[Big-Daddy]

I have a solution of H₃PO₄, 85% wt/wt purity, 1.69 gcm^-3 density and with a FW of 98.00 (I suppose I'd better acknowledge now I don't know what either "wt/wt purity" or "FW" mean). The solution is 250cm3 and contains 3.48 cm3 of concentrated H₃PO₄. (Again, I don't know what this means either.) The pKa values are as follows:

pKa1 = 2.15
pKa2 = 7.20
pKa3 = 12.44

When this is titrated with 40 cm3 of 0.80 M NaOH (take this as strong), calculate the exact concentrations of all species in the solution.

I take this as Na₃PO₄, Na₂HPO₄, NaH₂PO₄, H₃PO₄, H₂PO₄^-, HPO₄^2-, PO₄^3-, H⁺ (H₃O⁺) and OH^-. How would I even go about this? The difficult bit is predicting how much of each salt (Na₃PO₄, Na₂HPO₄, and NaH₂PO₄) will be produced, and how to calculate exactly how many moles of H₃PO₄ will be used up. Once I have the concentrations in new solution of the salts and the acid, it is easy to reach the exact [H₃O⁺] (using proton condition approach).

[twistedesoterix]

(I suppose I'd better acknowledge now I don't know what either "wt/wt purity" or "FW" mean).

Do you mean Weight/Weight Percent (% w/w)?  If so then % w/w is the weight in grams of a solute per 100 g of solution. 

FW is formula weight, which is the same as Molar Mass.  Molar mass of H₂) ~ 18 g/mol

[Big-Daddy]

(I suppose I'd better acknowledge now I don't know what either "wt/wt purity" or "FW" mean).

Do you mean Weight/Weight Percent (% w/w)?  If so then % w/w is the weight in grams of a solute per 100 g of solution. 

FW is formula weight, which is the same as Molar Mass.  Molar mass of H₂) ~ 18 g/mol

Thanks for the definition of FW - I suppose it should have been obvious given the Mr of phosphoric acid.

I gave you the question as it was written down, so the problem dictates 85% wt/wt purity, but unless you have a reason to think this is different from % w/w they are clearly close enough to be the same.

So if the solution of concentrated H3PO4 is 3.48 cm3 and has 85% wt/wt purity, I can work out the starting concentration of phosphoric acid in the total solution (250 cm3) as follows:

Mass of Conc. H3PO4 = 1.69*3.48=5.8812g
Mass of H3PO4 in Conc Solution = 0.85*5.8812=4.99902g
Moles of H3PO4 in Solution = 4.99902/98=0.0510104 mol
Concentration of H3PO4 in Solution = 0.0510104/0.250=0.204 moldm^-3

Is that correct?

[Borek]

Concentration of H3PO4 in Solution = 0.0510104/0.250=0.204 moldm^-3

Looks OK to me.

[Big-Daddy]

Concentration of H3PO4 in Solution = 0.0510104/0.250=0.204 moldm^-3

Looks OK to me.

Thanks - do you know how I might go about solving the rest of the problem? i.e. finding expressions for each of the other species in the solution?

[Borek]

See http://www.chembuddy.com/?left=pH-calculation&right=pH-polyprotic-acid-base

Equations 9.11-9.13 allow calculation of speciation in the solution of diprotic acid of known pH. For triprotic acid you would need to play a little bit with the derivation, but it is just a matter of following identical scheme. Sure, you need to calculate pH first - and I would say that's the more challenging part, as using Henderson-Hasselbalch equation will give only approximate result; H₃PO₄ is too strong for assumption H₂PO₄^- is produced only in neutralization. Generally speaking you should solve equation 11.16 for this page: http://www.chembuddy.com/?left=pH-calculation&right=pH-salt-solution - but as you can safely assume second and thoird dissociation steps don't play any role here, in fact equation 11.15 will be enough - and as the solution is acidic it can be simplified even more (one of teh terms is so low it can be ignored).

[Big-Daddy]

See http://www.chembuddy.com/?left=pH-calculation&right=pH-polyprotic-acid-base

Equations 9.11-9.13 allow calculation of speciation in the solution of diprotic acid of known pH. For triprotic acid you would need to play a little bit with the derivation, but it is just a matter of following identical scheme. Sure, you need to calculate pH first - and I would say that's the more challenging part, as using Henderson-Hasselbalch equation will give only approximate result; H₃PO₄ is too strong for assumption H₂PO₄^- is produced only in neutralization. Generally speaking you should solve equation 11.16 for this page: http://www.chembuddy.com/?left=pH-calculation&right=pH-salt-solution - but as you can safely assume second and thoird dissociation steps don't play any role here, in fact equation 11.15 will be enough - and as the solution is acidic it can be simplified even more (one of teh terms is so low it can be ignored).

