The `K_(SP)` of `Ag_(2)C_(2)O_(4)` at `25^(@)C` is `1.29xx10^(-11)mol^(3)L^(-3)`. A solution of `K_(2)C_(2)O_(4)` containing `0.152` mole in 500 mL water is shaken at `25^(@)C` with excess of `Ag_(2)CO_(3)` till the equilbirium is reached.
`Ag_(2)CO_(3)+K_(2)C_(2)O_(4)hArrAg_(2)C_(2)O_(4)+K_(2)CO_(3)`
Ar equilibrium the solution contains `0.0358` mole of `K_(2)CO_(3)`. Assuming degree of dissociation of `K_(2)C_(2)O_(4)` and `K_(2)CO_(3)` to be same, calculate `K_(SP)` of `Ag_(2)CO_(3)`.

Explanation

In the solution, we have the equilibrium,
`"(excess) 0.304M 0 0 Initial concn."`
`underset("solid") (Ag_(2)CO_(3))+underset(0.304-0.0716=0.2324)(K_(2)C_(2)O_(4)) Leftrightarrow underset(0.0716M ("ppt"))(Ag_(2)C_(2)O_(4))+underset("0.0716 Eqb. concn.")(K_(2)CO_(3))`
`[Ag^(+)]` in the solution may be calculated from the `K_(sp)` value of `Ag_(2)C_(2)O_(4)` (supposing `K_(2)C_(2)O_(4)` to be completely ionised)
`K_(sp) (Ag_(2)C_(2)O_(4))=[Ag^(+)] [C_(2)O_(4)^(2-)]`
`1.29 xx 10^(-11)=[Ag^(+)]^(2) (0.2324)`
`[Ag^(+)]=7.45 xx 10^(-6)M`
`(C_(2)O_(4)^(2-)` concn. from `Ag_(2)C_(2)O_(4)` is negligible)
Supposing `K_(2)CO_(3)` to be completely ionised,
`[CO_(3)^(2-)]=[K_(2)CO_(3)]`
=0.0716M
`K_(sp) (Ag_(2)CO_(3))=[Ag^(+)]^(2) [CO_(3)^(2-)] ([ICO_(3)^(2-)]" from "Ag_(2)CO_(3)" is negligible")`
`=(7.45 xx 10^(-6))^2 (0.0716)`
`=3.97 xx 10^(-12) ("mole/litre")^3`.