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On a generation ship, the people living on it need oxygen, lots of oxygen. More than they can realistically carry, thus a generation ship must be able to create breathable air from relatively common materials and elements found in space.
How does a generation ship provide oxygen to the people inside it? What common elements can the ship rely on to provide, let's say, 100 million people with breathable air for an indefinite amount of time?
Closed cycle life support systems (CLSS) are the only way to go for prolonged space travel or even space stations and colonies orbiting a Sun. As pointed out, the real problem is making up losses since no system is ever going to be 100% efficient.
One of the key elements for any sort of CLSS needs to be water. Water is essential for the life processes of everything aboard. The waste water stream is going to be processed to provide nutrients for the plants (both algae and food plants), then the cleaned water is going back for drinking and other uses in the human/animal side of the system.
So the ship needs millions of litres of water for the system. Extra water can be carried to buffer the system, act as radiation shielding or thermal heat sinks and other uses aboard the ship. One of the most convenient ways to carry all this water is as ice. Anthony Zuppero outlined a simple design in the shape of a doughnut or tire which uses ice as both a structural material and as the reservoir for all the water needs of the crew. Should there be some sort of disaster which cripples the CLSS, the ice can be melted and electrolysed to release hydrogen and oxygen, with the oxygen being added to the atmosphere. As an aside, the real danger in a closed environment isn't running out of Oxygen, but being poisoned by a buildup of CO2.
The most common solution to this approach is to try and mirror Earth's biosphere. Since you need water, food, and air, you need a solution that provides all 3.
Hydroponics (and its closely-related twin aeroponics) is an excellent way to provide food and air: You grow edible plants (especially those with green leaves), which as they are growing consume the CO2 you exhale and turn it back into breathable oxygen. Algae and certain bacteria are showing a lot of promise currently for filling this role very well. Water is recycled and reused, just as it is in Earth's water cycle.
You can supplement this with mechanical and/or chemical processes that also break down exhaled CO2 and release the oxygen back into the ship's systems.
Unfortunately, no man-made biosphere can possibly be 100% efficient, nor can any pressure vessel be made 100% sealed, so you will have to additionally carry tanks of compressed oxygen to replace losses on your travels. You will be able to collect small amounts of hydrogen and oxygen with something akin to a ram scoop, though without onboard recycling of CO2 and water you will not be able to collect enough for anyone to survive.
If your ship is traveling to the next star system, that's pretty much it; if you can stop at other stars along the way (which will always be a massive detour, but for a long enough journey may nonetheless be required), you can probably harvest base materials within the system, but it's going to be hard, expensive, time-consuming, and rely on carrying a lot of heavy equipment that is just taking up space and mass for greater than 99.999% of your total journey.
A generation ship carrying one million people for an indefinite period will need to carry or generate a minimum of 550,000,000 litres of $O_2$ per day. Mixed to match Earth atmosphere ratios, in which $O_2$ accounts for 21% of the air, that's a minimal atmosphere capacity of 2,620,000,000 litres.
Assuming a classic rotating cylinder ship, that is a cylinder roughly 3,000 km long and 1,050 km across. Such a ship would have an internal surface area of 11,700,000 $km^2$, so something between Canada and Antarctica in size. Again, that is the bare minimum – meaning zero redundancy, zero waste, zero loss – to support one million air-breathing adults.
99% of the interstellar medium by mass is gas, of which 70% is hydrogen and 28% helium. Oxygen makes up trace amounts – less than 1,000 molecules per $cm^3$ in the best case scenario; not enough to harvest en route to make a difference. Barring fusion of those H and He molecules, you're going to have to bring all the oxygen you need with you, in one form or another.
Fortunately, people – and plants – also need water to live, and water has oxygen.
To keep these people breathing you'll have to simulate or approximate the natural oxygen cycle found on Earth:
Specifically the Hydrogen and Oxygen steps. The only inputs this system needs are photosynthesis-permitting light, which can be achieved artificially, and topping up any lost hydrogen.
