Calculating CO2 Needed to Reduce pH of NaOH Solution to 12

[Sparky42]

I am trying to perform the calculation that will tell me how much (mass or moles) carbon dioxide (CO2) I would theoretically need to bubble through a 3 liter solution of a 2.6 M aqueous sodium hydroxide (NaOH) solution in order to reduce the pH of the solution to 12. I know that the reaction will be CO2 + NaOH ---> HCO3- + Na+ but I am unaware of how to calculate the mass of carbon dioxide I will need. Any help would be greatly appreciated. Please show all steps of work as I would like to understand this for future knowledge.

Thanks!

 

[Borek]

This is not an easy problem for several reasons. First, for such a concentrated solution you won't get precise results because of too high ionic strength. Second, carbonate that is produced during the neutralization is quite a strong base, so it can't be ignored.

As long as there is a huge excess of the strong base you can safely assume neutralization goes to the end (that is, your product is CO₃^2-, not HCO₃^-). Calculate how much NaOH has to be neutralized for the

Quick and dirty approach would be to calculate how much carbonic acid is needed to neutralize NaOH up to the moment excess NaOH has pH of 12. While it is not an exact result, it will give you order of magnitude.

 

[Sparky42]

Thanks Borek,

I am a little confused though. Here is what I think you are saying. Since the solution of NaOH is 2.6 molar and I have 3 liters of it that would be 7.8 moles. When the pH of the solution is 12 the [OH-] is 0.01 molar so 3 liters of this would be 0.03 moles. 7.8-0.03=7.77 moles meaning that 7.77 moles of carbonic acid is needed to neutralize the NaOH to the right concentration. This would also mean that 7.77 moles of carbon dioxide is needed to lower the pH to 12.

Is this correct? If not can you please show me some calculations to get me on the right track?

 

[Borek]

In general you are on the right track, but you forgot carbonic acid is diprotic.

 

[Sparky42]

Right, so it should be

CO2 +H2O--->H2CO3
H2CO3 +2NaOH--->Na2CO3 + 2H2O

meaning that for every 2 moles of NaOH only 1 mole of H2CO3 is needed thus only 3.8 moles of CO2 are needed?

 

[MrSid]

Borek means that the first step is reaction of two to one moles...
2NaOH + CO2 -> Na2CO3 + H2O (net eqn)
Since this product is still largely a strong base in water it will be the predominant specie before the next amount of CO2 can start reacting to make the NaHCO3...
Because of the high concentration of the species, solubility and ionization activities of the ions are going to make the calculation "wrong" going by simple stoichiometry in order to achieve pH 12. But at least you'll be somewhat close in pH to stop close to stoichiometric Na2CO3...

 

[Sparky42]

Thanks for your input everyone. Is there a way to figure out the amount of CO2 needed more accurately than the just stoichiometrically?

 

[Borek]

Thanks for your input everyone. Is there a way to figure out the amount of CO2 needed more accurately than the just stoichiometrically?

No.

That is, it can be calculated by ignoring solution ionic strength, but it is a waste of time, as the final result will be probably no more accurate then the approximated result you got from the stoichiometry.

 

[MrSid]

A theoretical treatment of the role of ionic strength and activity coefficients starts with Debye-Huckel. ... the Wiki article on the following search terms...

Debye-Huckel equation ; Debye-Huckel limiting law

It seems you are motivated by curiosity- follow where the Henderson- Hasselbach equation leaves off and involve some calculus!

 

[Borek]

Problem with Debye-Huckel theory is that is works correctly for solutions with ionic strength up to 0.1, solution in question is 26 times that.

 

[epenguin]

Sometimes if the student said why he is looking for a answer to the particular question we could answer more to the point.

I think some important things to realize are the following:

The stoichiometric calculation is quite close to right. What you have from half a mole of CO₂ per mole of NaOH is Na₂CO₃. For reference with a pK_a2 I found of 10.33, for reference the pH of 1M Na₂CO₃ works out as 12.06! 2.6 M I get 12.37. But if you are working with other pK’s (not clear to me what ionic strength the above are at) maybe it is exactly 12?

