Ocean Acidification: Can CO2 Release & pH Decrease Simultaneously?

[Studiot]

Hello Borek,
I really value your checking input as I am rather dashing things off, not preparing an essay or paper. In particular I forgot to thank you for the rather smarter titration curve.

Yes indeed natural waters contain many things.

Carbonates/bicarbonates are also introduced by direct solution from from carbonate rocks and the shells and skeletons of organisms. There is sufficient quantity and contact to maintain near saturation of calcium carbonate in most of the worlds waters.

Extra protons can be introduced via the oxides of sulphur and nitrogen in 'acid rain', and natural sulphurous process (vulcanicity).

Some more figures:

'Clean' Natural waters have a pH range of 7 - 9, oceanic pH is usually taken as 8.3

Clean rain has a pH of 5.6
Acid rain is defined as rain with a pH of less than 5

Major killing of fish commences at a pH of 4.5 and other life at a pH of 4

Carbon dioxide is the third most abundant dissolved gas, after nitrogen and oxygen but it is exceptional in that it does not dissolve in direct proportion to its atmospheric partial pressure.
There is widespread geographical difference in the ocean uptake, being supersaturated in tropical latitudes and undersaturated in temperate and polar ones.
There is resultant mass transport by the ocean current systems.

This comment is common to many environmental issues where there is an attempt to lump the whole of the Earth's surface under one value of some parameter, when in fact there is gain in one location and loss in another and transport between.

I think the original question amounts to "under what conditions (of pH and atmospheric %) could reaction 1 move to the left and release carbon dioxide to atmosphere?)

 

[billiards]

Some research suggests that ocean acidification is increasing at its fastest rate in 65 million years.

A new model, capable of assessing the rate at which the oceans are acidifying, suggests that changes in the carbonate chemistry of the deep ocean may exceed anything seen in the past 65 million years.

The model also predicts much higher rates of environmental change at the ocean’s surface in the future than have occurred in the past, potentially exceeding the rate at which plankton can adapt.

http://www.bris.ac.uk/news/2010/6835.html

 

[Studiot]

Interesting, I don't know if Professor Benton is still head of department at Bristol, but his work is exemplary and a really good read to boot.

 

[Studiot]

Since commenting that the underlying source of bicarbonate is the equilibrium with solid calcium carbonate, I need to add a couple of equations.

CaC{O_{3(s)}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over<br /> {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} Ca_{(aq)}^{2 + } + CO_{3(aq)}^{2 - }\quad - \quad [7]

If you combine equation 5 and equation 7 the net result of dissolving calcium carbonate in water is one ion each of calcium, bicarboante and hydroxyl

CaC{O_{3(s)}} + {H_2}O \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over<br /> {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} Ca_{(aq)}^{2 + } + HCO_{3(aq)}^ - + OH_{(aq)}^ - \quad - \quad [8]

The equilibrium constants for this can be algebraically manipulated to yield the kick-off pH for a natural water, saturated with calcium carbonate, at around 9.

If the interest is still there in this subject I will post the calculations. They are of interest because they show the inadequacy of Borek's concatenation method and the reason for not combining the constituent reactions.

 

[Borek]

If the interest is still there in this subject I will post the calculations. They are of interest because they show the inadequacy of Borek's concatenation method and the reason for not combining the constituent reactions.

Show them so that I can prove you are wrong

BTW, your last reaction can be also written as

CaCO₃(s) + H⁺ <-> Ca²⁺ + HCO₃^-

but it is not necessary - it is enough to add your reaction 7 (that is calcium carbonate dissolution and its equilibrium constant - solubility product) to the earlier set (the one already containing water ionization constant and bicarbonate dissociation).

Also, don't forget your earlier statement:

Conventional chemistry suggests that CO₂ will only be released if the pH falls below about 4.

which needs to be proven.

 

[Borek]

Carbon dioxide is the third most abundant dissolved gas (...) but it is exceptional in that it does not dissolve in direct proportion to its atmospheric partial pressure.

This statement is too general to be correct.

