What is the pH of a saturated zinc hydroxide solution?

From this site we learn that for zinc hydroxide, ##K_(sp)=3xx10^-16##.

How did they know this? How else but by measurement? The ##K_(sp)## value really should have been included with the problem.

So we write out the solubility equilibrium:

##Zn(OH)_2(s) rightleftharpoonsZn^(2+)+2HO^-##

So, here, ##K_(sp)=[Zn^(2+)][HO^-]^2=3xx10^-16##

Given the equilibrium expression, we call the solubility of ##Zn(OH)_2##, ##S##.

Thus ##K_(sp)=[Zn^(2+)][HO^-]^2=Sxx(2S)^2=3xx10^-16##

So we have a cubic to solve:

##4S^3=3xx10^-16##

##S=""^(3)sqrt((3xx10^-16)/(4))=4.22xx10^-6*mol*L^-1##.

To get a gram solubility, we simply multiply this by the molecular mass, ##4.22xx10^-6*mol*L^-1xx99.42*g*mol^-1##

##=0.42*mg*L^-1##, i.e. less than ##"1 ppm".##

But we are not finished yet. We were asked to find the ##pH## of a saturated solution.

Clearly, ##[HO^-]=8.44xx10^-6*mol*L^-1##; why clearly?

And thus ##pOH=-log_10(8.44xx10^-6)=-(-5.07)=5.07##.

And ##pH=14-pOH=8.93##

Does this value of ##pH## make sense, given that we dissolve an hydroxide? Why?