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Suppose a bath of water is in equilibrium with the air above it at temperature T and pressure P. The concentration of $\rm CO_2$ in the air is, say, 400ppm, and the concentration of $\rm CO_2$ in the water is whatever is needed to be in equilibrium.

Although they are in equilibrium (i.e. the concentrations in the air and water don't change), there will be transfer of $\rm CO_2$ molecules between the water and the air.

What is known about the rate of this transfer? Is there a formula for the half life, i.e. the time for half the molecules originally in the air to be in the water?

Equivalently, after 10 years say, what proportion of the $\rm CO_2$ molecules originally in the air will be in the water?

Edit added after David Bailey's excellent answer:

My motivation is to better understand calculation of how much $\rm CO_2$ added to the atmosphere is due to industrial output and how much is from ocean de-gassing.

For the purpose of this question we can assume perfect mixing in both the ocean and the air

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    $\begingroup$ The issue here is that this is not a simple process. When CO$_2$ dissolves in water it spontaneously combines with it to form carbonic acid, which also has an acid equilibrium in aqueous solution. So you not only need to account for the rate of dissolution of the CO$_2$, but the rates of all of the side processes that may “trap” a molecule into solution in another form. I don’t have any relevant kinetics data on hand, so I sadly can’t even attempt a quantifiable estimate of this for you. $\endgroup$ Aug 12, 2023 at 22:07
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    $\begingroup$ A simple analytic formula is certainly not possible. But given relevant kinetic data on the forward and reverse rates of every process I mentioned you could certainly estimate the rate of change of molecules from one species or state into another. I’m not aware of any explicit experimental data here either, sorry. $\endgroup$ Aug 12, 2023 at 22:12
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    $\begingroup$ Would Chemistry be a better home for this question? We can migrate it if you like. If you decide instead to cross-post, it is polite for each question to include a link to the other; it is also polite if your questions are slightly different and tailored for the skills of each community. $\endgroup$
    – rob
    Aug 12, 2023 at 23:42
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    $\begingroup$ After thought, I think the answer from David Bailey illustrates Physics is a good home for this for now $\endgroup$
    – Peter A
    Aug 13, 2023 at 8:21
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    $\begingroup$ Re your edit (v5): the ocean is currently absorbing excess carbon dioxide from the atmosphere, rather than degassing. $\endgroup$
    – rob
    Aug 13, 2023 at 15:53

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There is no formula for the half-life of a CO$_2$ molecule in the air that depends only on temperature and pressure and the equilibrium concentrations in the air and water.

At the very least, you also need to know the mixing time for CO$_2$ in the air volume above the water, since a CO$_2$ molecule has to reach the water surface before it has any chance of passing through it. For a perfectly static system with no air or water movement - which is hard to achieve experimentally - the average time for a CO$_2$ molecule to reach the water surface would roughly be the diffusion time $t=h^2/D$, where $h$ is the average height of the air above the water and $D=16\times10^{-6}\,\mathrm{m^2/s}$ is the diffusion constant for $\mathrm{CO_2}$ in air. For the Earth's atmosphere with a scale height of about $8.5$ km, $t\sim 10^5$ years. Because atmospheric mixing does not depend on diffusion, but instead is driven by winds, convection, storms, and other air movements, the actual atmospheric mixing time is only about a year.

In case you are trying to model CO$_2$ transfer between the Earth's atmosphere and ocean, note that simple models are hopeless. The transfer rate also depends on wave motion, surfactants, salinity, turbulence, bubble production, ocean layering, …. See, for example, this MIT Open Course lecture on "Marine Chemistry: Dissolved Gases and Air-sea exchange". The actual CO$_2$ atmospheric residence time is roughly 5-10 years.

Edit added after motivation and assumption of question clarified by OP.

Sea-air CO$_2$ exchange flux $F$ is typically modelled as $$F=k \Delta C$$ where $\Delta C$ is the air-sea CO$_2$ concentration difference. and $k$ is the "piston velocity" (i.e. the gas transfer velocity). This turns out to roughly depend on the wind velocity squared. Models for the gas-exchange process usually assume there is a thin boundary layer at the surface of the ocean where the gas exchange takes place, with mixed bulk atmosphere and ocean above and below. For example in a stagnant boundary layer model, the layer might be about 20 microns thick and $k\sim 5$ m/d, so it would take about a thousand days for a $5$ km deep ocean to exchange its CO$_2$. Better models include film replacement and eddy impingement.

Oceanic CO$_2$ partial pressures range from pCO$_2$~100 μatm to ~1000 μatm, so the ocean is locally a CO$_2$ source or sink depending on whether ocean pCO$_2$ is greater or less than the atmospheric pCO$_2$ of ~400 μatm, and the rate depends on how strong the wind is blowing and other factors. To accurately understand sea-air CO$_2$ exchange requires complex modelling based on lots of ocean and atmospheric measurements, and as shown in IPCC 2021 Figure 5.9(a) , the ocean is typically a CO$_2$ source in the tropics and a sink in higher latitudes.

Comparative regional characteristics of the mean decadal (1994–2007) sea-air CO2 flux (Fnet) and ocean storage of anthropogenic CO2

Figure 5.9(b) on the right shows the rate at which anthropogenic CO$_2$ is being stored in the oceans.

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  • $\begingroup$ Thank you for this excellent answer. I am going to edit the question to allow perfect mixing of the air and to mention my motivation. I will leave it open a little longer in case anyone can add to this $\endgroup$
    – Peter A
    Aug 13, 2023 at 8:18

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