See attached picture, it describes pure water in equilibrium with gaseous carbon dioxide. In such situation concentration of unreacted carbon dioxide is directly proportional to the gas partial pressure above the solution, while total concentration of dissolved carbon dioxide is a little bit higher, as some of the carbon dioxide reacted with water and dissociated lowering pH. Note, that it doesn't matter much what is equilibrium between carbon dioxide and non-dissociated carbonic acid, it is enough that the reaction between water and carbon dioxide is fast enough (it is).

analytical - means total analytical concentration of all forms of carbon dioxide - that is sum of CO₂, HCO₃^-, CO₃^2-
pH - obvious
[CO₂] - concentration of dissolved unreacted carbon dioxide (as explained above directly proportional to the partial pressure of carbon dioxide above the solution)
ratio - ratio of total analytical concentration to concentration of unreacted CO₂

As you see, for very low concentrations (low partial pressures) difference between total solubility and pressure is not linear, but it gets almost perfectly linear for higher pressures (even more linear than shown, I decided to cut off higher concentrations - even if they strongly supported my point, they are not that important in reality, we don't expect partial pressure of carbon dioxide to near 1 atm in a foreseeable future). But that's in pure water. I have a gut feeling that in sea water - in the presence of buffers - this dependency would be even closer to linear, as ratio from the right column is mainly function of pH and pH in buffered solutions changes very slowly. I can try to estimate it if anyone is interested.

Edit: numbers calculated with BATE, ionic strength of the solution ignored - but it won't change the general trend.

 

[Studiot]

First a couple of constants that can be found in tables. I am working at a typical water temperature of 10 deg C.

The reaction constant for reaction 7 is the solubility product

{K_7} = {K_{sp}} = 7.2x{10^{ - 9}}

The reaction constant for equation 5 is the base constant for the carbonate ion in water.

{K_5} = {K_b} = 1.06x{10^{ - 4}}

In equation 7 let s be the concentration of the carbonate ion. This is equal to the concentration of the calcium ion so

{K_7} = \left[ {C{a^{2 + }}} \right]\left[ {CO_3^{2 - }} \right] = {s^2}

s = \sqrt {72x{{10}^{ - 10}}} = 8.5x{10^{ - 5}}

However this is not the end of the story since the carbonate ion reacts further with the water as described by equation 5.
It is tempting to concatenate these equations to equation 8 so

{K_8} = {K_5}{K_7}
\left[ {C{a^{2 + }}} \right] = \left[ {O{H^ - }} \right] = \left[ {HCO_3^ - } \right] = s
{K_8} = \left[ {C{a^{2 + }}} \right]\left[ {O{H^ - }} \right]\left[ {HCO_3^ - } \right] = {s^3}
s = \sqrt[3]{{7.2x{{10}^{ - 9}}x1.06x{{10}^{ - 4}}}} = 9.1x{10^{ - 5}}

Unfortunately we now have two different estimates for s. Which is correct? Well neither. The first estimate (equation 7) assumes the carbonate ion does not react further, the second (equation 5) that all the dissolved carbonate reacts.
To get a better estimate let b be the concentration of the hydroxyl ion in equation 5.
Equation 5 says that for every ion of carbonate reacted one ion of hydroxyl is produced and one ion of bicarbonate.

Hence

Concentration of carbonate ion left is (s-b)

\left[ {CO_3^{2 - }} \right] = s - b

Concentration of bicarbonate = concentration of hydroxyl = b

And equilibrium of 5 becomes

{K_5} = \frac{{\left[ {O{H^ - }} \right]\left[ {HCO_3^ - } \right]}}{{\left[ {CO_3^{2 - }} \right]}} = \frac{{{b^2}}}{{\left( {s - b} \right)}}
This must be solved numerically to obtain a compatible set for s and b.

b \simeq 5.86x{10^{ - 5}}

Once we have an estimate for b we can calculate the pH since

pH = 14 - pOH = 14 + {\log _{10}}\left[ {O{H^ - }} \right] = 14 + {\log _{10}}\left( b \right)
pH = 9.8
Similar calculation at other temperatures yield
pH @ 5 deg C is 9.7
pH @ 10 deg C is 9.8
pH @ 25 deg C is 9.9
suggesting that pH for this system is relatively insensitive over the normal range.

This complexity is achieved by just a two phase system – water and solid calcium carbonate.
The next stage is to add an atmosphere with carbon dioxide to form three phase system.

 

[Borek]

Your approach is - in general - incomplete. That is, it yields relatively good results, but it is based on approximations, validity of which you are not checking - so in some unlucky cases you can be completely off.

Correct approach to the general equilibrium calculation is to:

1. Write equations describing all equilibria present in the solution.
2. Write all mass balances for the solution.
3. Write charge balance for the solution.
4. Solve.

