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I don't disagree with anything in your post so I would say we agree, but the bottom line is that if you are injecting pure oxygen into a body of water you are going to get significantly more dissolved oxygen than if you are injecting just air, and that is a benefit of using hydrogen peroxide. "Compete" is probably not the best technical term but the presence of nitrogen in the air does reduce the dissolved oxygen in water by reducing the partial pressure of the oxygen.What I wrote was exactly correct, but let's be sure we are agreeing on what scenario we are discussing before making additional claims about who was wrong. I think we are now saying the same thing with different word choices.
Henry's Law makes it clear that only the partial pressure of the gas involved determine how much dissolves. The amount of other gases present does not factor in.
"Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. "
Henry's law - Wikipedia
en.wikipedia.org
Thus, if the partial pressure of oxygen above seawater is about 0.2 atmospheres (normal O2 content), then the amount of O2 that dissolves is around 7 ppm.
What does not matter is how much N2 happens to be present. if the N2 is 0 atmospheres, 0.8 atmospheres (normal air) or 10 atmospheres (under substantial pressure), the amount of O2 that dissolves is the same. The N2 DOES NOT compete with the O2 for space.
If we agree on that basic tenant of gas laws, then we have no further disagreement.
But when you stated:
"Also keep in mind that you are providing pure oxygen with peroxide rather than 21% with air so it does not have to compete with nitrogen for solubility in water."
I was focusing on the word "compete", which literally is not what happens. O2 is not competing with N2 for space, it is just that the pressure of O2 is not 1 full atmosphere.
Again, if we agree on that we are all good to go.
The reason I an drilling in on this is because there are many other situations where solubility of one thing impacts another for a variety of different reasons, and these things periodically come up in chemistry forum discussions about things like dosing solutions and what can and cannot be combined (e.g., vodka in kalkwasser).