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About carbonation process in drinkable water pH adjustment

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Jablo:
Hi. First of all, sorry if this is not the right place to post this, but from all the options this seems the best. I'm a senior in Civil Engineering, and I'm working on my final project, which is a groundwater treatment station. Part of the process is lime softening, which results in a pH increase of up to 11. To reduce the pH to acceptable levels for distribution and consumption, a recarbonation process will be carried out, where gaseous CO2 will be injected into the water. What I cannot find out is the CO2 dosage needed to carry out this task. How can I estimate it? What other variables do I need to know? If someone could help me or guide me where I can get information on this topic I would be very grateful.

Borek:
You probably need to know more about the solution composition. What exactly is the "lime softening", how is it performed?

Basically it is about doing equilibrium calculations, as outlined here: https://www.chembuddy.com/?left=pH-calculation&right=toc but details will depend on the exact situation.

Jablo:
Thank you for your answer. I found this explanation of the pH very useful. For clarification, lime softening consists of adding a certain amount of lime to the water to react the Calcium Hydrogen Carbonate (which is highly soluble in water) into Calcium Carbonate which is less soluble.
 Ca(HCO3)2 + Ca(OH)2 -> 2CaCO3 + 2H20

Then through a sedimentation tank, the calcium carbonate is separated from the water. It is assumed that at the end of this reaction and depending on the amount of lime used you should have water without hardness, but with a pH of about 10.5.
Regarding the information you sent me where you explain what the pH is, if I have a hydrogen concentration of 10^-10.5 M. I have to increase the pH concentration to reach 10^-8.5 M.
If this is the carbonation reaction:

CO2+ H2O -> H2CO3

H2O -> H+ + HCO3-
HCO3- ->H+ + CO3-

Should I pass the molar weight to mass in all the equations to get the amount of CO2 needed?

Borek:
Beware, this is tricky.

Simple approach would be to assume at pH 8.5 solution is dominated by HCO3- and calculate amount of CO2 required to neutralize OH- from pH 10.5 to pH 8.5. It can be more or less right as a first approximation, but you will probably need more, as your solution contains undissociated Ca(OH)2 and some CaOH+, these will react with the CO2 too. So the second approximation will require taking into account these additional equilibria.

And then it is still best to experimentally check if the calculations were right.

Jablo:
Great. Thank you very much for your time. This is what I was looking for

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