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I know that part of the Sulfur cycle requires elemental Sulfur to be oxidized. Every example I've found uses Oxygen and becomes some combination such as Sulfur Trioxide or Disulfur Trioxide. If other oxidizers exist such as Chlorine and Bromine, could there be a sulfur cycle that includes Sulfur Tribromide or Disulfur Trichloride instead? The question breaks down into three related elements:

  1. At that point would there be no need for Oxygen?

  2. Is the Sulfur Cycle possible on a world without Oxygen?

  3. Is life possible in a world without Oxygen?

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I know that part of the Sulfur cycle requires elemental Sulfur to be oxidized. Every example I've found uses Oxygen and becomes some combination such as Sulfur Trioxide or Disulfur Trioxide.

The sulfur cycle on modern Earth involve elemental sulfur being oxidized, but that doesn't imply that elemental sulfur must at some be oxidized to participate in a biogeochemical cycle on some other world. If you delete sulfate from the diagrams on Wikipedia, a lot of links go along with it... but not all of them. Furthermore, elemental oxygen is not necessary for the production of sulfate. Sulfate is a useful ion biologically, and it's not obvious that halogenated sulfur compounds would be suitable replacements (maybe they would be, but it is not obvious, at least). But if life forms require sulfate, they can get the oxygen for it from other precursors; consider, for example, that sugars contain oxygen, but those oxygen atoms generally come from CO2, not from elemental oxygen.

It is convenient to have sulfur available in gaseous form sometimes, but SO2 and SO are certainly not required for that--H2S serves just fine.

If other oxidizers exist such as Chlorine and Bromine, could there be a sulfur cycle that includes Sulfur Tribromide or Disulfur Trichloride instead? At that point would there be no need for Oxygen?

Sure. Or sulfur hexachloride, or sulfur hexabromide.

Is the Sulfur Cycle possible on a world without Oxygen?

The sulfur cycle, no. A sulfur cycle, yes. And while something very much like the modern terrestrial sulfur cycle is not possible without any oxygen at all, it is certainly possible without gaseous atmospheric oxygen.

Is life possible in a world without Oxygen?

Again, you have to distinguish between oxygen atoms in general and free atmospheric oxygen. Oxygen is a sufficiently useful atom that I expect life is not possible entirely without it. But life is most certainly possible without free atmospheric oxygen. Plenty of life finds free oxygen to be horrendously poisonous.

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You can have a sulfur cycle where elemental sulfur is the most oxidized form.

https://en.wikipedia.org/wiki/Sulfur-reducing_bacteria

Elemental sulfur is oxidized as compared to hydrogen sulfide. There are bacteria that reduce elemental sulfur to hydrogen sulfide and oxidize foodstuffs using sulfur as the oxidant. Sulfur plays the role of oxygen here: instead of O2 you have S and instead of H2O you have H2S.

No need for any oxygen molecules to participate in the chemistry, although oxygen will be around as H2O.

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