From the same Wikipedia article linked above, the carbon cycle accounts for 99% of the oxygen, stored away in rock; your ship will need as much oxygen produced and cycling as possible, and cannot justify the space and mass a crust-substitute quantity of rock and minerals would require, or the time, so this step will need to be bypassed.
The "light-dependent reaction" in the diagram above is photosynthesis – plants combining $6CO_2$ (carbon dioxide) with $6H_2O$ (water) and light to produce $C_6H_{12}O_6$ (sugar – glucose) and $6O_2$ (oxygen).
On Earth the Amazon produces more than 20% of total oxygen from photosynthesis – 20% of 165,000,000,000,000 litres; see table 2 in link above – in an area roughly 5,500,000 $km^2$. That's tens of thousands times more oxygen than you need, produced in an area half the size of the cylinder described.
(There is another way to produce oxygen called photolysis – UV light breaking $H_2O$ into its constituent parts; $H_2$ to be absorbed/collected, free oxygen combining to $O_2$ – but on Earth it doesn't produce even 0.001% of the $O_2$ we breathe, so we probably shouldn't factor it into this situation.)
You absolutely will not collect fresh air along the way. Elements will be recycled and atoms (including oxygen) reused over and over. Ideally you only need to add energy. Other losses need to be managed so you carry enough atoms to last. At speed, resupply en route is impossible even if you were to pass something in interstellar space. Slowing down is the bulk of your energy budget so you only do that once, when you arrive.
A closed life support system is either an elaborate chemical factory or a small ecosystem. Since you specified 100 million humans, that is a rather large scale and is definitely an ecosystem.
At all costs, you want closed cycles that generally maintain themselves and regulate themselves, as well as repair and reproduce components. You want massive redundancy of small units, not huge machines.
So, we’re talking living organisms. If some nanofab is engineered it is a moot point of it being anything other than a bacteria or other cell, since it shares all the salient features with natural life.
You’ll want ponds that circulate themselves, not complex pumps and pipes. Everything will end up looking like wildlife, including “trees” and “fungi” that are actually engineered tech, and a large portion of oxygen production is done with sea algae.
Algea Aero/hydroponics grows fast enough, feeds your population, scrubs CO2, and produces oxygen...
But really, Hydrogen, Oxygen, and/or water is everywhere in space, Your ship might even run on a fusion/fission reactor that produces water, energy, and oxygen.
By definition, a generation ship is self-contained for its entire journey. With very few exceptions, if hard science is used, the system must be entirely closed, with little or no losses. That includes O2. Recycle. Every possible molecule of O2 needs to find its way back. Yes, you should also include rust in your assumptions...
Of course, all the recycling systems must also be closed-loop, with all waste products and catalysts being re-usable somewhere, otherwise THOSE would run out.
Hence the difficulty with closed loop systems...
OR, if you are stretching the science of your fiction, you can get Oxygen as a byproduct from fusion/fission...
Nuclear submarines today have systems that can remove carbon dioxide from the air, called scrubbers. However, they make their oxygen by electrolysis of water, and a spaceship cannot do that. There are a number of newer technologies being developed for capturing carbon dioxide from the atmosphere, including transparent tubes full of photosynthetic algae (at the University of Kentucky) to make the most efficient use of space.
Your spaceship will need to recycle the atoms of carbon and oxygen on board, because it can’t replace them. That means it uses some process that consumes energy to convert carbon dioxide and other waste products into oxygen plus useful organic molecules such as food. This could be plants, as on Earth, or it could be chemical engineering to produce essentially the same end products. For example, ultraviolet light can split carbon dioxide into carbon and oxygen, or it’s possible to use carbon, oxygen and hydrogen to synthesize complex organic molecules. Most refineries need fossil fuels as their source material, but it can also be done now with electricity, water, and carbon dioxide to produce “blue crude.”
Oxygen is not the only problem when creating a breathable environment.
This answer uses only currently feasible or existing technologies
While the astronauts in the Gemini and Apollo programs breathed 100 percent oxygen at reduced pressure for up to two weeks with no problems, breathing pure oxygen at earth level pressures is not medically advisable, (remember these astronauts were specially trained and in peak physical condition). The earth's atmosphere is only ~20% oxygen, as such our bodies have adapted.