I have seen this material before. In fact, when it comes to solving for pH, I would simply use the equation found in Robert de Levie's OCP on Acid-Base Equilibria. It is as exact as the one you linked me to, and almost identical.  As for the concentration equation, it is available for an n-protic acid on Wikipedia.

My problem here is that when you add a base to an acid a salt is produced with the anion of the acid and the cation of the base, so shouldn't all such problems require you to consider the creation of this salt in your calculations? To do so, you must know what concentration of each salt (Na3PO4, Na2HPO4 and NaH2PO4) is produced in the solution. And before you know that, how can you get exact [H₃O⁺ to use the concentration equations?

[Borek]

Have you checked what C_a and C_b mean?

[Big-Daddy]

As far as I know, they generally mean the starting concentration of the acid and base, but in this case Ca can mean the total starting concentration of all sources of the anion PO4^3- and Cb can be the total starting concentration of all of the sources the cation Na⁺. That doesn't really help, since we still don't know how to work out what these starting concentrations are individually.

I'm adding a certain amount of NaOH to pure H₃PO₄ - how do I figure out how much of each salt is produced? I can see now that Ca genuinely does refer to the starting concentration of the acid because any new salt formed capable of producing that anion can only add up to the same starting concentration. Meanwhile, I will take it without question that Cb retains the definition of the starting concentration of the base NaOH.

So essentially, when an acid (or multiple acids) is mixed with a base (or multiple bases), the salt(s) formed do not affect the pH of the solution; it can be calculated directly from the original starting concentrations of acid and base, as shown by the equation in the pH calculator used by ChemBuddy (as long as the new Ca and Cb in the solution formed by combining the acidic solution with the basic one is calculated from the previous Ca and Va, and Cb and Vb). Am I right?

That still doesn't solve the main part of this problem, though. I can find [H3O+] and [OH-] from the starting concentrations of acid and base, but how do I find Na₃PO₄, Na₂HPO₄, NaH₂PO₄, H₃PO₄, H₂PO₄^-, HPO₄^2-, and PO₄^3-? I'd like to learn what's going on in terms of the equilibria (as I need to calculate these concentrations exactly rather than with approximations).

[Borek]

As far as I know, they generally mean the starting concentration of the acid and base

No, not the starting concentrations but analytical concentrations of base and acid in the mixture.

I'm adding a certain amount of NaOH to pure H₃PO₄

And you can use this information to calculate Ca and Cb.

I can find [H3O+] and [OH-] from the starting concentrations of acid and base, but how do I find Na₃PO₄, Na₂HPO₄, NaH₂PO₄, H₃PO₄, H₂PO₄^-, HPO₄^2-, and PO₄^3-? I'd like to learn what's going on in terms of the equilibria (as I need to calculate these concentrations exactly rather than with approximations).

Once you know pH you can use formulas for ion speciation that I pointed you to earlier.

[Big-Daddy]

Once you know pH you can use formulas for ion speciation that I pointed you to earlier.

I'm not trying to work out the concentration of any ions - I'm trying to work out the concentration of salts formed. Individually. And this in turn, I imagine, will affect the concentration of the ions left in pure ionic form in solution after some are converted into salts of Na⁺.

I can easily work out the total concentration of H3PO4 removed, i.e. the total concentration of salt formed, but there are 3 different types of salt - how do I know what concentration of each will be formed?

As for the left over ions, I could believe from the calculator that in any mixture of acids and bases the salts formed have no relation to the concentrations of the ions, so once I have pH those ion concentration calculations will always hold true regardless of how many acids/bases are in the mixture.

But the problem is still - how much Na3PO4 is formed, how much Na2HPO4 is formed, how much NaH2PO4? If you could provide me with the formula for this one example in the necessary form, perhaps I could understand where you're getting it from?

[Borek]

I'm not trying to work out the concentration of any ions - I'm trying to work out the concentration of salts formed.

Salts are made of ions, don't they?

At first you put your question as

calculate the exact concentrations of all species in the solution.

For me that means concentrations of individual ions, not salts.

Besides, concentrations of salts are impossible to calculate.

