogen combine to produce 26.7 MeV of energy. So the production of that 0.025 cubic meters of helium produces 26.7 MeV times Avogadro's Number of energy, which Google conveniently converts to Joules for us without asking: 2.6 × 1012 joules.

Wikipedia says annual hydrogen consumption was 15 million kilograms in 2000. Let's roll with that, because whynot. Helium's atomic mass is 4.003, let's just call it 4, so the number we just ran in Joules was for 4 grams. So we have to multiple that Joule count by 3,750,000,000 (15M kg / 4g), yielding 9.75 × 1022J of excess energy to produce the world's helium use for 2000. Human energy consumption in 2000 (because why not stay consistent) was 117.687 petawatthours, which is 4.23 × 1020J.

So, to produce the amount of helium we used in 2000, if I've got all my numbers correct (and it's been a while, but the tools I'm using here handle units automatically so that's a plus for my chances of being right), would produce as a byproduct approximately ~230 times the amount of energy humans consumed in that year.

That's actually enough energy to be a serious problem, even before we discuss the fact that it comes out as positrons and gamma rays. (And neutrinos, but that's not a problem.)

In conclusion, we are not about to start producing helium for commercial purposes from fusion, even if we could.


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