Calculate Equilibrium Constant for Polyprotic Acid Reaction

[i_love_science]

The principal equilibrium in a solution of NaHCO₃ is
HCO₃^-(aq) + HCO₃^-(aq) <-> H2CO₃(aq) + CO₃^2-(aq)
Calculate the value of the equilibrium constant for this reaction.

My solution:

This overall reaction is the same as the sum of the following reactions:
HCO₃^-(aq) <-> H2CO₃(aq) + OH^-(aq)
HCO₃^-(aq) <-> CO₃^2-(aq) + H⁺(aq)

The overall equilibrium constant is (K_b)(K_a2) = [10^(-14) / 4.3*10^(-7)]*[4.8*10^(-11)] = 1.1^10^(-18).

The correct answer is 1.1*10^(-4).

Can anyone explain where I went wrong? Thanks.

 

[Borek]

The overall equilibrium constant is (K_b)(K_a2)

Is it?

 

[mjc123]

If you multiply the expressions for the equilibrium constants of the two reactions, you will find you have [H⁺] and [OH^-] in the numerator. To get the equilibrium constant for the original reaction, you need to divide by [H⁺][OH^-].