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Suppose for whatever reason the sun's power has weakened. The Earth gets colder. All the water has frozen, and the temperature is getting to around the boiling point of oxygen.

Can an "oxygen cycle" be possible, in which oxygen reaches it's boiling point and begins raining, forming lakes, evaporating, and forming clouds, like the water cycle today? Where perhaps in the summer, oxygen rains, and in the winter, nitrogen/oxygen and other trace gases rain and maybe in the polar regions, the atmosphere itself starts snowing?

Would this be possible, or is the changing air pressure, temperatures and properties of the gases prevent that kind of phenomenon from happening?

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    $\begingroup$ It is kind of impossible. Oxygen is highly reactive and maintained by photosynthesis. In your scenario photosynthesis would cease and free oxygen would start dropping. No idea if it would be fast enough to prevent "oxygen cycle" but pretty sure it would not be sustainable. $\endgroup$ – Ville Niemi Mar 26 at 18:27
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    $\begingroup$ This is possible, but the window of physical conditions to achieve that would be very narrow. Nitrogen boils at 77K, Oxygen at 90K - and that's at 1 atm pressure. If the Earth gets that cool, atmospheric pressure would be lower, which would further narrow the temperature window. $\endgroup$ – Alexander Mar 26 at 18:35
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    $\begingroup$ Required reading as regards these matters - A Pail of Air. You will think about this story for the rest of your life. $\endgroup$ – Willk Mar 26 at 19:59
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    $\begingroup$ I love that story. I first read it in 1972, when I was 10. $\endgroup$ – pojo-guy Mar 26 at 21:40
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For oxygen, just maybe - oxygen will freeze out before nitrogen, and the triple point of oxygen is at a very low pressure (0.152 kPa or much less than 1% of normal sea level pressure), so it will still have a liquid phase at the about 80% of current atmospheric pressure that would exist if most of the oxygen had rained out of the atmosphere (water vapor and CO2 would freeze out much earlier, but they are much more minor components).

However, as Alexander points out in the comments, the available liquid range would be very narrow. As the pressure drops, the liquid range narrows, and once the nitrogen starts to freeze out the atmospheric pressure will drop super low very quickly.

So - theoretically physically possible, but not likely to happen on Earth. If the temperature was stabilized within a narrow range, though...

Titan (the moon of Saturn) does have a "methane cycle", with methane rain and lakes, analogous to the water cycle on Earth. But that cycle isn't using the primary constituent of its atmosphere -- Titan's atmosphere is ~95% nitrogen, which stays gaseous.

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