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If a container filled with pure oxygen and fitted with leak-proof valves were to be dumped into the exosphere at night, would the contained oxygen condense?

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According to this article from the NASA web site, in the vicinity of the earth the temperature in the shade is -156C. The boiling point of oxygen at atmospheric pressure is -183C, so if the pressure in your container is at or less than atmospheric pressure the oxygen will not condense.

The NASA article doesn't say what is keeping the temperature at -156C, rather than the 2.7K of the microwave background. Possibly it's thermal radiation from the Earth.

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The net gave me the impression night-time temperatures approach absolute zero ... –  Everyone Jun 26 '12 at 16:55
    
If you go far enough away, the temperature will be about the temperature of the cosmic microwave background i.e. 2.7K. However the NASA article is clear that it doesn't get this cold close to the Earth. –  John Rennie Jun 26 '12 at 17:11

Since you have not given any parameters, I suggest you have a look at this question.

From the phase diagram you can deduce the boiling point given the conditions you assume inside the container.

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