How much water to raise ph of diet coke to 5.5

[tiger_striped_cat]

This is certainly NOT a homework assignment. I am a medical student and we are learning about oral flora, I read that the safe zone for enamel is a pH of 5.5, and as a diet soda drinker I am curious if this is even possible.

I probably should remember how to do something like this.

The ph of most sodas are between 2.5-4.7. I'm taking the extreme end of 2.5. How much water would I have to add to raise the solution up to a ph of 5.5. The ingredients of diet coke is:

Carbonated water, caramel E150d, sweeteners (spertame), flavoring (caffeine), phsophoric acid, citric acid.

I got this off the UK coke site but it should work well enough. I assume that the phosphoric acid and citric acid are what drop the pH. Both are weak acids, correct? I'm sure I could probably look up an example of this, but I'm not sure I could find one that uses a solution like this. Is there any way to calculate this based on the pH of a solution, rather than the pka of some particular component.

Any help would be appreciated.

 

[Borek]

This is certainly NOT a homework assignment.

For consistency we have to treat all such questions the same way - if it looks like a homework, it lands in the homework section.

This is tricky. You have a system containing three weak acids, and 8 Ka constants, of these several quite close to the desired pH. Calculating pH of such solution is equivalent to finding a root of a 10th degree polynomial, not a thing you want to do manually. Using some assumptions problem can be simplified, but I doubt it can be simplified enough to get a meaningful answer without some kind of numerical approach.

Quick (and lousy, but worth of trying) way of finding the order of magnitude is to assume you have a strong acid of pH 3 and check how much water has to be added to raise pH to 5.5. This is pretty easy - and the result is telling.

I did much more precise simulation using pH calculator built into one of my programs. First, I checked (by trial and error) how much water must be added to 1 mL of 0.001 M HCl to change its pH to 5.5, then I checked how much will pH of just a phosphoric acid change if diluted in the same ratio - difference was 0.01 pH unit, so neglectable.

I am not sure after such dilution it makes sense to still call it a coke.

 

[tiger_striped_cat]

Just to prove it's not a HW assignment. Feel free to make any assumptions you want:

I just looked at the coke here in the US and it lacks citric acid, so maybe that improves things.


Yes, go ahead and assume a ph of 3.
I think phosphoric acid is a weak acid, but I wouldn't mind seeing both calculations--weak or strong.

How would I go about calculating something like this? Again, not a HW assignment so no need to take the typical oh-no-I-might-be-helping-him-cheat mentality. Feel free to make *any* *any* *any* assumptions you want (order of magnitude would be fine) so that I can 1) help remember some of my strong acid/weak acid gen chem and 2) learn what it would take.

 

[Borek]

For strong acid it is just application of pH definition and calculation of dilution.

 

[tiger_striped_cat]

Isn't phosphoric acid a 'weak' acid? If it were a weak acid, how would I calculate that?

 

[Borek]

There is no simple and universal method, especially in the case of multiprotic acid.

For phosphoric acid at pH 5.5 you can assume 1st step to be dissociated 100%, 2nd step not dissociated yet (i.e. - it behaves like a strong monoprotic acid). You are about three pH units from pKa1, so ratio of concentration of H₂PO₄^- to H₃PO₄ is about 10³, and you are about 2 pH units from pKa2, so ratio of concentrations of H₂PO₄^- to HPO₄^2- is about 10² - in other words, H₂PO₄^- is a dominating form.

See weak acid pH, polyprotic acid pH and polyprotic acid - simplified methods of pH calculation for details.

 

[tiger_striped_cat]

Thanks for your help Borek.

just a few more questions:

* What do you mean by 'steps'? 1st order and 2nd order approximations?
* The difference between strong and weak acids are if they completely disassociate in water. I assume this has to do with the chemical structure (in particular, based on the bonding, how readily the acid will lose a proton, which I assume is it's pka1.)
* We don't talk about 'strong acids' and 'weak acids' 'having a low ph' do we? pH is for a solution, and the strength of the acids have to do with the pk values, correct?
* In your explanation I'm assuming you're using the pka values for the first and second disassociation of h3p04 but you're doing this at a ph of 5.5 (my desired ph), why are we not assuming it at a ph of 2.5 (the initial ph of coke)?

* I don't have time to look into too much further at the moment, but I will look into it later (your links are exactly what I'm looking for). I have a ph meter in the mail and I'm going to do a little experiment to see if they match my calculations
* Maybe a easier way to get a little more information is to ask the question of how much phosphoric acid would need to be added to 2 liters of water to get a ph of 2.5. Do you have a webpage explaining this calculation?

I know I've done things like this before, but it's been a while, so I'll have to take it slow.

 

[Borek]

What do you mean by 'steps'? 1st order and 2nd order approximations?

Not approximations, dissociation steps - we are talking about multiprotic acid.

The difference between strong and weak acids are if they completely disassociate in water. I assume this has to do with the chemical structure (in particular, based on the bonding, how readily the acid will lose a proton, which I assume is it's pka1.)

Something like that. Note that strong acids don't dissociate completely, they just have much larger dissociation constants, so the equilibrium of the dissociation reaction is shifted far to the right.

We don't talk about 'strong acids' and 'weak acids' 'having a low ph' do we? pH is for a solution, and the strength of the acids have to do with the pk values, correct?

Yep. pH is property of the solution, strength (expressed as dissociation constant) is property of the acid.

In your explanation I'm assuming you're using the pka values for the first and second disassociation of h3p04 but you're doing this at a ph of 5.5 (my desired ph), why are we not assuming it at a ph of 2.5 (the initial ph of coke)?

We were not talking about pH 2.5 yet, so I was not saying anything about the way it should be treated. 2.5 is close to pKa1, so concentrations of H₃PO₄ and H₂PO₄^- are comparable.

Maybe a easier way to get a little more information is to ask the question of how much phosphoric acid would need to be added to 2 liters of water to get a ph of 2.5. Do you have a webpage explaining this calculation?

No. At pH 2.5 you can treat phosphoric acid as monoprotic, you can start with the weak acid pH page, equations derived there can be used to calculate concentration as a function of pH, you just have to rearrange them.