[Rutherford]

In some soda-water which has a pH of 5.65, only 0.2% of the dissolved CO₂ is present as non-ionized acid. What is the pK_a1 of H₂CO₃?

I suppose that the reaction is:
CO₂+H₂O HCO₃^-+H⁺
[H⁺]=2.24*10^-6M
Don't know what to do now. I don't have the data of the concentration of CO₂. How can the 0.2% help?

[fledarmus]

I believe you are supposed to assume that any CO₂ that dissolves will be present as H₂CO₃. The reaction that you've presented should have H₂CO₃ rather than CO₂+H₂O.

[Rutherford]

Okay. Is the degree of dissociation 1-0.002=0.998? I tried to put this, but didn't get the right result (which is 3.77). So I need help here because the concentration of H₂CO₃ in unknown.

[vmelkon]

perhaps it's 99.8% * 2.24*10-6M

[Rutherford]

No, in that case it should be 2.24*10^-6/0.998, but I don't get the right result.

[Dan]

Raderford, can you please show your working?

I seem to be getting 2.95 when I assume K_a2 is not relevant (perhaps this is a poor assumption).

[Rutherford]

Of course I can. I think that K_a2 has to be neglected, because it is not given in the problem (in the source that I got this problem from, it is always written what is required to solve given problems). Last time I was trying to solve this, I did the following:
Wrote the reaction:
H₂CO₃ HCO₃^-+H⁺
[HCO₃^-]=[H⁺]=2.24*10^-6
I thought that α is 1-0.002=0.998
[H₂CO₃]=[H⁺]/α-[H⁺]=4.489*10^-9
Now putting these values in the eq. constant expression:
k=2.24*10^-6*2.24*10^-6/4.489*10^-9=1.1976*10^-4
pKa=3.92, but it is 3.77 in the answer so I am probably wrong somewhere.

[AWK]

Something is wrong in data (degree of dissociation should be  at least ten times less)

[Rutherford]

When I put a smaller α, I get a bigger pKa, so I don't know what is wrong here, but something probably is.

[fledarmus]

When I run your numbers through your equation, I get Dan's answer of 2.95 rather than your 3.92, but either way it isn't 3.77. I'm agreeing with AWK on this one.