Zn(OH)2 (K_sp = 3.0x 10^-17) is amphoteric and also forms [Zn(OH)4]^2- (Zn^2+ + 4OH- [Zn(OH)4]^2-, beta_4 = 2.0x 10^15). Its total dissolved zinc is s = [Zn^2+] + [[Zn(OH)4]^2-]. At what pH is the solubility a minimum?

JEE Advanced levelSingle correct optionDifficulty 3 of 3ia158

$Zn (OH) X_{2}$ ( $K_{s p} = 3.0 \times 1 0^{- 17}$ ) is amphoteric and also forms $[Zn (OH) X_{4}] X^{2 -}$ ( $Zn X^{2 +} + 4 OH X^{-} [Zn (OH) X_{4}] X^{2 -}$ , $β_{4} = 2.0 \times 1 0^{15}$ ). Its total dissolved zinc is $s = [Zn X^{2 +}] + [[Zn (OH) X_{4}] X^{2 -}]$ . At what pH is the solubility a minimum?

[Zn X^{2 +}] = K_{s p} / x^{2}

and

[[Zn (OH) X_{4}] X^{2 -}] = β_{4} K_{s p} x^{2}

, with

x = [OH X^{-}]

. So

s = K_{s p} (x^{- 2} + β_{4} x^{2})

d s / d x = 0

gives

x^{4} = 1/ β_{4}

x = 1.495 e - 04 M

pH = 14 + lo g x = 10.17