Oxygen toxicity is a very real risk in human spaceflight (Pure oxygen is also highly explosive). Accordingly, most modern spaceflight missions (such as ISS missions, Skylab, Space Shuttles, and Orion MPCV) place a high importance on introducing diluents nitrogen as well as oxygen (exception: pressure suits use pure oxygen at the lowest inflation pressure).
Oxygen is largely recyclable, many other comments have outlined good techniques for recycling it. Oxygen is the 3rd most common element in The Milky Way at ~1.04% of mass, so smaller ships could be dispatched to gather it and return to the larger habitat ship with 100 million people in it.
There are a some realistic solutions to diluting oxygen but they have trade offs
Nitrogen, the 7th most common element, is 1/10th as common as oxygen at ~0.096% of mass which would make it unrealistic to collect during space travel and is relatively heavy compared to Helium and Hydrogen, ~7 and 14 times more heavy respectively.
Helium is used as dilutent in some deep diving equipment (aka Heliox). Helium is very common in the milky way (2nd most common element at ~24% of mass). However there are 3 significant but solvable problems with helium due to its low molecular weight. 1) It is not suitable for dry suit inflation (poor thermal insulation). 2) It impairs but does not inhibit communication (the speed of sound is faster in a lower molecular weight gas, which increases the resonance frequency of the vocal cords). 3) Probably the most problematic but still solvable is Helium leaks are much more common than other gases. Atoms of helium are smaller allowing them to pass through smaller gaps in seals.
Hydrogen is both the most common element at ~74% of mass in the milky way and has been used in deep diving gas mixes (aka Hydrox) but is very explosive when mixed with more than about 4 to 5% oxygen. The 4 to 5% oxygen mixtures usage is limited to deep dives (also has the same communication problems as Helium). You could use a safe 5% oxygen mix in a high pressure cabin environment (side effects will occur) or have some sort of mask that filters out some of the hydrogen when breathing so that mixture entering the body has enough oxygen but the environment is not highly unstable. (You could use a ~20% oxygen mix and just make sure absolutely nothing sets it off)
Oxygen Only if a dilutent absolutely cannot be used there are a couple options. Running it at about 1/3rd of earth atmospheric pressure would sustain life in some of the passengers in peak physical condition but others would experience either respiratory failure or conditions such as: venous/vascular air embolisms, pulmonary oxygen toxicity, oxidative stress exasperating existing conditions, retinopathy, hypoxia, obstructive lung disease. The pure oxygen is highly explosive even at lower pressures.
Small note about moisture content
Gas is generally compressed for storage which removes moisture from it, while not usually deadly, it causes dehydration, so probably best re-moisturize the air.
Disclaimer: I am not a diving instructor, if you intend to use this information for diving, please consult a diving instructor first. The presented medical information is summarized, consult a doctor. Do not use this as a medical advice or opinion
https://en.wikipedia.org/wiki/Life_support_system#Atmosphere https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements#Universe https://en.wikipedia.org/wiki/Breathing_gas#Helium https://en.wikipedia.org/wiki/Breathing_gas#Hydrogen https://www.sae.org/publications/technical-papers/content/2005-01-2896/
You Have to Stretch some Laws of Physics
If you want a true deep-space ship (eg. nothing around you, not even asteroids, for hundreds of light-years) your crew is doomed. At some rate your ship will emit energy to space around you in the form of radiation, which will add up over time. In addition, versus the vacuum of space you will lose small molecular gasses straight through your hull! That loss will primarily be in the form of Hydrogen, which you won't miss much until your water supplies start to run low.
If we stretch science a bit...
Perhaps you could find a way to spot and track asteroids and even rogue planets outside of solar systems. You pick up and process those sources as you go along to replenish supplies. Being a generational ship the planets you come across are presumably not fit for human habitation, but they could still contain ice and other elements/minerals you need. A real-life example of such a planet would be Europa.
As for the basic problem that your power sources are going to run low at some point... well, I won't mention it if you don't.
