how do I find ion concentration at equilibrium from the dissociation/formation of a complex ion?

Question

I have the formation constant and the molarity of the simple metal ion and the polyatomic ion that forms the complex ion. How do I find the ion concentration of the simple metal ion?

Answer 1

First, write the dissociation equation. For example

MX₃ = M²⁺ + 3X

Now set up and ICE table

                   MX₃         M²⁺     X

Initial         [MX₃]       0        0

Change      -z             +z       3z

Equil.         [MX₃-z]    z        3z

Write the equilibrium expression for dissociation

Kd =   z • (3z)³/ {[MX₃] - z}

Kd = 1/Kf

If z << [M],

Kd = z • (3z)³ / [MX₃] which can be solved for z using a scientific calculator. The free ion concentration is equal to z.

If z can’t be neglected in the denominator, a computer algorithm may be needed to solve the problem.

Answer 2

If I am understanding your question right, you are saying you have the initial concentrations of both the metal ion and the polyatomic anion with which it forms the complex. You also know the K_f of the complex. To solve this you would create an ICE table (stands for "initial", "change" and "equilibrium" concentrations). The chemical equation, assuming the simplest case of 1:1 ratio of reactants, and the ICE table are shown below:

M+ + A- <-> MA K_f = [MA]/[M⁺][A^-]

I [M]_o [A]_o 0 These are the initial concentrations

C -x -x +x x is the concentration of each reactant that forms the complex

E [M]_o -x [A]_o - x x These are the concentrations at equilibrium.

You simply substitute the equilibrium concentrations into the K_f expression above to get x, like so:

K_f = x/ ([M]_o -x])([A]_o -x) x is your only unknown so you should be able to solve for it.

If the ratio is not 1:1, you would simply adjust x according to the stoichiometry of the formation reaction. You may also be able to consider x negligible compared to initial concentrations depending on their value relative to the value of K_f. Consult your professor on what the concentration vs K_f limits are for being able to use that approximation.