What makes the two gases compressible, but does not apply to a liquid like water?
[Physics] Why are oxygen and hydrogen compressible, but water is barely compressible
water
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(source: engineeringtoolbox.com)
The graph(blue lines are what you want) only goes till $50^o C$, but we can extrapolate and say that the oxygen level will become a fifth or so by the time the water reaches its boiling point.
The solubility comes from Henry's law, but I don't know the temperature dependance of the proportionality constant $k_H$--I'll check it out tomorrow and edit it in..
Air is a mixture of gases, and the concentration of oxygen dissolved in solution (in the water) is proportional to the partial pressure of oxygen in the air. Raoult's law states: The vapour pressure of an ideal solution is directly dependent on the vapour pressure of each chemical component and the mole fraction of the component present in the solution.
Best Answer
The force between two (non-reacting) atoms is approximately given by the Lennard-Jones potential, and this varies with the separation of the atoms something like this:
(this image is from the Wikipedia article I linked above). In the diagram the parameter $\sigma$ can be thought of as the size of the atom, so the value on the $x$ axis of $r/\sigma = 1$ is the point where the atoms come into contact. When the atoms are far apart there is a very slight attraction, but as soon as the atoms come into contact there is a strong repulsion and it's very hard to push the atoms any closer together.
Be cautious about taking this too iterally as atoms are somewhat fuzzy objects and don't have an exact size. nevertheless the point remains that there is a distance between the atoms at which they suddenly start to strongly repel each other.
Now back to your question. For nearly ideal gases like oxygen and hydrogen at standard temperature and pressure one mole (that is $6.023 \times 10^{23}$ molecules) occupies about 22.4 litres. This means the average spacing between molecules is around 3nm. The size of an oxygen molecule is very roughly (they aren't spherical) 0.3nm, so the spacing between the molecules is about 10 times their size. That's way off to the right on the graph above, and it means the forces between them are low and it's very easy to push them together. This is why gases can be easily compressed.
Now conside water. One mole of water (0.018kg) occupies about 18ml, so the spacing between the molecules in water is about 0.3nm - in other words they are in contact with each other. This is the point where the molecules start to repel each other stringly, and that makes it hard to push them closer together. That's why water is not easily compressed.
You ask about compressing a mixture of (unreacted) oxygen and hydrogen. Well if you compress oxygen enough it liquifies, and the density of liquid oxygen is about 1140 kg/m$^3$. This makes the spacing between oxygen molecules about 0.35nm. This spacing is about the same as the size of the O$_2$ molecules so it's hard to compress liquid oxygen. You can repeat this calculation for liquid hydrogen (density about 71 kg/m$^3$) and you get a very similar result. Actually I would expect liquid hydrogen to be more compressible than liquid oxygen and water because the H$_2$ molecule is significantly smaller. However a quick Google failed to find values for the bulk modulus of liquid hydrogen.