So, in the case of calcium carbonate solution, we have 4 equilibria present:

K_{sp} = [Ca^{2+}][CO_3^{2-}]

K_w = [H^+][OH^-]

K_{a1} = \frac {[H^+][HCO_3^-]}{[H_2CO_3]}

K_{a2} = \frac {[H^+][CO_3^{2-}]}{[HCO_3^-]}

Mass balance for the calcium carbonate:

[Ca^{2+}] = [H_2CO_3] + [HCO_3^-] + [CO_3^{2-}]

and charge balance for the solution:

2[Ca^{2+}] + [H^+] = [OH^-] + [HCO_3^-] + 2[CO_3^{2-}]

This is set of equations that describes the solution. 6 equations, 6 unknowns. They don't have to be easy to solve (heck, they AREN'T ease to solve), but once solved, they give you exact information about what is going on in solution.

If I understand correctly, your approach (the better one) doesn't contain full mass balance - that is, equations you wrote are equivalent to assumption that

[Ca^{2+}] = [HCO_3^-] + [CO_3^{2-}]

This is not a bad approximation, so your final results are close to the reality, but it is still approximation only, while the general approach doesn't need any approximation.

Note, that general approach can use any set of equilibrium constants, as long as equations are independent, and there are four of them. So I can replace K_a2 with overall dissociation constant K_a12 (see my earlier post) and I will get exactly the same result.

(side note: your equation K5 = b^2/(s-b) is a simple quadratic polynomial, so it doesn't require numerical approach).

--

 

[Studiot]

Hey c'mon I'm putting my pen and paper calculations where my mouth is. I don't have the resources some command.

Nevertheless I'm simply trying to develop a sufficiently accurate chemical model so that all can use it to discuss the question at hand.
I'm totally open to anyone correcting or improving the model.

My K_b already includes constants K_w; K_a1; K_a2 yes you need these but I have done that bit for you to supply some actual numbers.

So how about you supply some numbers and come up with a better estimate.

(side note: your equation K5 = b^2/(s-b) is a simple quadratic polynomial, so it doesn't require numerical approach).

And please show me how to solve a single equation in 2 unknowns you know neither s nor b.

 

[Borek]

I'm totally open to anyone correcting or improving the model.

That's what I am trying to do - I am trying to explain to you what is the general approach, that yields always correct results.

My K_b already includes constants K_w; K_a1; K_a2 yes you need these but I have done that bit for you to supply some actual numbers.

So how about you supply some numbers and come up with a better estimate.

For the record: model we deal with still ignores many things, like CaOH⁺ and CaHCO₃⁺ complexes present in the solution. But if we limit ourselves just to the equilibria I have listed I got pH of 9.88 (see attached image). This is not for any particular temperature - I have used just pKw = 14, pKa1 = 6.35 and pKa2 = 10.03 (the latter is equivalent to the value you used in your calculations, pKa+pKb=pKw). At this stage this is the same result you got, but when we start to saturate solution with carbon dioxide which will lower pH, first step of acid dissociation will start to play an important role, and your approach will be giving worse and worse results - or you will be forced to modify your model.

Note that my approach - solving full system of equations for all variables - yielded immediately all concentrations of all ions involved. Also note that it yielded the same result you got, even if you have claimed that it is inadequate.

And please show me how to solve a single equation in 2 unknowns you know neither s nor b.

My mistake - I thought you are trying to solve just one equation for b, which is a standard approach when you try to solve simplified systems. But what you meant was that whole system of equations has to be solved numerically, right? But now I understand even less, as if you are solving system using numerical approach, why do you start with approximations, instead of solving full system in a general way?

Explanation to the image with calculation results: first, there is a list of substances present and their equilibrium concentrations (don't pay attention to concentration of CaCO₃(s), it is just lousy reporting of calculation results for solids). Things below are just to check if the numerical result is correct. Balances of mass and charge show numbers of moles of each element expected in 1L of solution, differences are just rounding errors. Then comes list of equilibria given for the system - these are same 4 I have listed in my previous post. Equilibrium constants are given for reactions as written on the right, so these are not dissociation constants but protonation constants, which are just reciprocals. Water dissociation constant is not 1e-14, as for mass balances I had to take water presence into account, that in turn means Kw is not just [H⁺][OH^-] but [H⁺][OH^-]/[H₂O]. But these are just technical minor points, related to implementation, system and model used is exactly as described in my previous post.

