Question

What is the solubility of `AgBr` in a solution of `AgBr` and `AgSCN`? Given `rarr` `K_("sp AgSCN") =10^(-12)` `K_("sp AgBr")=5xx10^(-13)` Thank you:)

Answer 1

`x = sqrt((1xx10^(-12)xx12)/18`

Explanation:

I would represent the dissociated ions in equal amounts as they are in equilibrium.

`"AgSCN" rightleftharpoons "Ag"^+ + "SCN"^-`
`color(white)(xxxxxxxx)xcolor(white)(xxxxx)x`

`"AgBr" rightleftharpoons "Ag"^+ + "Br"^-`
`color(white)(xxxxxxl)ycolor(white)(xxxm)y`

`K_("sp AgSCN") = ["Ag"^+]["SCN"^-]`

`K_("sp AgSCN") = (x+y)(x) = 10^(-12)color(white)(xxxxxxl)(1)`

`K_("sp AgBr") = ["Ag"^+]["Br"^-]`

`K_("sp AgBr") = (x+y)(y) = 5xx10^(-13)color(white)(xxxmll)(2)`

`(x+y) = (10^(-12))/xcolor(white)(xxxxxxmmmmmmmll)(1)`

`(10^(-12))/x xx y = 5xx10^(-13)color(white)(xxxxxxmmmllll)(2)`

`y/x = 5/12`

`y = (5x)/12`

`((5x)/12 + x)(x) = 1xx10^(-12)`

`(17x^2)/12 = 1xx10^(-12)`

`x = sqrt((1xx10^(-12)xx12)/17`

`"solubility of " [Ag^+] " in this solution"= 8.4016805xx10^-7`

Answer 2

Well, if the solution contains both, then since concentration is a state function, we can construct two sequential reactions.

`"AgSCN"(s) rightleftharpoons "Ag"^(+)(aq) + "SCN"^(-)(aq)`
`"AgBr"(s) rightleftharpoons "Ag"^(+)(aq) + "Br"^(-)(aq)`

We can define the solubility of `"AgBr"` as given by `["Ag"^(+)]`, as it is the ion with a coefficient of `1` and is shared between the two compounds.

Assuming the solution was saturated with `"AgSCN"(aq)` already, `["Ag"^+]_i ("AgSCN") = sqrt(10^(-12)) = 10^(-6) "M"`, while `["Br"^(-)]_i = "0 M"`.

The ICE Table would give for `"AgBr"` in saturated `"AgSCN"(aq)`:

`5 xx 10^(-13) = ["Ag"^+]["Br"^-] = (10^(-6) + s)s`

`=> color(blue)(s = ["Ag"^(+)])`

`= color(blue)(3.66 xx 10^(-7) "M")`

(The solution with the small s approximation is `5.00 xx 10^(-7) "M"`, which is 36.6% error, so the approximation would not be good.)

This is in contrast to its solubility by itself:

`"Solubility"_("AgBr"(s,"pure")) = sqrt(5 xx 10^(-13))`

`= 7.07 xx 10^(-7) "M"`

So the solubility of `"AgBr"` in aqueous solution has decreased, in accordance with the common ion effect.