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In the near future, the last of the human race has been forced to retreat underground due to extreme global warming or some solar flare event.

Vertical hydroponic farms provide the food for the survivors, but do not produce enough oxygen for breathing.

The three most abundant minerals in the earth's crust are:

  1. Oxygen
  2. Silicon
  3. Aluminum

Oxygen composes around 460 000 molecules out of every million (ppm). This is around 46%. Assuming that the survivors have had time to prepare for the apocalypse, and have a state of the art facility in which to work.

Would it be feasible for us to extract the bare elements from the crust?

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    $\begingroup$ As with most confined spaces, the more immediate problem will be with sinking (scrubbing) the carbon dioxide, not running out of oxygen. $\endgroup$ – Samuel Nov 22 '15 at 0:53
  • $\begingroup$ Good point. I'll take this into account! $\endgroup$ – Quiquȅ Nov 22 '15 at 1:00
  • $\begingroup$ And once you have the time the most effective way to get rid of the carbon dioxide is to feed it to a hydroponic farm that uses it to produce food. And oxygen. Also since you need to secure a source of water anyway, getting the oxygen from that might actually be easier. $\endgroup$ – Ville Niemi Nov 22 '15 at 2:03
  • $\begingroup$ @Ville Niemi And then we hopefully have a nice and tidy cycle. Except chemistry is a fickle thing and it might not work out evenly. I'll put some more research into this. Thanks for the help! $\endgroup$ – Quiquȅ Nov 22 '15 at 2:05
  • $\begingroup$ It's called a terrarium.... $\endgroup$ – bowlturner Nov 22 '15 at 2:07
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Yes

But I foresee two issues:
1) Crustal rocks will be deficient in a several elements more common on the surface of the Earth than occur in the rocks. The two most important of these are hydrogen and nitrogen.
2) Refining rocks into their constituent minerals or elements will require a great deal of energy. So you'll need powerful power generators. You'll also need some where to dump your waste heat from those generators.

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Possible, but you're unlikely to need much of it. A vegetation capable of feeding a population would also generate enough oxygen to support it. Here's why.

Humans and other animals use oxygen to oxidize organic matter (i.e. food) into CO2 and H2O. On the other hand, plant use CO2 and H2O to synthesize organic matter, and release O2 as a byproduct. So it can be written as

1) CO2 + H2O + light -> food + O2↑
2) food + O2 -> CO2↑ + H2O

No matter what real formula of food is and how you transform it, the O2 parity will hold.

That said, some oxygen may be trapped in metal oxides, like rust. This will have to be replenished. Luckily enough, you'll still need metals (say Al and Fe) to keep the colony going. Those will have to be reduced from oxides, using up a lot of energy and again releasing O2 (or CO2, if you plan using current technology to make iron/steel, but then the plants will capture it) as a byproduct.

Another thing mentioned in the comments is that you'll have to buffer CO2 somewhere, because large CO2 concentrations are bad for humans. The present Earth uses ocean for that, limestone and similar rocks are another great buffer. But you don't have oceans at hand. Of course, you don't want to store CO2 indefinitely, but rather feed it to the plants.

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It sounds possible.

The most abundant mineral with oxygen in the earth's crust is silicon-dioxide (silica).

The energy of the bond in this molecule is about 400kj/mol, so this is the amount of energy you need to expend to release one mole of 02 from silica.

One mole of oxygen will last one person about 23 minutes of breathing. So one person will use 438 moles of oxygen per week. It takes 175,000 KJ of energy to make 438 moles of oxygen from silica. http://www.bromicacid.com/lonelychemist/Questions/093008.htm

175,000 KJ is equivalent to 6kg of coal. http://www.kylesconverter.com/energy,-work,-and-heat/kilojoules-to-tons-of-coal-equivalent

So, one person needs to supply enough silica, plus 6kg of coal (or its equivalent) to use as energy to release enough oxygen to last them a week.

I think its more likely you would want to treat the dirty air above you though.

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    $\begingroup$ Wouldn't burning the coal require a fairly large amount of oxygen? $\endgroup$ – a CVn Nov 23 '15 at 9:31

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