Chapter 11, Problem 68E

An electrochemical cell consists of a nickel metal electrodeimmersed in a solution with $[{\text{Ni}}^{\text{2+}}]=1.0M$ separatedby a porous disk from an aluminum metal electrodeimmersed in a solution with $[{\text{Al}}^{\text{3+}}]=1.0M$ . Sodium hydroxideis added to the aluminum compartment, causing $\text{Al}{\left(\text{OH}\right)}_{\text{3}}\text{(}s\text{)}$ to precipitate. After precipitation of $\text{Al}{\left(\text{OH}\right)}_{\text{3}}$ has ceased, the concentration of ${\text{OH}}^{\text{–}}$ is $1.0\times {10}^{-4}$M,and the measured cell potential is 1.82 V. Calculate the $K_{\text{sp}}$ value for $\text{Al}{\left(\text{OH}\right)}_{\text{3}}$ .
$\text{Al}{\left(\text{OH}\right)}_{\text{3}}\text{(}s\text{)}\rightleftharpoons {\text{Al}}^{\text{3+}}\text{(}aq{\text{)+3OH}}^{\text{–}}\text{(}aq\text{)}{K}_{\text{sp}}\text{=?}$