Question

What is the concentration of \[A{g^ + }\] in a solution made by dissolving both \[A{g_2}Cr{O_4}\] and \[A{g_2}{C_2}{O_4}\] until saturation is reached with respect to both the salts?
[\[{K_{sp}}(A{g_2}{C_2}{O_4}) = 2 \times {10^{ - 11}}\], \[{K_{sp}}(A{g_2}Cr{O_4}) = 2 \times {10^{ - 12}}\]]
(A) \[2.80 \times {10^{ - 4}}\]
(B) \[7.6 \times {10^{ - 5}}\]
(C) \[6.63 \times {10^{ - 6}}\]
(D) \[3.52 \times {10^{ - 4}}\]

Answer 1

Hint: In order to find the concentration of \[A{g^ + }\] in a solution, we must know about the solubility product equation. Solubility product constant is similar to equilibrium constant but this involves the dissolution of the solid substance in aqueous solution.

Complete Solution :
Before moving onto the problem, we must know what a solubility product constant is. Solubility product constant is a type of equilibrium constant for dissolving the solid substance into aqueous solution. It is represented by the symbol \[{K_{sp}}\].
Let us now consider \[A{g_2}Cr{O_4}\],
\[A{g_2}Cr{O_4}\] will get dissociated into \[2A{g^ + }\]and \[CrO_4^{2 - }\]ions. The reaction is given below:
\[AgCr{O_4}(s) \rightleftharpoons 2A{g^ + }(aq) + CrO_4^{2 - }(aq)\]
The solubility product for \[A{g_2}Cr{O_4}\] is written as
\[{K_{sp}} = {[A{g^ + }]^2}[CrO_4^{2 - }]\]
Where \[{K_{sp}}(A{g_2}Cr{O_4}) = 2 \times {10^{ - 12}}\]
\[2 \times {10^{ - 12}} = {[A{g^ + }]^2}[CrO_4^{2 - }]\]……… (1)

- Let us now consider \[A{g_2}{C_2}{O_4}\],
\[A{g_2}{C_2}{O_4}\] will be dissociated into \[2A{g^ + }\]and \[{C_2}O_4^{2 - }\] ions. The reaction is given below:
\[Ag{C_2}{O_4}(s) \rightleftharpoons 2A{g^ + }(aq) + {C_2}O_4^{2 - }(aq)\]
The solubility product of \[A{g_2}{C_2}{O_4}\] is written as
\[{K_{sp}} = {[A{g^ + }]^2}[{C_2}O_4^{2 - }]\]
Where \[{K_{sp}}(A{g_2}{C_2}{O_4}) = 2 \times {10^{ - 11}}\]
\[2 \times {10^{ - 11}} = {[A{g^ + }]^2}[{C_2}O_4^{2 - }]\]………. (2)

Now consider,
 \[[CrO_4^{2 - }] = x\]
\[[{C_2}O_4^{2 - }] = y\]
\[[A{g^ + }] = 2x + 2y\]
Substituting these values in equation (1) and (2)
From equation (1), we get
\[2 \times {10^{ - 12}} = {(2x + 2y)^2}x\]………. (3)
\[2 \times {10^{ - 11}} = {(2x + 2y)^2}y\]……… (4)
Divide equation (3) and (4), we get
\[\dfrac{{2 \times {{10}^{ - 12}}}}{{2 \times {{10}^{ - 11}}}} = \dfrac{{{{(2x + 2y)}^2}x}}{{{{(2x + 2y)}^2}y}}\]
\[0.1 = \dfrac{x}{y}\]
\[x = 0.1y\]

Substituting the above value in equation (4)
\[2 \times {10^{ - 11}} = {(2 \times 0.1y + 2y)^2}y\]
\[2 \times {10^{ - 11}} = 4.84{y^3}\]
Therefore, we get
\[y = 1.6 \times {10^{ - 4}}\]
\[x = 0.16 \times {10^{ - 4}}\]
Total \[[A{g^ + }] = 2x + 2y\]
\[[A{g^ + }] = 2 \times 0.16 \times {10^{ - 4}} + 2 \times 1.6 \times {10^{ - 4}}\]
\[[A{g^ + }] = 3.52 \times {10^{ - 4}}\]
Hence the concentration of \[A{g^ + }\] in a solution is \[3.52 \times {10^{ - 4}}\]
So, the correct answer is “Option D”.

Additional information- Solubility of a substance depends on the following parameters
- Solubility of a substance will depend on the temperature. The temperature will be varying for all salts.
- In order to dissolve a solute in a solvent, the solvation enthalpy should be greater than the lattice enthalpy.
- Non-polar solvents will have only low solvation enthalpy.
- solvation enthalpy of ions is always negative i.e.; it means energy will be released during the process.

Note: Solubility of a substance is determined based on the table given below:

Soluble	Solubility > 0.1M
Sparingly Soluble	0.001M < Solubility <0.1M
Insoluble	Solubility < 0.1M