Ocean Acidification: Can CO2 Release & pH Decrease Simultaneously?

[Borek]

Not really.

Without the other ions being present the dissociation constants are quite different.

Can you give example of what you mean? Assuming - for now - that there is no other acid/base systems present water/carbon dioxide system contains H₂O, H⁺ (well, no such thing as a free proton, but let's not complicate things further), OH^-, CO₂, H₂CO₃, HCO₃^- and CO₃^2-. You have listed 6 reactions between them - it is possible to list further ones, but only 5 are independent. For example you have listed:

HCO₃^- <-> H⁺ + CO₃^2-

and

CO₃^2- + H₂O <-> HCO₃^- + OH^-

When you take water autodissociation into account, this is in fact the same equilibrium (that's what Brønsted-Lowry theory is about). Acid dissociation constant for the first reaction is

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

base dissociation constant for the second is

$$K_{b1} = \frac {[HCO_3^-][OH^-]} {[CO_3^{2-}]}$$

(water concentration is assumed to be constant and ignored by convention)

and

$$K_{a2} K_{b1} = K_w$$

where K_w is water ionization constant.

The situation is further complicated by all the other ions p[resent in seawater.

Yes, but we can ignore them for now to make sure we talk about the same system.

Also these equations carry the reason for the buffering at pH = 8.8 and again at pH = 4.5, which was the other question asked about my posts.

I can't see it, please elaborate. Carbonates have a significant buffering effect at pH around 6.37 (pK_a1) and 10.25 (pK_a2). See attached image. This is a plot of a carbonate buffer capacity, horizontal scale is pH, black arrow points at pH 8.8 (where buffering capacity is relatively low). For low pH this plot assumes high pressure of carbon dioxide, but it is irrelevant in the range we are talking about.

--

 

[Studiot]

I will post some more theory as soon as I get a chance,
meanwhile I offer this experiment.

Take a beaker of water, with a pH meter inserted. What is the pH?

Stir in a spoonful of common salt what now is the pH?

Stir in a spoonful of bicarbonate of soda what now is the pH?

Add a knob of dry ice what now is the pH?

 

[Borek]

Results will depend on water source. Assuming for simplicity we start with a distilled water - at first it will be slightly acidic, something around 6 or even less, as atmospheric carbon dioxide dissolves pretty fast and keeping water pure is very difficult. Addition of NaCl - increase of ionic strength will shift pH up a little bit, but change will be below 0.1 pH unit. Bicarbonate - pH goes up, how far depends on concentration, we are talking about pH 8 range. Dry ice - pH goes down, but it is rather impossible to calculate how far, as this is a dynamic situation and final concentration of dissolved gas depends on many factors.

 

[aquitaine]

How can the ocean simultaneously release CO2 and decrease in pH?

Currently the ocean is acidifying, as it absorbs about a third of the fossil-carbon dioxide that we emit, which then in part assumes the form of carbonic acid. But in the future, if the increasing atmospheric greenhouse effect continues to also warm the ocean enough, we expect this absorption will be reversed and vast quantities of CO2 will distil out from the ocean.

Nonetheless, apparently we do not expect warming to cause any reverse to the acidification. (I asked one paleoceanographer/marine-chemist, and heard there is no contradiction for water to be simultaneously decreasing in pH and liberating CO2.) But naively, if warmed water begins losing carbon, then shouldn't the concentration of carbonic acid fall (and hence the pH start to rise back again)?

Edit: The topic of this thread is not "global warming or climate change". It is purely an ocean-chemistry question. Regardless of what is actually happening to our ocean (or rather, regardless of what external factors may be controlling the temperature of and the partial pressure of CO2 above a hypothetical test-ocean) the question is simply whether in principle such an ocean hypothetically could ever be driven (by adjusting those two parameters) to release CO2 while simultaneously to decrease in pH, and how exactly? (So this is what self-censorship is like..)