--

 

[Studiot]

Note that my approach - solving full system of equations for all variables - yielded immediately all concentrations of all ions involved. Also note that it yielded the same result you got, even if you have claimed that it is inadequate.

Thanks for the independant check by more sophisticated means. I see your calculator required 17 iterations.
I don't remember claiming any end result as inadequate. But I am also conscious of the length of the path yet to be trod. I am just trying to build up from small beginnings in simple steps.
The issue is really not one of "is the model as complex and comprhensive as possible?", but
"is it up to supplying the desired results correctly?"
You seem to have confirmed that all these sundry august institutions have got is right, so the next step is to examine the effect of three big inputs.

1) The effect of acidifying gases in the atmosphere
2) The effect of biologcal agents
3) The effect of chemicals in solution, other than calcium carbonate



Please remember that this is not 'my theory'. I lay no claim to originality. This is the Earth Sciences part of the forum so I am aware that many readers will not be chemists (nor am I actually) so I am trying to carry out forum policy and expound and explain conventional thinking in the subject area.
By conventional thinking I mean the equations and theory you will find in publications and papers from leading Oceanographic organisations around the globe. My sources in particular come from the National Oceanographic Centre, Southampton, NOAH and the Woods Hole Institute, University of Ontariao Environmental Science Unit and the institution where I was a postgrad many centuries ago and then called the Plymouth School of Maritime Studies ( now Plymouth University).
So I am trying to help others, mostly environmentalists, understand the output of learned institutions.

But now I understand even less, as if you are solving system using numerical approach, why do you start with approximations, instead of solving full system in a general way?

One form of numerical approach is to have a seed approximation for at least one of the variables. This is used to calculate approximations for other variables, which are then recycled to improve the first approximation.

You may not be aware that Oceanographers have several definitions of ocean alkalinity,
Here is the relevant one to our equations, the carbonate alkalinity

{A_{carb}} = \left[ {HCO_{_3}^ - } \right] + 2\left[ {CO_3^{2 - }} \right]

Using our equations it is possible to explain the apparent paradox that the pH can simultaneously decrease with whilst the alkalinity increases. Obviously not indefinitely though.

 

[sylas]

Thanks for the independant check by more sophisticated means. I see your calculator required 17 iterations.
I don't remember claiming any end result as inadequate. ...

What you claimed is that Borek's method was inadequate, back in [post=2696999]msg #54[/post].

If the interest is still there in this subject I will post the calculations. They are of interest because they show the inadequacy of Borek's concatenation method and the reason for not combining the constituent reactions.

Borek is pointing out that in fact, he is using the general method; which is not "inadequate" at all.

You may not be aware that Oceanographers have several definitions of ocean alkalinity

Or maybe he is. Borek is (I believe) our most competent and well informed science advisor on chemistry. I'm trying to say this gently... but frankly it is getting a bit old your trying to imply Borek is in need of your help to understand the relevant chemistry. Just make the points you feel relevant, and no doubt we'll all learn something working through the discussion.

Cheers -- sylas

 

[Borek]

I will post the calculations. They are of interest because they show the inadequacy of Borek's concatenation method and the reason for not combining the constituent reactions.

I don't remember claiming any end result as inadequate.

Sorry, but you have lost me here. You claimed you will show inadequacy of my method but now you say that it can produce adequate end results?

Note that I don't claim originality of the method I present either. This approach is about as old as modern chemistry. And while there are many simplified approaches that stem out from the general model, and while many of these simplified models are used in different branches of the scientific world (be it Earth sciences, biology, agricultural sciences and so on), they are just that - simplified approach to partial problems. Simplified - which means they work only in a limited range of concentrations/conditions. That was the price paid to make them usable before computing power became so cheap.

At the moment any PC with GHz processor (have you seen a weaker one in the last few years?) have enough power to calculate equilibrium of system like sea water in a reasonable time using general approach (given you have enough data about all equilibria present, but that's another can of worms). See for example

http://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/

(they have modified the model to reduce number of variables and make calculations faster, but it is still the same method, based on all equilibria and mass/charge balances). There are also other programs like MINTEQA and MINEQL (here I am quoting names from memory, so I can be off) all based on the same general approach.

Edit: Sylas answered while I was editing the post, it took me much longer than expected because of several phone calls in the meantime.

--
methods

 

[Borek]

Borek is (I believe) our most competent and well informed science advisor on chemistry.

I am not, but thank you