[Big-Daddy]

Problem:
A 10.00 mL sample of an aqueous solution containing NH₃ gas is added to 15.00 mL of 0.0100 M H₂SO₄. The resulting solution was titrated with 0.0200 M standard NaOH solution and the equivalence point was reached at 10.64 mL. K_b(NH₃) = 1.8 · 10^-5, K_a2(H2SO4) = 1.1 · 10^-2.

Comments:
Never seen a back-titration involving equilibrium constants. Both guidance for this problem, and any links to more questions with back-titrations involving equilibrium constants (or general problems involving equilibrium constants where the initial conditions for each of the reactions are not always known, as in here), would be appreciated.

[danteOne]

Can you post the question that you are trying to answer? You only posted the setup of the problem and forgot to include the unknown that you need to find.

[Big-Daddy]

My apologies. Here it is:

Calculate the pH of the solution at every stage of the titration. (Footnote: This means starting with the 10.00 mL sample (Solution 1), then the resulting solution after addition of sulphuric acid (Solution 2), then the solution when the equivalence point is reached after NaOH titration (Solution 3).)

[MrTeo]

Ok, let's start: what species do you have after the titration with NaOH?

[Big-Daddy]

NH₄OH? We'll have Na⁺ ions, sulphate ions and NH₄⁺ ions in solution.

[MrTeo]

Sulphate ions? The second dissociation is not complete, so there is also hydrolysis.

Let's change the question a little: what is the equivalence point of the titration with NaOH?

[Big-Daddy]

That is what we have to find (the pH at that equivalence point) as well as the pH at the two other main stages.

2.128 * 10^-4 mol of NaOH were required to reach the equivalence point. This corresponds to an NaOH concentration of 5.9708 * 10^-3 M in this solution. So I suppose we could say [Na⁺] = 5.9708 * 10^-3 M. Then what?

[MrTeo]

That is what we have to find (the pH at that equivalence point) as well as the pH at the two other main stages.

Actually I was thinking about a broader question like "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?"

2.128 * 10^-4 mol of NaOH were required to reach the equivalence point. This corresponds to an NaOH concentration of 5.9708 * 10^-3 M in this solution. So I suppose we could say [Na⁺] = 5.9708 * 10^-3 M. Then what?

This doesn't really help.
Again: what species do you have in solution after the titration with NaOH?

Sulphate ions? The second dissociation is not complete, so there is also hydrolysis.

[Big-Daddy]

Actually I was thinking about a broader question like "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?"

As far as I know, the mathematical definition of the equivalence point is as the point of inflexion on the titration curve. However, we aren't shown a titration curve here. So I'm unsure how to answer your question.

Perhaps you want me to say HSO₄^- (the prevalent species in acidic conditions) is titrated up to the equivalence point at which point [HSO₄^-] = [SO₄^2-]? I'm not confident about defining the equivalence point by [HSO₄^-] = [SO₄^2-] ...

Again: what species do you have in solution after the titration with NaOH?

After the titration I would suspect the solution to be more alkaline so sulphate ions should be the more prevalent? Before the titration the solution will be more acidic so HSO₄^- ions should be prevalent.

[Borek]

http://www.titrations.info/titration-basic-terms

[Big-Daddy]

So in the case of the m-protic acid titrated with the monoprotic base, what is the equivalence point: m*n₀[Acid]=n₀[Base] (where n₀[Base] is the number of moles of the base added, and equivalent for the acid)? If not this, then I don't know what "stoichiometric amount of titrant to titrated substance" means.

And even if this is the equivalence point definition for a diprotic acid, how does it help, when we're considering a mixture of an acid (H2SO4) and a base (NH3)?

MrTeo's two major questions - "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?" - would be nice to answer. But even after reading that page I do not understand equivalence point well enough to answer them. I assume it comes down to an issue of HSO₄^- and SO₄^-, or maybe NH4+, but as you can see my thinking is unclear here as I do not properly understand the situation. What happens when a mixture of NH3 and H2SO4 is titrated with NaOH?

[magician4]

in my opinion, the only reasonable definition of "equivalent point" here would be, that n₀(NaOH) + n₀ (NH₃) = 2 n₀(H₂SO₄)

from this, I would start my calculation, regarding H₂SO₄ (undissociated) and OH^- (unreacted) as inexistant

hence, equilibrium NH₃ / NH₄⁺ and SO₄^2- / HSO₄^- would be the pH-determinating particles


regards

Ingo

[MrTeo]

in my opinion, the only reasonable definition of "equivalent point" here would be, that n₀(NaOH) + n₀ (NH₃) = 2 n₀(H₂SO₄)

Hm. I would actually say (I'm thinking, for example, of a titration followed with a pH-meter) that the equivalence point is when all the HSO₄^- has been titrated by NaOH. There you should "see" (actually this is quite a nasty mixture I'd say, so I'm not sure whether this inflection point will be very clear on the diagram) the second equivalence point of the sulphuric acid titration. But I suppose it's mainly a terminology issue. [Edit: Ok, I think we're saying exactly the same thing  ]

from this, I would start my calculation, regarding H₂SO₄ (undissociated) and OH^- (unreacted) as inexistant

hence, equilibrium NH₃ / NH₄⁺ and SO₄^2- / HSO₄^- would be the pH-determinating particles

Exactly. That's what I wanted to hear from BD. At the equivalence point the problem is similar to the one of the pH of a salt with hydrolysis of the anion and the cation at the same time (which btw under certain approximations that I should check has a result independent of concentration, but I don't know if this applies here).

[Borek]

At the equivalence point the problem is similar to the one of the pH of a salt with hydrolysis of the anion and the cation at the same time (which btw under certain approximations that I should check has a result independent of concentration

http://www.chembuddy.com/?left=pH-calculation&right=pH-salt-simplified

And let's not forget about

http://www.titrations.info/titration-equivalence-point-calculation

[Big-Daddy]

Ok from magician4's equation for the equivalence point I gather that both NH3 and NaOH are to be seen as bases and n₀(NaOH) reaches the point where the equation is satisfied, at the equivalence point. n₀(NaOH) for Solution 3, at the equivalence point, is given. n₀(H2SO4) (for Solutions 2 & 3) is given. n₀(NH3) is then calculated from the other two, for all 3 Solutions.

So now we just set up mass balance, charge balance and equilibrium equations for each of the three situations and solve them for [H⁺]? Well that's the exact (and un-insightful) method anyway. I tried this for the first solution, Solution 1, with only n₀(NH3) present. It worked. This leads me to think it would work for the others as well.

So what method might we follow which would enable us to solve it for Solution 1, at least, without needing to handle a cubic or higher order equation?  I'm talking about approximations