[cruckshank]

I'm really struggling with the following problem, and would appreciate some guidance about how to solve it!

"A large volume of 0.150M H2SO3(aq) is treated with a solid, strong base to adjust the PH to 5.50. Assume that the addition of the base does not significantly affect the volume of the solution. Estimate the concentrations of H2SO3, HSO3-, and SO3(2-) present in the final solution. Ka1= 1.5*10^-2 and Ka2= 1.2*10^-7."

I've tried to use the two Ka values to write equations and solve but I keep getting the wrong answer.

Thanks in advance!

[Borek]

I've tried to use the two Ka values to write equations and solve but I keep getting the wrong answer.

That's more or less a correct approach, hard to say what went wrong not seeing details of your work. Please post them and we will start from there.

Also: what was the answer you got, and how do you know it was incorrect? Do you know what is the expected answer?

[Babcock_Hall]

Cruckshank,

You might also give some thought to the two values of the pK_a's.  You might be able to draw some inferences from them.

[cruckshank]

Hi, thanks for your responses. I've attached my working, I think maybe my assumption that the initial concentration of H2SO3 being 0.15M is incorrect, but I'm not sure how else to solve if I don't use this initial concentration.

I know these answers are wrong, as the books specifies answers of:
[HSO3-] = 0.14
[SO3 2-] = 0.0054
[H2SO3] = 3.2*10^-5

Thanks again.

[Babcock_Hall]

Borek is more of an expert, but I might suggest an approximate approach.  What does the difference in the two pK_a values suggest to you?

[cruckshank]

Borek is more of an expert, but I might suggest an approximate approach.  What does the difference in the two pK_a values suggest to you?

The difference between them is quite large? I'd say ignore the second dissociation but then I'd be unable to work out the concentration of the sulfite?

[Borek]

Your first equation doesn't take into account fact pH is 5.5, what you did would make sense if the question was about just a dissociation of the acid.

Many ways to skin that cat. [H⁺] and both Ka are given (try to plug them into dissociation constant definitions, after slight rearranging you will get equations with ratio of concentrations of unknowns on one side and known values on the other), sum of concentrations of [H₂SO₃]+[HSO₃^-]+[SO₃^2-] is that of initial H₂SO₃ (that's what mass balance tells you), it yields three equations in three unknowns. It doesn't have to be easy to solve, but it is definitely doable this way.
 
Also, take a look here:

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

see equations 9.11 to 9.13 (and their derivation).

[Babcock_Hall]

Borek is more of an expert, but I might suggest an approximate approach.  What does the difference in the two pK_a values suggest to you?

The difference between them is quite large? I'd say ignore the second dissociation but then I'd be unable to work out the concentration of the sulfite?

To me it suggests that the two dissociations are pretty much stepwise, meaning that one happens almost to completion, then the other one happens.  However, why don't you give Borek's method a try a see where you go?

[AWK]

I think that an iterative method like to Borek suggestion may work quite easy. Using starting value 0.15 for [HSO₃^-] and [H₃O⁺] calculated from pH you may put them into K₁ and K₂ expressions and calculate the first iterations of [H₂SO₃] and [SO₃^2-] respectively. After recalculation of [HSO₃^-] from mass balance you can obtain better values of [H₂SO₃] and [SO₃^2-]. Put these new values to acidity constants in the next iteration. I do hope that next iteration step will be the last one.
Note your so called correct answer is not correct from mass balance point of view, but of course these may be the final approximation (in such a case I always show non-abridged numbers in brackets to prove mass balance).

[Borek]

I think that an iterative method like to Borek suggestion

No need for iterations, you get an exact linear solution immediately.

[cruckshank]

Your first equation doesn't take into account fact pH is 5.5, what you did would make sense if the question was about just a dissociation of the acid.

Many ways to skin that cat. [H⁺] and both Ka are given (try to plug them into dissociation constant definitions, after slight rearranging you will get equations with ratio of concentrations of unknowns on one side and known values on the other), sum of concentrations of [H₂SO₃]+[HSO₃^-]+[SO₃^2-] is that of initial H₂SO₃ (that's what mass balance tells you), it yields three equations in three unknowns. It doesn't have to be easy to solve, but it is definitely doable this way.
 
Also, take a look here:

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

see equations 9.11 to 9.13 (and their derivation).

Hi, thanks for this, I've given it another go and 2 of my concentrations are extremely close/the exact answer in the back of the book. I'm struggling to see where I've gone wrong with the sulphite concentration. Also, I tried it two different ways as I wasn't entirely sure which method was correct (I think the second?). Would appreciate if you could help me out further!

[Borek]

What you did is hard to follow as it is not entirely clear what do the symbols mean and when [H₂SO₃] is intended to mean total (initial or analytical) concentration and when the equilibrium concentration. I suggest using just C (or C_A) for the analytical concentration of all forms of the sulfite (0.15M in your case).

I can be wrong because of that, but as far as I can tell both approaches are equivalent and should yield exactly the same result (a bit different from what you got though, and a bit different from the answer given as correct).

Best answer I got with BATE is [H₂SO₃]=3.073e-05, [HSO₃^-]=0.1445 and [SO₃^2-]=5.439e-03.

[AWK]

I assumed for [HSO₃^-] as 0.150 since we are much closer to pKa₂ than to pKa₁ but any value >0 and ≤0.15 will work. Putting this data to both Ka₁ and Ka₂ I got 3.1623^.10^-5 for [H₂SO₃] and 5.6921^.10^-3 for [SO₃^2-] in the first iteration, and finally 0.144276 for the new [HSO₃^-]. 3 More iterations give results very close to Borek program with calculator only

[Borek]

I will repeat what I wrote earlier: you don't need any iterations, as it is pretty simple to solve analytically. Iterating is just complicating a simple problem IMHO.

And it is hardly surprising that you got good results after one iteration step as the problem is linear. Actually because of that I doubt you will get different result in the next steps.