What [Ag1+] is required to start precipitation of AgOH from a saturated solution of Mg(OH)2? Ksp = 2.4 ✕ 10-8 for AgOH and Ksp = 2.6 ✕ 10-11 for Mg(OH)2

AgOH ==> Ag^+ + OH^-

Mg(OH)2 ==> Mg^2+ + 2OH^-

Ksp Mg(OH)2 = (Mg^2+)(OH^-)^2
Ksp AgOH = (Ag^+)(OH^-)

Calculate (OH^-) in a saturated solution of Mg(OH)2. That is
.........Mg(OHO2 ==> Mg^2+ + 2OH^-
I........solid........0.......0
C........solid........x.......2x
E........solid........x.......2x
Ksp = (Mg^2+)(OH^-)^2
Ksp = (x)(2x)^2 = 4x^3
Solve for x = (Mg^2+) and twice that will be OH^-.
Then set up Ksp = (Ag^+)(OH^-). substitute OH and solve for (Ag^+).

To determine the [Ag^+] required to start precipitation of AgOH from a saturated solution of Mg(OH)2, we need to compare the solubility products (Ksp) of AgOH and Mg(OH)2.

The solubility product expression for AgOH is:
AgOH ⇌ Ag^+ + OH^-

The solubility product expression for Mg(OH)2 is:
Mg(OH)2 ⇌ Mg^2+ + 2OH^-

Given the Ksp values:
Ksp(AgOH) = 2.4 × 10^-8
Ksp(Mg(OH)2) = 2.6 × 10^-11

We can use these Ksp values to calculate the concentration of OH^- in the saturated solution of Mg(OH)2. Since AgOH will precipitate when the concentration of Ag^+ exceeds the concentration of OH^-, we need to find the concentration of OH^- when the concentration of Ag^+ is equal to or slightly greater than its solubility product.

Let's assume x is the concentration of OH^- in the solution of Mg(OH)2. For every mole of Mg(OH)2 dissolved, we get 2 moles of OH^- ions. Thus, the concentration of OH^- is 2x.

The solubility product expression for AgOH can be written as:
Ksp(AgOH) = [Ag^+][OH^-]
Substituting the values:
2.4 × 10^-8 = [Ag^+][2x]

Therefore, [Ag^+] = (2.4 × 10^-8)/(2x)

However, the concentration of Ag^+ should be equal to or slightly greater than its solubility product. So, we approximate [Ag^+] = Ksp(AgOH) / 2x

Using the given Ksp values, we can determine x:

[Ag^+] = (2.4 × 10^-8)/(2x) = (2.4 × 10^-8) / (2(2.6 × 10^-11))
[Ag^+] = 4.62 mol/L

Therefore, the concentration of Ag^+ required to start precipitation of AgOH from a saturated solution of Mg(OH)2 is approximately 4.62 mol/L.