[Rutherford]

Problem: When saying ''the solution of carbonic acid'', it is actually meant "the solution of carbon-dioxide in water". Most often, the pK_a1 of this acid is said to be 6.4, which is not expected from its structure. If you know that in a soda-water, whose pH is 5.65, only 0.2% of the dissolved carbon-dioxide is present in the form of the undissociated acid, how much is the real first constant of acidity of H₂CO₃?

Attempt: I mark the number of moles of the dissolved CO₂ at the beginning with x. 0.002x is the number of moles of carbonic acid. I have two reactions:
1° CO₂+H₂O HCO₃^- + H⁺      pK_a1=6.4
  0.998x              y    2.24*10^-6

y=0.177x

2° H₂CO₃ HCO₃^- + H⁺        pK_a1=z
  0.002x       0.177x   2.24*10^-6

z=3.70 while the answer is 3.77. Where am I wrong?

[Borek]

1° CO₂+H₂O HCO₃^- + H⁺      pK_a1=6.4
  0.998x              y    2.24*10^-6

Why 0.998x? pKa=6.4 was calculated assuming there is NO free CO₂ present.

I would start calculating total concentration of carbon dioxide from known pH and pKa (assuming it is 100% converted into carbonic acid).

[Rutherford]

How do you mean NO CO₂? The first pKa was calculated assuming that this reaction represents carbonic acid:
CO₂+H₂O HCO₃^-+H⁺

"I would start calculating total concentration of carbon dioxide from known pH and pKa (assuming it is 100% converted into carbonic acid)."

How would I do that? I need the concentration of HCO₃^-?

[Borek]

How do you mean NO CO₂? The first pKa was calculated assuming that this reaction represents carbonic acid:
CO₂+H₂O HCO₃^-+H⁺

No FREE CO₂. All is present in the form of H₂CO₃, and the reaction taking place is

H₂CO₃ H⁺ + HCO₃^-

Thats the definition of Ka, isn't it?

How would I do that? I need the concentration of HCO₃^-?

Stoichiometry of dissociation.

[Rutherford]

No FREE CO₂. All is present in the form of H₂CO₃, and the reaction taking place is

H₂CO₃ H⁺ + HCO₃^-

I calculated H₂CO₃ when assumed that [H⁺]=[HCO₃^-] then the actual [H₂CO₃]=0.002*the one I calculated earlier. Then again, I get the same pKa, 3.7  .

[Borek]

Wording is slightly ambiguous. I took

only 0.2% of the dissolved carbon-dioxide is present in the form of the undissociated acid

to mean 0.2% of the carbon dioxide is converted to carbonic acid (and dissociation follows), while it is intended to mean "in the equilibrium mixture concentration of carbonic acid is 0.2% of the total analytical concentration of CO₂".

And with this approach I got 3.77.

Which is kind of stupid - final answer is given with three sigfigs, while the least number of sigfigs in the input data is one.

[Rutherford]

Got the same.
Calculated the concentration of CO₂ from the first reaction, then added the amount that dissociated (2.24*10^-6) to get the analytical concentration and I multiplied it by 0.002 to get the concentration of carbonic acid. Then the answer is really 3.77.

Thanks for the help.