A version of this question was posted in Chemistry SE and Worldbuilding SE but I've been told it wasn't appropriate to either SE. I was told to give this SE a chance. If you think it doesn't belong on this SE please let me know how to appropriately edit or otherwise another SE to try.

Note: This problem comes from my interest in exoplanet sciences but is applicable to earth sciences.

Something that's unclear to me is the exact relationship between gases within an atmosphere and the concentration of those gases within the bodies of water under that atmosphere (like an ocean).

For example, the solubility of $CO_2$ at $298K$ and 1 bar of atmospheric pressure is ≈ 1.496 g/L in water. My understanding is that number indicates how much $CO_2$ can be dissolved within water before the water is saturated with $CO_2$ and won't accept any more. However this doesn't tell me what the concentration of $CO_2$ would be under given conditions, it just sets a maximum cap. So 1.496 g/L of $CO_2$ might be dissolved in water, but it could be much much less.

Key Question: What are the calculations I need to go through to figure out the amount (total and ratios) of gases in both a planet's oceans and atmosphere assuming I know either:

  1. The total amount of gases within the ocean/atmosphere system, or...
  2. The amount of gases present in the atmosphere but not the corresponding amount present in the waters.

I want to be able to figure out all the ratios of gases within an atmosphere and dissolved in a planet's waters. Ball-parking is fine if that's the best I can do.

I know this is probably a very complex topic, so if it's helpful you could look at the idealized earth situation I've outlined (below) and rift off those numbers. Not necessary, but there if you need it.

You could also take the approach of looking at two gases with very different solubility in water, like $CH_4$ than $CO_2$ and look at what would happen to the ratios if there was a heck of a lot more of the less-soluble gas (Methane in this case) in the system than the more soluble gas.

Whatever approach you take, I'd appreciate any insight you can provide. What I really want to learn is both general principles as well as how to establish those principles in actual calculations.

Scenario: If useful refer to these numbers

Assume uniform temperatures at water surface. Assume no minerals dissolved in water.

SURFACE (Idealized Earth)

  • Total Surface Area: 510 million $km^2$
  • Water Surface Area: 361 million $km^2$
  • $15C$ Uniform Surface Temperature

ATMOSPHERE (Idealized Earth)

  • By Volume: 79% Nitrogen, 20.95% Oxygen, 0.049% Carbon Dioxide, 0.001% Methane
  • Surface Pressure: $1$bar

OCEAN (Idealized Earth)

  • Water Ocean (What should be the corresponding dissolved gases?)
  • Volume: $1.4$ billion $km^3$

Bonus: What would change if we lowered/increased atmospheric pressure and/or lowered/raised temperature?

  • 1
    $\begingroup$ Previous iterations of this question: chemistry.stackexchange.com/questions/118512/… and worldbuilding.stackexchange.com/questions/151647/… $\endgroup$ – n_bandit Jul 26 '19 at 15:30
  • 2
    $\begingroup$ For what it's worth, I think this is a good question and squarely on-topic. I'm not expert enough in the area to answer it, but I think you have a good chance. THere are a few different things being asked, so you might try to condense it to the "learning general principles" part, and leave out things like "bonus" questions ;-) Welcome to the site! $\endgroup$ – Semidiurnal Simon Jul 26 '19 at 17:43
  • $\begingroup$ butane.chem.uiuc.edu/pshapley/GenChem1/L23/web-L23.pdf $\endgroup$ – Keith McClary Jul 27 '19 at 17:29
  • $\begingroup$ Also take into account fractionation of isotopes of the gases. E.g. more light ones in the atmosphere in warmer times, more in the ice in colder times, ... $\endgroup$ – user18411 Dec 24 '19 at 9:38

The law that governs the pressure of gas at equilibrium over a dissolved liquid is called Henry's Law. Note, the equilibrium assumption, as there is some time dependence. That is, if you start with no dissolved gas in pure water, then in a relatively short amount of time, the amount of dissolved gas will be less than equilibrium.

Let's talk in abstractions. Henry's law's temperature dependence resembles the Clausius-Clapeyeron equation, so we can draw some parallels. From the Wikipedia link, for a given gas, the Henry's law constant is $$H(T)=H(T_0)\exp\left[-\frac{\Delta K}{R}\left(T^{-1}-T_0^{-1}\right)\right]$$ .$-\frac{\Delta K}{R}$ is Wikipedia's version of $-\frac{\Delta_{sol} H}{R}$, but I used $K$ to not cause confusion. For pure water, this link shows the solubility of different gases at different temperatures, but most relevant is the temperature at 0$^\circ$C (that is, it gives $H(T_0)$).

But all of this doesn't answer the two questions that your question boils down to.

  1. Raising the temperature decreases the solubility of a gas.
  2. If the pressure increases, then the amount of gas that can be dissolved increases per Dalton's Law. If the system is not in equilibrium, then the amount of solute will increase in proportion to Henry's Law.

This is a very complex question, for reasons I shall explain. Firstly. we need to establish what gases there are in the atmosphere before we worry about how much is dissolved in the ocean. Argon deserves a mention at 1 percent, and there are a few other gases in trace amounts, some of them more abundant than methane, some less, but as, like neon, they are only present in trace amounts, we can ignore them.

The oxygen content of the ocean is very important, because almost all marine life except for plants depends on it, but where does it come from? Only some of it is absorbed from the atmosphere, the rest is produced in situ by phytoplankton, blue-green bacteria, algae and sea grass etc. It is just as well that CO2 readily dissolves in water, because all marine photosynthetic life depends on it, though as you say, there are barely more than trace amounts. This breaking down of CO2 and production of oxygen in the ocean is extremely important from many points of view, including climate warming. More CO2 is disposed of by marine plant life than by all the world's rain forests.

The amount of free methane in the ocean is negligible, but the ocean nevertheless contains fairly large amounts in the form of methane clathrates. This is a sort of methane ice, where methane has combined with water at very cold temperatures, and it lies on the sea bed, mostly in deep water where the temperature is about 5 C even in the tropics. Clathrates are also found in the arctic tundra, and climate change fanatics are worried about the prospect of it thawing out.

  • $\begingroup$ For simplicity we don't need to consider the Earth's atmosphere as is, but we could look to the simplified example I've provided of 4 gases. They aren't in the true Earth ratio exactly, but working off of them should illustrate the principles at play. Also, I suppose an answer could take into account methane and carbon dioxide sequestration especially if that's a major factor. That would be great if doable, but I totally don't expect it. I'm cool looking over that in the name of simplicity. $\endgroup$ – n_bandit Jul 27 '19 at 15:40
  • $\begingroup$ In other words, I don't expect anyone to go to the trouble to provide a truly comprehensive answer involving gas sequestration within ices, rock, etc. Awesome if they do, but not needed. We could simply think of the "earth" as an impermeable surface/material and the world's waters as a single giant uniform pool of water on top of that impermeable idealized surface. Basically simplifying it to a thought experiment. Again, if you or someone wants to provide more, great! But I'm happy for a super simplified model working on basic principles. $\endgroup$ – n_bandit Jul 27 '19 at 15:44

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