Imagine you have 1 L of 1 M solution of Na₂HPO₄. What is concentration of Na₃PO₄ in this solution? Of NaH₂PO₄ in this solution? And before you will say question doesn't make sense, as these salts are not present, think twice - 1M solution of Na₂HPO₄ can be for example prepared by dissolving 0.5 moles of Na₃PO₄ and 0.5 moles of NaH₂PO₄ in 1 L of solution, or 0.5 moles of Na₂HPO₄ and 0.25 moles of each other salt an so on. So not only these salts "are" present, but actually there are infinitely many answers to the question.

At the same time you can calculate equilibrium concentrations of all ions present - and this answer is unambiguous and unique.

[Big-Daddy]

I'm not trying to work out the concentration of any ions - I'm trying to work out the concentration of salts formed.

Salts are made of ions, don't they?

At first you put your question as

calculate the exact concentrations of all species in the solution.

For me that means concentrations of individual ions, not salts.

Besides, concentrations of salts are impossible to calculate.

Imagine you have 1 L of 1 M solution of Na₂HPO₄. What is concentration of Na₃PO₄ in this solution? Of NaH₂PO₄ in this solution? And before you will say question doesn't make sense, as these salts are not present, think twice - 1M solution of Na₂HPO₄ can be for example prepared by dissolving 0.5 moles of Na₃PO₄ and 0.5 moles of NaH₂PO₄ in 1 L of solution, or 0.5 moles of Na₂HPO₄ and 0.25 moles of each other salt an so on. So not only these salts "are" present, but actually there are infinitely many answers to the question.

At the same time you can calculate equilibrium concentrations of all ions present - and this answer is unambiguous and unique.

OK, so how would I go about calculating these equilibrium concentrations? Do I only need the triprotic version of the equations on the website you linked to (i.e. the salt production has no effect on the equilibrium concentrations of each of these ions, only the starting concentration of the acid and base)?

As for the salts themselves, what if I then added more of the salt Na₂HPO₄ (after the acid-base reaction)? Do I not need to know the concentration of each of the salts present individually to create the expression needed for pH calculation?

[Borek]

OK, so how would I go about calculating these equilibrium concentrations? Do I only need the triprotic version of the equations on the website you linked to (i.e. the salt production has no effect on the equilibrium concentrations of each of these ions, only the starting concentration of the acid and base)?

Yes, triprotic versions of these equations will be enough.

As for the salts themselves, what if I then added more of the salt Na₂HPO₄ (after the acid-base reaction)? Do I not need to know the concentration of each of the salts present individually to create the expression needed for pH calculation?

If any salt is added you can easily convert it to new concentrations of the acid and the base, after all adding a salt is not different from adding acid and base separately. As explained earlier, individual salts concentrations are irrelevant and ambiguous.

[Big-Daddy]

OK, so how would I go about calculating these equilibrium concentrations? Do I only need the triprotic version of the equations on the website you linked to (i.e. the salt production has no effect on the equilibrium concentrations of each of these ions, only the starting concentration of the acid and base)?

Yes, triprotic versions of these equations will be enough.

As for the salts themselves, what if I then added more of the salt Na₂HPO₄ (after the acid-base reaction)? Do I not need to know the concentration of each of the salts present individually to create the expression needed for pH calculation?

If any salt is added you can easily convert it to new concentrations of the acid and the base, after all adding a salt is not different from adding acid and base separately. As explained earlier, individual salts concentrations are irrelevant and ambiguous.

I now see what you might be saying. Am I right to interpret this: regardless of how many reactions the H3PO4 undergoes, I do not need to consider the salts produced, because I can simply take the analytical concentration of H3PO4 from before (modified only for the volume of base or salt with which is titrated, not for anything else) and the analytical concentration of NaOH (in the new solution which is a mixture of both) - Ca and Cb respectively - and then, if new salt is added thereafter to the solution from the H3PO4-NaOH titration, I have to add a term for that salt alone but I do not have to consider the salts previously in solution and can continue to consider the Ca and Cb of H3PO4 and NaOH as before. In other words, for the first titration, of H3PO4 by NaOH, I can use Ca and Cb (as shown by the pH Calculator equation or any others), and then, let's say I add some Na2HPO4 to the resultant solution as the second titration - all I will need to calculate the pH is to find the pH of the first solution's titration and the volume of that solution (which can be found from Ca, Cb, Va, Vb and Ka/Kb values) and then titrate that with an equivalent expression for the salt Na2HPO4 - I still don't need to know how much salt was produced for using the pH calculator equation, as I can just refer back to the original Ca of H3PO4. How accurate is this?

Sorry it was so difficult to try and explain what I meant. I was trying my best.