What Are the Equilibrium Concentrations in a Cadmium and Chromium Solution?

[Zipzap]

Homework Statement

Cadmium metal is added to 0.350 L of an aqueous solution
in which [Cr3+] = 1.00M. What are the concentrations
of the different ionic species at equilibrium?
What is the minimum mass of cadmium metal required
to establish this equilibrium?

2Cr3+(aq) + Cd(s) <--> 2Cr2+(aq) + Cd2+(aq) where Kc = 0.288

Homework Equations

ICE box, and Kc = [Products] / [Reactants]

The Attempt at a Solution

I know that Cd(s) is not included in the equilibrium expression. I also know that the initial concentration of Cr3+ is 1.00 M. Setting up the ICE box, I get this:

2Cr3+(aq)
I = 1.00M
C = -2x
E = 1-2x

2Cr2+(aq)
I = 0
C = +2x
E = 2x

Cd2+(aq)
I = 0
C = +x
E = x

And setting up my equilibrium expression, I get the following:

Kc = ( [Cr2+]^2 * [Cd2+] ) / [Cr3+]^2

0.288 = ( (2x)^2 * x ) / (1-2x)^2

Unfortunately, going further gives me a cubic function, and my gut is telling me that I have screwed up somewhere or misinterpreted the question, for every other question I have seen in the textbook does not require solving a cubic function. Where did I screw up?

 

[Borek]

This question is badly worded. Doesn't matter how much metallic Cd you add, system will be at equilibrium. It may be not able to dissolve more Cd, but that's another thing. By "equilibrium" they probably mean there is an excess solid Cd present, but it suggests that when there is no solid, system is not at equilibrium, which is a nonsense.

But then - assuming you are looking for the solution with solid Cd present - all you did is correct, and yes, you get a cubic equation.