What precipitates form by mixing 2 saturated solutions?

[sneakycooky]

Homework Statement

A saturated solution of cobalt(III) hydroxide (Ksp = 1.6 x 10^-44) is added to a saturated solution of thallium(III) hydroxide (Ksp = 6.3 x 10^-46). What is likely to occur?
a. both salts remain stable in solution
b. cobalt(III) hydroxide precipitates, thallium(III) hydroxide remains stable in solution
c. thallium(III) hydroxide precipitates, cobalt(III) hydroxide remains stable in solution
d. both salts precipitate

Relevant Equations

IP =[A^n+]^m * [B^m-]^n for degree of saturation, perhaps

At first I thought it was C. A few sources agree with me. The book says D is correct. Here was my reasoning for C:

With both salts having a formula of MX3, their [OH-]'s can be compared according to their Ksp values. Tl(OH)3 has a smaller Ksp, so it has a smaller [OH-] at saturation than Co(OH)3. I think mixing the two salts would make this be true:
[OH-] at Tl(OH)3 saturation < [OH-] of mixing both saturated salt solutions < [OH-] at Co(OH)3 saturation
With Tl(OH)3 being vastly supersaturated, I think it will all precipitate. I don't think Co(OH)3 would have any way to precipitate after this

The book says D is the answer. Here is its justification:

"When the solutions are mixed, [OH-] is above saturation levels for both the cobalt and the thallium in the solution. Since Tl(OH)3 has a smaller Ksp than Co(OH)3, it will react first. The ion product of the mixed solution is higher than the Ksp for Tl(OH)3, and the system will shift left to precipitate solid Tl(OH)3. After the Tl(OH)3 precipitates, a small excess of OH- will remain, which gives an ion product slightly above the Ksp of Co(OH)3. This will cause a small amount (1% - 3%) of Co(OH)3 to also precipitate."

The book's logic isn't making sense to me. Everything I underlined seems wrong to me; mixing the 2 solutions wouldn't make [OH-] be higher than in the Co(OH)3 solution as far as I can tell. I wonder if the "small excess of OH-" refers to the leftover 10^-7 M of OH- from the water of the Tl(OH)3 solution, but that seems like the wrong answer. If there were an excess, the precipitation of Co(OH)3 could make sense, but I guess that leads to the question: is there an excess of OH- after Tl(OH)3 precipitates?

Could someone tell me if the mistake is mine or the book's?

 

[Borek]

You are both wrong and none of the explanations given is correct. Both salts will stay in the solution.

OH^- in both solutions doesn't come from dissolution of hydroxides, but from water autodissociation and is - for all practical purposes - constant at 10^-7 M (it doesn't have to be this way for every hydroxide, but the two listed have solubilities so low they won't change the pH). Concentration of cations in both solutions is defined by that. After mixing of the solutions concentration of OH^- won't change at all and concentrations of cations will go down (each solution gets diluted by the other one) - leaving the saturation state.

 

[sneakycooky]

Interesting! I think I understand. Hydroxide concentration could be much higher, but it is the metal ions that are unstable in water (leading to a low Ksp for both salts). If say 1 L of each saturated solution is combined, the concentration of each metal is halved, and neither will precipitate. I'm not surprised I missed it, but the book has no excuse. Thanks!