[Borek]

Concentration of A^2- in the solution of a weak, diprotic acid H₂A, was determined to be 5.1×10^-10 M. Drop of a methyl orange added to the same solution colors it red. Can you calculate pK_a2 of the acid?

[Rutherford]

Not sure, but after a lot of approximations I got pK_a2≈9.3.

[Borek]

after a lot of approximations

Care to list them?

[Rutherford]

Methyl orange became red=very acidic solution. The concentration of A^2- is very low, so it means that K_a1 is enough big (therefore the solution is acidic).
H₂A HA^-+H⁺ K_a1 reasonable big
HA^- A^2-+H⁺ K_a2=?
If I mark the concentration of H₂A by x, then the concentration of HA^- and H⁺ are both x, too.
K_a2=[A^2-]*[H⁺]/[HA^-]
The concentration of A^2- is very low, meaning that K_a2 is very low, too, so the amount of H⁺ produced can be neglected to the amount of H⁺ produced in the first step. The amount of HA^- that dissociated can also be neglected, so:
K_a2=5.1×10^-10*x/x=5.1×10^-10
pK_a2≈9.3.

[Borek]

If I mark the concentration of H₂A by x, then the concentration of HA^- and H⁺ are both x, too.

This part is wrong. You are right K_a1 is much higher than K_a2, but you don't know if it is high enough for H₂A to be completely dissociated. Besides, what you wrote suggests [H₂A]=[HA^-]=[A^2-] - are you sure that's what you mean?

[Rutherford]

If I mark the concentration of H₂A by x, then the concentration of HA^- and H⁺ are both x, too.

This part is wrong. You are right K_a1 is much higher than K_a2, but you don't know if it is high enough for H₂A to be completely dissociated. Besides, what you wrote suggests [H₂A]=[HA^-]=[A^2-] - are you sure that's what you mean?

I am not sure how did you mean that the thing I wrote suggests that. If K_a2 is very low, that can't be true.

[Borek]

Sorry, a typo.

[H₂A]=[HA^-]=[H⁺]

[Rutherford]

Okay, using now a different approach:
H₂A HA^-+H⁺ K_a1
y-x        x     x
K_a1=x²/(y-x), from this y-x=x²/K_a1

K_a1*K_a2=[H⁺]²[A^2-]/[H₂A]
K_a1*K_a2=x²*5.1*10^-10/(y-x)
When I change x²/K_a1, derived above, for y-x, K_a1 and K_a1 are negating each other, and x² and x² do the same thing so again I get: K_a2=5.1*10^-10, pK_a2≈9.3.
Is this way maybe correct?

[Borek]

Stop worrying about H₂A, you were much closer earlier. There is a single assumption needed - and you already did it, you are just unnecessarily making thing more difficult than they are.

[Rutherford]

I am not sure what's wrong in this way, but I will re-write the first way from beginning with the correction (without introducing K_a1):
H₂A HA^-+H⁺
y-x        x    x
HA^-             A^2-     +     H⁺
x-5.1*10^-10     5.1*10^-10     x+5.1*10^-10
Because 5.1*10^-10 is probably much less that x, it can be neglected for [H⁺](without this assumption it couldn't be solved, but that is way the methyl orange is introduced I think) and for [HA^-].
K_a2=5.1*10^-10*x/x=5.1*10^-10
pK_a2=9.29

[Borek]

Let's say it is OK now.

Basically it is all about rewriting Ka2:

[tex]K_{a2} = [A^{2-}] \frac {[H^+]}{[HA^-]}[/tex]

If the second dissociation step is much weaker than the first, we can safely assume [H⁺] = [HA^-] - and they cancel out. This is the only required assumption, and it holds reasonably good for a surprisingly wide range of K_a1 and K_a2 values. Methyl orange was there just to make sure K_a1 is reasonably large and solution is not too diluted.