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As I understand it, our current breathable atmosphere on Earth is the result of early lifeforms transforming huge amounts of CO2 into O2. Before early life, even Earth's atmosphere wasn't livable for modern humans, let alone that of current Mars, Venus, etc.

But is there any reason why a planet couldn't develop naturally, without the aid of life, to have an atmosphere that humans could survive breathing? Obviously it wouldn't be common, but it is possible, or is there some chemical process I'm not aware of that would prevent the formation of a breathable atmosphere through non-biological means?

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It's actually not that far-fetched, but non-biological processes involved with creation of such breathable atmospheres likely wouldn't be sustainable for a very long time. Let's imagine a largely water ice world that was first formed beyond its host star's snow line but later swapped orbit with another body that orbited within the star's Goldilocks zone. Surface ice would start melting because it was exposed to more heat from its sun, water vapor would increase atmospheric pressure, and stronger UV radiation would start splitting weak Hydrogen bond to $\require{mhchem}\ce{H_2O -> H+ + OH-}$ and then $\ce{2OH- -> O- + H_2O}$ in its ionosphere.

Now, if the planet moved at just about right distance to its parent star and the star isn't too violent (our Sun for example isn't), the atmosphere would start losing Hydrogen to space with solar wind while the remaining Oxygen would be too heavy and sink towards the surface. Planet would still also be losing Oxygen to space, but at a slower rate than it's losing Hydrogen. And nearly 8/9 the mass of water is Oxygen. Some of this Oxygen would recombine into its molecular state $\ce{O_2}$ and some would form Hydrogen Peroxide $\ce{H_2O_2}$ when recombining with the rest of liquid water on planet's surface, leaving mostly Oxygen rich atmosphere above it which would be technically breathable.

This couldn't be considered a self-sustainable process though, due to Hydrogen loss to outer space. There are also other non-biological processes that could enrich planet's atmosphere with Oxygen, like e.g. naturally occurring Sabatier and Bosch reactions, but these reactions require high temperature and large quantities of Hydrogen so such atmosphere would be less likely and too hot and quite possibly too dense (pressure, violent weather,...) for comfort.

Another problem with no life present is that such atmospheres could contain high concentrations of toxic elements and molecules in them, such as too high concentrations of Carbon Oxides ($\ce{CO}$ and $\ce{CO_2}$) which are both toxic to the human body, cyanides (containing $\ce{CN-}$), even too high concentrations of gaseous state Hydrogen Peroxide above the planet's surface. So called filler gases like all the non-reactive, inert gases, e.g. Nitrogen, Argon, even Helium would be fine though. So it gets complicated and it's a lottery. But it's not out of the realm of possibility.

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