Calculating the solubility of calcium carbonate at set pH values

Question

Calculate the solubility of CaCO3 Ksp=3×10^-9 in a buffered solution with a pH=6.82 and 8.98, please? The working behind this would be appreciated.

Answer 1

Hi Thomas L.

When working a problem such as this, you have to ask yourself the right questions, to advance the solution. Here, you are given Ksp for CaCO3, which you know represents [Ca⁺²][CO₃^-2] . But, those two product ions then undergo very different fates! Ca⁺² is subject to complexation by water, but that is ignored in writing the reaction for the dissolution, because Ca⁺² does not undergo additional changes (such as to Ca(OH^-1)⁺¹ ion) to any significant extent (unlike Al³⁺ , for example!). OK for that ion. But CO₃^-2 : ah, a different story! You already know that that ion is in the dissociation series for H₂CO₃ in water. Carbonic acid has (and you should be able to look this up yourself!) pK_a1 = 6.35, and pK_a2 = 10.33 . So you know that at pH 6.82, essentially all the CO₃^2- has at least monoprotonated, and a significant bit of that has even diprotonated. ONLY the still-unprotonated CO₃^2- is available to satisfy the Ksp equation! So look first at the diprotonated/monoprotonated ratio: that's 10^-(6.82-6.35) or 10^-0.47 ~~ 0.3388 . Hold that value for a moment, and calculate the monoprotonated/unprotonated ratio, that's 10^{-(6.82-10.33)} = 3235.94 . Multiply that by 1.3388 = 4332.27 . That's a ratio, so only 1 part of 4333.27 is still present as CO₃^2- of the originally dissolved carbonate!

Now you are ready to set up your calculation. You have established the "still present":"originally dissolved" ratio for CO₃^2- . The "originally dissolved" was the same as the molar concentration of Ca²⁺ ; so write the Ksp calculation in terms only of the originally dissolved Ca²⁺ . So Ksp = [Ca²⁺][Ca²⁺ /4333.27 ] = 3*10^-9 . Solve that through for Ca²⁺ and there you have the answer (for pH = 6.82).

A similar set of calculations are needed for the pH = 8.98 case, but at 8.98, you may neglect the 2nd protonation, it contributes insignificantly to the removal of CO₃^2- .

Hope that helps you.

-- Cheers, --Mr. d.