There is a misconception in the pool industry that the way you add acid to the pool can affect the lowering of pH vs. TA differently. This is not true and some of the methods that are proposed can be damaging to pool surfaces. The acid column or slug method does not lower the TA any better and this method was debunked in this paper.
Adding a strong acid lowers both the pH and the Total Alkalinity (TA) no matter how you add it -- slowly, quickly, in one place, around the pool, etc. 25-1/2 fluid ounces of full-strength Muriatic Acid (31.45% Hydrochloric Acid) in 10,000 gallons will lower the TA by 10 ppm. However, the pH also gets lowered and people often then make the mistake of adding a pH Up product to then raise the pH, but pH Up is sodium carbonate that increases TA even more than a pure base so the net result is that the TA can get higher than when you started! This is the classic see-saw effect in trying to lower the TA through improper methods. One should not add a base to increase the pH when trying to lower the TA.
So how does one lower the TA without lowering the pH? The answer is that you can't in one step, but what you CAN do is to raise the pH without changing the TA. How? By increasing the rate of carbon dioxide outgassing from the pool by having the overall pool pH be lower and by aerating the water. The outgassing of carbon dioxide is equivalent to removing carbonic acid from the pool and that increases the pH (by removing an acid) without changing the TA (because it's a weak acid so essentially carbonate/bicarbonate ions are removed at the same time as hydrogen ions). An efficient procedure for lowering the TA is the following where I indicate what happens at each step. In the following, it is assumed that 6.8 is the lowest measurement on the pH test so that 7.0 is the next to lowest reading (so that you always know your true pH rather than <= 6.8).
ACTIVITY .......... pH .... TA
Acid .................... - ........ - ... Add enough acid to bring pH down to 7.0
Aeration ............. + ....... 0 .... Aerate until pH rises to 7.2
Acid .................... - ........ - ... Add enough acid to bring pH down from 7.2 to 7.0
Aeration & Acid .. 0 ....... - ... Continue this combination until TA is at the target you want
then AFTER you have reached your target TA,
Aeration ............. + ....... 0 ... Aerate until the pH rises to your target pH (say, 7.5).
Net of Above ...... 0 ....... -
It is the combination of acid addition AND aeration of the water that has the TA drop without continuing to lower the pH. Some people use jets to aerate the water. Others turn on waterfalls, spillovers, fountains, point returns up with the pump on high speed. The key is to increase the air/water surface area. Using smaller bubbles is more efficient as the surface area to volume ratio is higher. Even if you don't aerate, there will still be carbon dioxide outgassing, but it will be slower. This is why lowering the entire pool pH speeds things up as well. Having the pool pH be 7.0 for some hours or even a day or two is not a problem, but slugging acid in one place having the pH potentially plummet to very low levels near plaster or vinyl can be harmful to such surfaces.
Adding any concentrated pool chemical, but especially acid or chlorine, should be done by slowly pouring over a return flow with the pump running, preferably in the deep end. For a larger pool, this can be done over multiple returns. To ensure thorough mixing, one can lightly sweep the side and bottom where the chemical is added by using a (nylon) pool brush.
There is another misconception in the pool industry that adding carbon dioxide for pH control increases the TA of the water. It does not. Over time the TA may be seen to rise if one is using a hypochlorite source of chlorine, but the TA rise is from the sodium hydroxide "excess lye" in chlorinating liquid or from calcium hydroxide and calcium carbonate in Cal-Hypo. The TA does not change when carbon dioxide is added nor does it change when carbon dioxide is outgassed.
The Total Alkalinity (TA) test adds acid to the solution until the pH gets to 4.5 at which point the indicator changes from green to red in color. ANY ions that can accept a hydrogen ion (from the acid) will get measured. Look at the formula for alkalinity here and notice the "[OH-] - [H+]" part of the equation. Any weak acid, such as carbonic acid from carbon dioxide, added to water will lower the pH with no change in TA because the (negatively charged) anion increases TA since it can accept a hydrogen ion but the (positively charge) cation which is hydrogen ion lowers TA since it is a hydrogen ion itself (so is already "on the way" towards the transition point in the TA test). A strong acid lowers both the pH and TA because there is no anion remaining that can accept a hydrogen ion (i.e. chloride ion remains chloride ion and does not combine with hydrogen).
Also, see this link (from Rutgers University) which describes (total) alkalinity as a net increase in weak acid anions and says "Because dissolution of CO2 by itself adds equal concentrations of HCO3- and H+ and does not affect alkalinity".
