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So, if I understood it correctly, I was right that injecting CO2 raises KH by adding bicarbonate ions to water, but since equal amounts of H+ ions are being added, total alkalinity doesn't change.
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Yes and no.
It is not correct to say that injecting CO2 raises KH.
KH is carbonatic hardness and, being a hardness, it has to do with Calcium and Magnesium as much as (and even more than) with Bicarbonates.
I try to explain.
Total hardness (GH) is the total amount of Calcium and Magnesium in water.
It can be divided in two parts:
Carbonatic hardness (KH)= the amount of Calcium and Magnesium that we can think balanced (in eq/l) by bicarbonates
Permanent hardness = the amount of Calcium and Magnesium that we can think balanced (in eq/l) by all the other anions (Cl-, SO4--, NO3- etc.)
Carbonatic hardness can be brought to zero just boiling the water; since thermal energy breaks down Bicarbonates into Carbonates and CO2 (that evaporates).
Carbonates instead precipitate with Calcium and Magnesium (as Calcium and Magnesium Carbonates are almost insoluble in water) and can be eliminated (filtered out for example).
For this reason Carbonatic Hardness (KH) is often referred to as "Temporary Hardness"
What remains after the boiling is the residual amount of Calcium and Magnesium that can be considered balanced by other anions.
This residual amount cannot be removed with such simple tricks and is therefore defined as "Permanent Hardness".
In this way it is always: Total Hardness (GH) = Temporary Hardness (KH) + Permanent Hardness.
So now I hope is clear the concept of KH.
Practically there is no easy and straightforward way to measure KH; nor we are particularly interested in measuring this parameter.
What we call KH is instead another parameter (much more important for us) which is Alkalinity (also called Buffer Capacity of water).
The liquid colorimetric tests we use and call KH tests are instead Alkalinity tests.
So, now that is clear (I hope) what KH is, is easy to understand that there is no way we can increase KH by adding just CO2 or just Bicarbonates into water.
For increasing KH we should add instead Calcium and/or Magnesium Bicarbonates.
Thus, adding just Bicarbonates we raise instead Alkalinity!
In partcular:
1) If we add Sodium or Potassium Bicarbonates we raise just Alkalinity (besides obviously Sodium or
Potassium concentration)
2) If we add Calcium or Magnesum Bicarbonates we raise at once Alkalinity, KH and GH (besides
obviously Calcium or Magnesium concentration)
It has to be pointed out that we could rise Alkalinity also by adding some other weak acids like acetic, fulvic, citric, gluconic etc.
All of these will be red by our colorimetric Alkalinity (KH) tests.
That's why in waters with a high content of such acids (from peat extract for example) we cannot rely on the reading of our Alcalinity (KH) tests to calculate CO2 concentration (from equations or tables and in conjunction with pH value).
CO2-Bicarbonates-pH relationship always holds true;
but in these cases we cannot know Bicarbonates concentration because our tests give us Alkalinity (which in these cases will be an overextimation of just Bicarbonates).
So,...Yes; you are right regarding the CO2, Bicarbonates and Alkalinity relationship.
Adding CO2 raises Bicarbonates, but not Alkalinity nor KH.
Most of the confusion comes indeed from using the terms "KH" (or carbonatic hardness) and "KH test" instead of the correct ones "Alkalinity" and "Alkalinity test"
Hope I helped.

Cheers,
Fabrizio