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By Henry's Law ($K=\frac{x}{p}$), we know that increasing the partial pressure of a gas also increases its solubility on liquids.

Now a lot of sources (this, for instance) illustrate this by presenting "the bends", a condition that divers may experience when ascending too quickly due to the rapid decrease of the N$_2$ solubility on the blood.

So, my question is: why is the N$_2$ solubility greater deep on the ocean than on the surface?

Henry's Law says nothing about the total pressure, only about the partial pressure of the gas in question. When deep in the ocean, the pressure rises thanks to the water above, not due to a increase in N$_2$ partial pressure.

Bonus: does this video illustrate a greater CO$_2$ solubility in coke because of the greater total pressure? I feel like this is the same situation. Why would the CO$_2$ partial pressure be greater?

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The pressure underwater increases by roughly 1 bar every 10 meters, that is, ambient pressure is doubled every 10 meters compared to standard atmospheric pressure.

When you breathe air underwater using a SCUBA tank, your lungs expand against the surrounding ambient pressure in order to increase their volume. As the lungs expand they fill with gas which props up the lungs (balances pressure within the lungs with the surrounding pressure). The SCUBA device adjusts the pressure of the delivered air so that it balances the ambient pressure. The partial pressure of the component gases in the gas mixture inhaled from the SCUBA tank is not the same at the water surface and when a diver is submerged. That means the pressure of the air in the diver's lungs is not ~1 bar, it can be far greater, within limitations imposed by the potential toxicity of high blood concentrations of oxygen and nitrogen.

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