A little late to this topic, but perhaps something to consider looking into is how does the current ocean pH now compare with 50+ million years ago, back when CO2 levels were significantly higher than today (and there were no polar ice caps at all)? Perhaps it could be a good insight as to where the acidification issue is going as CO2 levels rise. Just a thought.

 

[sylas]

A little late to this topic, but perhaps something to consider looking into is how does the current ocean pH now compare with 50+ million years ago, back when CO2 levels were significantly higher than today (and there were no polar ice caps at all)? Perhaps it could be a good insight as to where the acidification issue is going as CO2 levels rise. Just a thought.

I'm not familiar with that work. There are attempts to estimate pH in the past; but there are all kinds of additional uncertainties. I had a quick look and do not feel competent to summarize all the questions and variables for such reconstructions.

You are quite right that paleo studies are an important part of the whole picture and a useful test for various theories, but in general, my impression is for the most part, that we get a better view of what is going on in the present and in the immediate future from more direct data, and then this helps us interpret the records from the past.

Cheers -- sylas

 

[Studiot]

drat, I was going to add numbering to my list of equations so I could refer to them, but can't seem to edit that post now.

 

[Evo]

drat, I was going to add numbering to my list of equations so I could refer to them, but can't seem to edit that post now.

Just paste the corrected version into another post and I can edit your old post, if you'd like.

 

[Studiot]

Yes please any numbering you can do would be great.

 

[Evo]

Yes please any numbering you can do would be great.

Just copy your old post into a new post, edit it, then I will move it back into the old post for you.

 

[aquitaine]

I'm not familiar with that work. There are attempts to estimate pH in the past; but there are all kinds of additional uncertainties. I had a quick look and do not feel competent to summarize all the questions and variables for such reconstructions.

You are quite right that paleo studies are an important part of the whole picture and a useful test for various theories, but in general, my impression is for the most part, that we get a better view of what is going on in the present and in the immediate future from more direct data, and then this helps us interpret the records from the past.

Cheers -- sylas

Just a coincidence but this just happened to come up in physorg recently.

In a paper published April 26 in the Proceedings of the National Academy of Sciences, a team of researchers led by a Stanford geologist said that as carbon dioxide gas dissolved in the oceans, it raised the acidity of seawater.

The research team said it was a deadly combination - carbon dioxide in the atmosphere and higher acidity in the oceans - that eventually wiped out 90 percent of marine species and about three-quarters of land species, in a cataclysmic event 250 million years ago known as the "end-Permian extinction."

Back then, the ocean teemed with corals, algae, clams and snails. Soon after, however, there was an abrupt change to a thick layer of bacteria and limestone, a "slime-world," dominated by bacteria.

Just in case you change your mind. :)

 

[Gokul43201]

Here:

$$C{O_{2(gas)}} \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} C{O_{2(aq)}} ~~~-[1]$$

$${H_2}O + C{O_{2(aq)}} \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} {H_2}C{O_3}~~~-[2]$$

$${H_2}C{O_3} \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} {H^ + } + HCO_3^ -~~~-[3]$$

$$HCO_3^ - \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} {H^ + } + CO_3^{2 - }~~~-[4]$$

$$CO_3^{2 - } + {H_2}O \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} HCO_3^ - + O{H^ - }~~~-[5]$$

$$HCO_3^ - \mathbin{\lower.3ex\hbox{\buildrel\textstyle\rightarrow\over {\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}}} C{O_{2(aq)}} + O{H^ - }~~~-[6]$$

No need to edit old posts.

Also, call dissociation of water into (for simplicity) H+ and OH- equation [7].

Equations 4,5 and 7 involve a redundancy (4+5=7), as do equations 2,3,6 and 7 (2+3+6=7). Therefore you are left with no more than 5 independent equations.

 

[Studiot]

Many thanks Gokul for doing that, as well as to Evo for offering. I did try last night but got really tangled up.