How to get these results? Calculations with unbuffered solutions are just as tricky if not more so than with buffered ones so student should do some practice with them. (for another example see https://www.physicsforums.com/showpost.php?p=3890835&postcount=7).

Write out conservation of mass equations and electroneutrality equations. Cross out all concentrations that are going to be negligible and hence show that in sodium carbonate solution [HCO^-₃] = [OH^-]

Hence show that [OH^-] = √([CO₃^2-].K_w/K_a2)

And/or get an equation for [H⁺] directly.

In conclusion I find that in 2.6 M Na₂CO₃ solution about 1 % of the carbonate is actually in the form bicarbonate HCO₃^-. Now if we still want to get that down from pH 13.37 to 12.0 I find from Henderson-Hasselbalch (maybe there is a better way) that at pH12 about 2% is in form HCO₃ So I would have to remove about 1% of the CO2 to get from 12.37 to 12 I think. Or in the original terms of the problem the amount of CO2 I have to put in NaOH is about 99% of stoichiometric.

About the reservations made re ionic strength etc. OK. However calculations of amounts to be added or subtracted to change pH by a certain amount are reasonable, as much the same influences will be applying before and after such a change.

If still of interest it would be nice to see some actual calculations set out.

Of rather more useful interest than the pH of sodium carbonate is that of sodium bicarbonate. It is useful to realize that its pH is exactly half way between pK_a1 and pK_a2 of carbonic acid and so about 8.3, and a student should be sure he can show this.

 

[MrSid]

I suspect that if we were writing this as a problem to solve for a student, we would steer away from the complications of solubility and ionic strength by focusing on a weaker solution of NaOH, where the assumptions in the Henderson Hasselbach equation are followed.

However, I do sometimes see in these questions the quizzical nature that suggests that someone is trying to solve a real world problem in engineering or industry, and they vaguely remember that something in their chemistry background spoke to how to approach the problem. Additional questions speak to me of an interest in whether there might be reasons to deviate from the simple methods-I am always one to take the simple method as a reality check (like Borek), and then look at the additional factors that can affect the result.

One reality check (from my days in industry) is that a concentrated solution of NaOH (2.6 M is over 10% by wt NaOH) generally cannot hold any appreciable amount of Na2CO3, and so unless it is kept warm some of the Na2CO3 will crash out. Even at 20C for pure Na2CO3, the saturation is 21 % or just at 2M.

We can see that using formal concentrations in the equations are going to deviate from real dissociated ionic concentrations. Though the DHLL and EDHLL describe activity coefficients only up to 0.1 M they do set the trend that activity coefficients will be less than 1 for the ionic species and so dissociation deviates even more so for this concentrated NaOH, saturated Na2CO3, and H2O ionizations from theoretical based on formalisms. Theory that extends the deviations for activity coefficients into high molarity regions relies on empirical factors (see Wiki for SIT), but the trend is there that formal concentrations can be wrong for concentrated solutions.

In industry, a packed bed contactor (gas scrubber) would plug off with Na2CO3 long before reacting out most of the NaOH at this concentration unless run above ambient temperatures. Too hot and the reaction reverses and CO2 is evolved from the NaHCO3. The second ionization and reaction with CO2 will be of a saturated Na2CO3 solution in contact with an excess of solid Na2CO3. To determine the conditions, they have to do the experiment...

 

[Borek]

To be honest, I am too lazy to try to solve it on paper. I haven't spent substantial time writing this pH calculator to do these things by hand

pK_a1 = 6.37, pK_a2 = 10.25, for NaOH pK_a = 13.8. Ionic strength ignored completely.

You need 1.2859 M of carbonic acid (that was done by trial and error, but when you are shown pH immediately after entering concentration it takes almost no time to converge on the right answer).

Edit: converge is not the word I was looking for, but for the life of me I can't remember WHAT IS THE WORD I WANT TO USE.