Question

\[{K_{sp}}\] of $CuS$ , $A{g_2}S$ and $HgS$ are ${10^{ - 31}}$ ,${10^{ - 44}}$ and ${10^{ - 54}}$ $mo{l^2}$ $litr{e^{ - 2}}$ respectively. Select the correct order for their solubility in water.
1. $A{g_2}S$>$HgS$>$CuS$
2. $HgS$>$CuS$>$A{g_2}S$
3. $HgS$>$A{g_2}S$>$CuS$
4. $A{g_2}S$>$CuS$>$HgS$

Answer 1

Hint: The solubility product constant is the equilibrium constant for the dissolution of a solid substance into an aqueous solution. It is denoted by the symbol \[{K_{sp}}\]. It is found out by using the solubility product of the components given.

Complete Step by step answer:
Equilibrium reaction for dissolution reaction of $CuS$ is:
$CuS \rightleftharpoons [C{u^{ + 2}}] + [{S^{ - 2}}]$
So, \[{K_{sp}}\]= $[C{u^{ + 2}}][{S^{ - 2}}]$ = ${10^{ - 31}}$
Hence, we get ${S^2}$ =${10^{ - 31}}$
Therefore S= 3.16 $ \times {10^{ - 16}}$
Equilibrium reaction for dissolution reaction of $A{g_2}S$ is:
$A{g_2}S \rightleftharpoons 2A{g^ + } + {S^{ - 2}}$
So, \[{K_{sp}}\]=${[A{g^ + }]^2}[{S^{ - 2}}]$ = ${10^{ - 44}}$
Hence, we get ${(2S)^2}(S)$ = ${10^{ - 44}}$
Therefore, 4${S^3}$ = ${10^{ - 44}}$
So, S= 1.357$ \times {10^{ - 15}}$
Equilibrium reaction for dissolution of $HgS$ is:
$HgS \rightleftharpoons H{g^{ + 2}} + {S^{ - 2}}$
So, \[{K_{sp}}\]=$[H{g^{ + 2}}][{S^{ - 2}}]$ =${10^{ - 54}}$
Hence, we get ${S^2}$= ${10^{ - 54}}$
Therefore, S= $1 \times {10^{ - 27}}$

Therefore solubility of $CuS$ , $A{g_2}S$ and $HgS$ are 3.16$ \times {10^{ - 16}}$ , 1.357$ \times {10^{ - 15}}$and$1 \times {10^{ - 27}}$. This implies that the highest solubility is that of $A{g_2}S$ and lower than this is of $CuS$. And lowest solubility is that of $HgS$ according to the values found. So the correct order according to the solubility in water is $A{g_2}S$>$CuS$>$HgS$.

So the correct option is 4th.

Note: The solubility product is a kind of equilibrium constant and its value depends on temperature. \[{K_{sp}}\] Usually increases with an increase in temperature due to increased solubility. It should be kept in mind that solubility products represent the extent of dissolution of any compound.