Let's look at some actual examples of adding carbon dioxide which produces a weak acid (carbonic acid) to lower pH vs. adding a strong acid such as Muriatic Acid. To simplify this example I assume zero Cyanuric Acid (CYA) and Free Chlorine (FC) in the water and ignore ion pairs (e.g. CaCO3o, CaHCO3+). What I show below is what happens starting with a TA of 100 ppm and going from a pH of 7.5 to a pH of 7.0 in terms of each of the carbonate (and some other) chemical species. In the following I am using common units of ppm CaCO3 that is used in measuring Total Alkalinity (TA) so converting from mole/liter to this ppm is a factor of 50,043.5 (1000 times half the molecular weight of calcium carbonate since it counts twice towards alkalinity -- that is, the TA ppm measurement is the equivalent carbonate, not bicarbonate).
Carbon Dioxide - CO2(aq) 5.4699 ppm
Carbonic Acid - H2CO3 0.0084 ppm
Bicarbonate Ion - HCO3- 99.5215 ppm
Carbonate Ion - CO32- 0.2260 ppm
Hydroxyl Ion - OH- 0.0279 ppm
Hydrogen Ion - H+ 0.0017 ppm
Total Alkalinity (TA) = 99.5215 + 2*0.2260 + 0.0279 - 0.0017 = 99.9997 ppm
Total Carbonate = 5.4699 + 0.0084 + 99.5215 + 0.2260 = 105.2258 ppm
Adding CO2 to Lower pH
Carbon Dioxide - CO2(aq) 17.3369 ppm
Carbonic Acid - H2CO3 0.0267 ppm
Bicarbonate Ion - HCO3- 99.8518 ppm
Carbonate Ion - CO32- 0.07190 ppm
Hydroxyl Ion - OH- 0.0088 ppm (not exactly same as H+ due to temperature)
Hydrogen Ion - H+ 0.0055 ppm
Total Alkalinity (TA) = 99.8518 + 2* 0.0719 + 0.0088 - 0.0055 = 99.9989 (same as before, ignoring rounding error since molar concentrations had 5 significant digits)
Total Carbonate = 17.3369 + 0.0267 + 99.8518 + 0.07190 = 117.2873 ppm
Adding Muriatic Acid to Lower pH
Carbon Dioxide - CO2(aq) 15.5676 ppm
Carbonic Acid - H2CO3 0.0240 ppm
Bicarbonate Ion - HCO3- 89.5679 ppm
Carbonate Ion - CO32- 0.0643 ppm
Hydroxyl Ion - OH- 0.0088 ppm
Hydrogen Ion - H+ 0.0055 ppm
Total Alkalinity (TA) = 89.5679 + 2*0.0643 + 0.0088 - 0.0055 = 89.6998 ppm
Total Carbonate = 15.5676 +.0240 + 89.5679 + 0.0643 = 105.2238 (same as before, ignoring rounding error)
So you can see from the above that what happens when carbon dioxide is added to the water is that it mostly stays in the water as aqueous carbon dioxide (dissolved gas) and a very small amount of it directly lowers the pH and this causes some of the carbonate to shift towards bicarbonate. Overall, the TA does not change. Essentially, carbon dioxide addition has two effects that cancel each other out -- 1) it increases carbonates which if the pH didn't change would increase TA (just as adding baking soda increases TA with little change in pH) and 2) it lowers pH which also lowers the TA (just as a strong acid does). These two effects cancel each other out. When carbon dioxide outgasses, the exact opposite occurs with the pH rising, but the TA still remains the same.
When a strong acid is added, both the pH and the TA get lowered because you get a shift from bicarbonate ion to dissolved (aqueous) carbon dioxide. That is, a strong acid just shifts equilibrium and this not only lowers the pH but also lowers the TA. When you add a weak acid, such as carbonic acid from carbon dioxide, the lowering of the TA from the lower pH cancels out the increase in TA from the increased total carbonate.
Conceptually, because the amount of hydrogen ion is so much smaller than the amount of bicarbonate and aqueous carbon dioxide, very little of the added carbonic acid needs to dissociate and the resulting lower pH prevents any further dissociation hence the pH is lowered while the TA is not. With a strong acid, the dissociation is complete so affects the equilibrium reactions so that the TA is lowered along with the pH.
I just want to try and clarify a little. When one adds carbon dioxide to water, it equilibrates to about 98.85 % aqueous carbon dioxide and 0.15 % carbonic acid (which Richard's numbers above indicate).
When acid is added to water, it can react with bicarbonate in the water to produce or create carbonic acid. Then the newly formed carbonic acid will equilibrate to the above percentages. In other words, most of the newly formed carbonic acid (from the acid addition reacting with alkalinity) will transform or change to aqueous carbon dioxide (98.85%).
When sodium carbonate (soda ash) is added to water with a low pH, it reacts with existing carbonic acid (not the aqueous carbon dioxide) and forming bicarbonate and thus adding more alkalinity to the water and raising the pH. Since carbonic acid has been removed by this reaction, some of the existing or remaining aqueous carbon dioxide will convert over to carbonic acid (0.15%) to re-establish the same equilibrium as cited above. But since the overall amount of carbonic acid has been reduced (by the addition of soda ash), the pH rises higher.