Anyway the reason for posting the list of equations is that we are trying to discuss the acidification of at least the surface layers of the ocean.

Also there is another measured parameter, the alkalinity, which is a measure of the ability of the water sample to act a base by reacting with protons.

An apparent paradox occurs in this system as both the acidity (pH) and the alkalinity can simultaneously increase.

To discuss all this we need some to understand how the acidity (pH) and alkalinity is measured, along with how that affects the equilibrium position of the reactions. The point being that natural waters can act as either a base or an acid according to circumstance.

It is not possible to determine the concentrations of the reactants without disturbing the reactions so the standard methods involve titrating to an endpoint.
The standard endpoint indicators in use are either Methyl Orange or Phenolphthalein.
The former changes colour at pH = 4 and the latter at pH = 10 so the choice of titration indicator will influence the results.

A plot of a typical titration is attached.

Additionally whilst the bicarbonate reactions are very fast in the lab, ocean mixing times mean that the response of a large inhomogeneous mass of water is much slower.

With reference to my equations it is important not to combine them as has been suggested, none are ‘redundant’.

Equations 3 and 4 form the reactions of the so called ‘carbonate buffer’.

Equation 4 is the first of two steps in the formation of carbonic acid: the carbonate ion in solution removes one proton forming the bicarbonate ion. Thus the pH rises or is prevented from falling if protons are being added.

Equation 3 is the second step in the formation of carbonic acid by removal of another proton.

Now these reactions occur around points C and D of the titration curve. Thus a titration with methyl orange as an indicator will reach DE on the curve at a pH of 4-5.
This corresponds to an endpoint where most of the bicarbonate and carbonate has been converted to carbonic acid.
Whilst a titration with phenolphthalein as indicator will end somewhere on BC and corresponds to and endpoint for the situation where most of the carbonate ion has been converted to bicarbonate but little of the bicarbonate has been converted to carbonic acid.


This is the basic chemistry of the ‘buffer’.
Most seawater is actually in contact with calcium carbonate and other ion sources, which replenish the participants in the carbonate buffer.
Set against this is the entry of carbon dioxide from the atmosphere.

I will discuss how that affects the situation in the light of the other equations, in my next post.

 

[Borek]

With reference to my equations it is important not to combine them as has been suggested, none are ‘redundant’.

No. Equilibrium in this system (water/carbon dioxide) is described by 5 equilibrium constants. You may write as many reactions as you want, but equilibrium constant for every other reaction but the five basic ones can be calculated from these initial ones. In this sense every other reaction is redundant.

I will give another example. Let's concentrate on just acid dissociation for a moment. Carbonic acid dissociates in two steps:

H₂CO₃ <-> H⁺ + HCO₃^-

and

HCO₃^- <-> H⁺ + CO₃^2-

with respective dissociation constants

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

and

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

Someone may say these are not all reactions describing dissociation of carbonic acid, as there is also overall dissociation reaction

H₂CO₃ <-> 2H⁺ + CO₃^2-

with overall dissociation constant

$$K_{a12} = \frac {[H^+]^2[CO_3^{2-}]} {[H_2CO_3]}$$

However, this part of the system is NOT described by three reactions and three constants, as

$$K_{a12} = \frac {[H^+]^2[CO_3^{2-}]} {[H_2CO_3]} = \frac {[H^+][HCO_3^-]} {[H_2CO_3]} \frac {[H^+][CO_3^{2-}]} {[HCO_3^-]} = K_{a1}K_{a2}$$

- so this reaction and its equilibrium constant doesn't add new information about the system. Knowing any two of three these constant - K_a1, K_a2, K_a12 - we can calculate third one. That means one of these reactions is redundant. Same logic can be applied to the reactions you have listed. There are five independent equilibrium constants describing full system, if you have more than five reactions - some of them are redundant, as their equilibrium constants can be calculated from the basic 5.

