Question

Consider the solubility of $AgCN$ in water. If $x$ is amount of $C{N}^{-}$ that gets hydrolysed by water and $s$ is the solubility of $AgCN$ when two equilibriums are coexisting :
$AgCN\left(s\right)\stackrel{K_{sp}}{\rightleftharpoons }A{g}^{+}\left(aq\right)+C{N}^{-}\left(aq\right)$
$C{N}^{-}\left(aq\right)+H_{2}O\left(l\right)\stackrel{K_h}{\rightleftharpoons }HCN\left(aq\right)+O{H}^{-}\left(aq\right)$

Which of the following relation is correct?

• A. $\frac{K_w}{K_a}=K_h=\frac{s+x}{x^2}$
  
• B. $\frac{K_w}{K_a}=K_h=\frac{s}{x^2+s}$
  
• C. $\frac{K_w}{K_a}=K_h=\frac{s-x}{x^2}$
  
• D. $\frac{K_w}{K_a}=K_h=\frac{x^2}{s-x}$
  

Answer 1

The correct option is D $\frac{K_w}{K_a}=K_h=\frac{x^2}{s-x}$


Theory:
Solubility of salt in water :
Solubility of salt in water containing anion from weak acid . Example - $AgCN$

$AgCN\left(s\right)\stackrel{K_{sp}}{\rightleftharpoons }A{g}^{+}\left(aq\right)+C{N}^{-}\left(aq\right)$

$C{N}^{-}\left(aq\right)+H_{2}O\left(l\right)\stackrel{K_h}{\rightleftharpoons }HCN\left(aq\right)+O{H}^{-}\left(aq\right)$

Adding the above two reactions the overall reaction is:

$AgCN\left(s\right)+H_{2}O\left(l\right)\rightleftharpoons A{g}^{+}\left(aq\right)+HCN\left(aq\right)+O{H}^{-}\left(aq\right)$

$AgCN\left(s\right)\rightleftharpoons A{g}^{+}\left(aq\right)+C{N}^{-}\left(aq\right)$
$s(s-x)$
$C{N}^{-}\left(aq\right)+H_{2}O\left(l\right)\stackrel{K_h}{\rightleftharpoons }HCN\left(aq\right)+O{H}^{-}\left(aq\right)$
$s-xxx$

$K_{sp}=s(s-x)$

$\frac{K_w}{K_a}=\frac{x^2}{s-x}=K_h$

Also, $K_{sp}\times K_h=s.x^2$