This is a post I’ve been meaning to write for some time, and is partly motivated by a discussion Drikan and I have been having on another blog. It will probably end up being two posts, with this one simply laying out the basic chemistry associated with the uptake of CO2 by the oceans. It will probably be a pretty dull post, so this is more for my benefit than anything else. I’m also not a chemist, so I don’t claim that I won’t make any mistakes. If anyone notices any, feel free to point them out.
The partial pressure of carbon dioxide, , in the atmosphere is related to the the concentration of in the oceans, , through Henry’s law
where is the Henry’s law constant. The units of is typically mol/kg, is typically atm – or ppm – and is then atm/mol/kg. Henry’s law constant depends on Temperature and salinity, and at a temperature of C, and salinity, , of 34.78 g/kg, is 35.18 atm/mol/kg (or 0.02839 mol/kg/atm).
However, most of the dissolved inorganic carbon () in the oceans is not in the form of dissolved . Dissolved reacts with water, , to form bicarbonate, , and ionised hydrogen, ,
with a dissocation constant given by
Similarly, bicarbonate dissociates to form ionised hydrogen, and carbonic acid, ,
with a dissociation constant given by
Henry’s law constant, , and the two dissociation constants, and , depend on temperature and salinity and are given in this file. Given these constants, we now have 3 equations and 5 unknowns. There are, however, two other quantities that we need to know and that are also largely unaffected by changes in pH, pressure, temperature, or salinity. They are the , and the titration alkalinity, , given by
In surface waters, nitrate ions – – are an important nutrient and so their concentration is very low and can be ignored. Similarly, the contribution due to and , can also be neglected. We do, however, have to consider the dissociation of boric acid
with a dissociation constant
As with the other constants, also depends on temperature and salinity and is given in this file.
You may note that by adding this we’ve added another equation with two more unknowns. In the oceans, however, the residence time of boron is very long, depends on the Salinity, , and so the total amount of boron () is given by
which – if I’ve counted correctly – means we now have 7 equations with 7 unknowns.
To now determine the partial pressure of , , in equilibrium with the ocean, we first need values for the and for . Pre-industrial global averages for these two quantities are , and . The next step is to determine . This is done iteratively by considering how the and the carbon alkilinity, , depends on . From the equations for and we can write the DIC as
We can also use the equations and to write as
Dividing the equation for the by the equation for gives a quadratic that can solved for
To determine , we simply make a first guess, then solve equation the equation above using
where we’ve used that
This process is repeated until the new value for matches the previous value, to within some small tolerance.
Once has been determined, we can determine using
The partial pressure of , , is then given by
where is the inverse of Henry’s constant, , and has units of mol/kg/atm. We’ve also converted from , to . We can also determine the ocean pH from
That’s where I’m going to stop. Essentially, if you specify values for the dissolved inorganic carbon () the titration alkilinity () the salinity (), and the temperature, you can determine by first making a guess and then iterating to a solution. Given this, you can then determine the partial pressure of in the atmosphere, , and the pH of the ocean. What you can then do is see how various quantities vary with temperature or with DIC, which I will look at in the next post.
Tans, P., Why Carbon Dioxide from Fossil Fuel Burning Won’t Go Away in MacAladay, J. (ed), Environmental Chemistry, Oxford University Press, 1998, pp. 271-291.
I also found this site useful, although I couldn’t get the code to actually run.
Eli Rabett, Quadratic coke.
Eli’s post about alkalinity.
A similar post by Nick Stokes.
Nick Stokes’s online calculator.