Equations 3 and 4 form the reactions of the so called ‘carbonate buffer’.

I would say these are two independent buffers, based on two different acid/conjugate base pairs.

Equation 4 is the first of two steps in the formation of carbonic acid: the carbonate ion in solution removes one proton forming the bicarbonate ion. Thus the pH rises or is prevented from falling if protons are being added.

Equation 3 is the second step in the formation of carbonic acid by removal of another proton.

Now these reactions occur around points C and D of the titration curve.

No. Reaction 3 takes place between points A and B, while reaction 4 takes place between points C and D.

I happen to have a little bit better version of the titration curve you refer to:

(used by permission, see www.titrations.info/acid-base-titration-polyprotics-and-mixtures)

This is not exactly the same situation, as this is titration curve for Warder titration - so there is not only carbonate, but also NaOH present initially, hence end point at pH 11.34 - but after that moment it is the titration you are talking about, even with color ranges marked.

 

[Studiot]

No. Equilibrium in this system (water/carbon dioxide) is described by 5 equilibrium constants

Actually I wasn't describing the carbon dioxide/water system.
I was describing the actual chemical mix you are likely to find in natural waters, before introducing carbon dioxide.

I probably didn't present the equations in the most sensible order and I apologise for this.

The carbonate ions are present from other sources and the question is what happens if we now introduce carbon dioxide?
Also what happens if there is replenishment of carbonate?

I would say these are two independent buffers, based on two different acid/conjugate base pairs.

Yes that is true but I am only using the name granted to the pair of buffer reactions by all the authorities I have read.

No. Reaction 3 takes place between points A and B,

The natural waters I have described cannot attain the high pH between A an B so neither reaction can be here.

I think you have reactions 3 and 4 the wrong way round, 4 must come before 3 as you add protons.

I agree with your concatenation of reaction constants, but have not yet reached the point where they are relevant.

 

[Borek]

Actually I wasn't describing the carbon dioxide/water system.
I was describing the actual chemical mix you are likely to find in natural waters, before introducing carbon dioxide.

Yes and no. "Natural mix" would contain many different ions as well (with their own buffering capabilities, think borates, phosphates, ammonia and so on, even humic acids in some cases), so far you have concentrated on system built around carbonates equilibria - and technically any system containing carbonates already contains carbon dioxide. When you add CO₂ you will be just shifting equilibria present, but the chemistry will be the same.

I probably didn't present the equations in the most sensible order and I apologise for this.

Order is not a problem (well, obviously it is - I have mistaken numbers when referring to the reactions and titration curve). Selection of the reactions - is. And you should really add water autodissociation.

The carbonate ions are present from other sources and the question is what happens if we now introduce carbon dioxide?
Also what happens if there is replenishment of carbonate?

What is the source of carbonates? If it is calcium/magnesium carbonate, in both cases pH goes down, although presence of carbonates slows the process down. And if that's the case pH goes down and alkalinity goes up, I have signalled before that there is no simple dependency between both.

Yes that is true but I am only using the name granted to the pair of buffer reactions by all the authorities I have read.

As far as I can tell one of these systems is called carbonate buffer, the other bicarbonate buffer. Probably calling them collectively "carbonate buffers" is OK, my English ducked behind the table and pretends to be not here.

The natural waters I have described cannot attain the high pH between A an B so neither reaction can be here.

OK, but when you state reaction 4 takes place at C-D it is confusing. After first end point concentration of CO₃^2- is neglectable. At the first end point CO₃^2- constitutes about 1% of all forms of carbonates present. This part of the titration curve (C-D) is dominated by the protonation of bicarbonate (or by the bicarbonate buffering).

I think you have reactions 3 and 4 the wrong way round, 4 must come before 3 as you add protons.

Yes, sorry - my mistake.

--

 

[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.

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

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

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

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.

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[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.

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methods

 

[Borek]

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

I am not, but thank you