I hope this helps.
I would qualify that slightly by saying that when one adds a strong acid to the water, one is really mostly shifting bicarbonate ion into aqueous carbon dioxide. Carbonic acid always remains a relatively small component. Though a technicality, it is an important one since this is why lowering the pH results in faster carbon dioxide outgassing. These two species (bicarbonate ion and aqueous carbon dioxide) are by far the most dominant in the buffer system. So I would simplify what goes on by writing the following:
Adding a strong acid such as Muriatic Acid or dry acid supplies hydrogen ions shown in bold:
HCO3- + H+ ---> CO2(aq) + H2O
Bicarbonate Ion + Hydrogen Ion ---> Aqueous Carbon Dioxide + Water
Result is lowering in pH since some of the added hydrogen ions remain (not shown above) while the TA is lowered due to the removal of bicarbonate ions.
Adding sodium carbonate (soda ash) supplies carbonate ions shown in bold which consumes hydrogen ions:
2CO32- + 3H+ ---> HCO3- + CO2(aq) + H2O
Carbonate Ion + Hydrogen Ion ---> Bicarbonate Ion
Result is raising of pH as hydrogen ions are consumed and the TA rises as well as more bicarbonate ions are formed.
Adding a strong base such as lye supplies hydroxyl ions shown in bold:
CO2(aq) + OH- ---> HCO3-
Aqueous Carbon Dioxide + Hydroxyl Ions ---> Bicarbonate Ion
Result is raising of pH since some of the added hydroxyl ions remain (or equivalently, some of them reduce the amount of hydrogen ions to form water) while the TA is raised due to the creation of bicarbonate ions.
Carbon Dioxide Outgassing results in a very small amount of the following:
HCO3- + H+ ---> CO2(aq) + H2O
Bicarbonate Ion + Hydrogen Ion ---> Aqueous Carbon Dioxide + Water
The pH rises as the amount of hydrogen ion decreases. However, very little of the above reaction needs to occur because the equilibrium relationship of these chemicals is a constant:
K = [CO2(aq)] / ([HCO3-] * [H+])
At pool pH, the [H+] term is much, much smaller than the other two so it takes a very small reduction in the two components in the denominator to result in a large required change of the component in the numerator. Essentially, one can consider the bicarbonate ion concentration to be fixed and having, for example, the hydrogen ion concentration cut in half resulting in the aqueous carbon dioxide concentration being cut in half. So the pH goes up and down with the aqueous carbon dioxide concentration, all else mostly staying the same (well, except for hydroxyl ion, but that's not very relevant here since that relationship is separately fixed). Another useful way to look at this is to solve for hydrogen ion concentration in the above and take the log10 to get pH:
pH = log10( [CO2(aq)] / ([HCO3-] * K) ) = log10([CO2(aq)]) - log10([HCO3-]) - pK
So having carbon dioxide either outgas or get injected just reduces or increases the aqueous carbon dioxide which increases or decreases the pH. Adding a strong acid or base shifts the amounts of aqueous carbon dioxide and bicarbonate ion which affects both the pH and the TA. Adding sodium carbonate is exactly like adding both a strong base and sodium bicarbonate because
OH- + HCO3- ---> CO32- + H2O
Hydroxyl Ion (lye) + Bicarbonate Ion (baking soda) ---> Carbonate Ion (pH Up) + Water
By the way, adding sodium bicarbonate (baking soda) mostly results in an increase in TA with little increase in pH unless the pH is low to begin with. This is because the added bicarbonate ion mostly stays as bicarbonate ion with some becoming aqueous carbon dioxide and some becoming carbonate ion essentially populating all three. The pH at which adding bicarbonate has no effect on the pH is around 8.0 (it's not 8.2 due to ion pairs with calcium). However, increasing the TA from 50 to 100 ppm starting from a pH of 7.5 only has the pH rise to 7.7. Starting from 7.0, however, the pH rises to 7.3. In practice, one may see a greater pH rise than this because of carbon dioxide outgassing, especially if the starting pH is lower. Baking soda added to the water too quickly without sufficient mixing has a localized high concentration of bicarbonate that outgasses carbon dioxide more readily. This is a case where how you add the chemical does have some small effect on the net pH result.
I think I stated in my post what you responded with. I stated that the final result of adding acid (that reacts with bicarbonate), ultimately ends up as aqueous carbon dioxide. My point is that it is carbonic acid that is first produced, and yes, (as you also stated), it then converts and ends up as aqueous carbon dioxide (99.85%). That was my way of clarifying the relationship of carbonic acid and dissolved carbon dioxide. Your first equation does simplify it and short cuts it to the final result. I believe that your use of the formula CO2 (aq), recognizes that it also includes the small portion of actual carbonic acid.
Your second formula again simplifies it.
In regards to your last and final paragraph and the effect of adding bicarb, I am in complete agreement.