Solubility. {AGAIN?!?!}

When we talked about solubility, we said it was a dynamic process.  "Dynamic process" is super-secret science code for equilibrium.  If we want to think about solubility in its most generic terms, it is the reaction:

"ionic solid" ↔ "component ions"

Since pure solids do not appear in equilibrium constant expressions, the equilibrium constant for this type of reaction is just the product of the concentrations of the component ions, and is called a solubility product constant, with the symbol K_sp.  Although K_sp refers to a specific type of reaction, it's just another equilibrium constant so it follows all the same rules and has the same meaning as any other equilibrium constant.
For a specific example, let's consider the "insoluble" salt calcium sulfate.  Writing a K_sp-type chemical equation for calcium sulfate:

CaSO₄(s) ↔ Ca²⁺(aq) + SO₄^2-(aq)

K_sp = [Ca²⁺]_eq[SO₄^2-]_eq

"Insoluble" salts have reactant-favored K_sp's,
If we think about making a precipitate rather than dissolving a salt, we might think of a reaction like:

Ca(NO₃)₂(aq) + K₂SO₄(aq) ↔ CaSO₄(s) + 2 KNO₃(aq)

Equilibrium is really all about the chemical processes that are happening, not all the random fluff that might also be included like catalysts or spectator ions.  This means that we can always think about the net-ionic equation for a process at equilibrium rather than the full-molecular or full-ionic equation.  In fact, we can often completely eliminate terms from the equilibrium constant expression by using a net-ionic equation.  For the calcium sulfate equilibrium above, the full-ionic and net-ionic equation is:

Ca²⁺(aq) + 2 NO₃^-1(aq) + 2 K⁺(aq) + SO₄^2-(aq)  ↔  CaSO₄(s) + 2 K⁺(aq) + 2 NO₃^-1(aq)

Ca²⁺(aq) + SO₄^2-(aq)  ↔  CaSO₄(s)

That's the reverse of the K_sp equation for dissolving calcium sulfate, so we can manipulate the equilibrium constant expression and value to use in this situation.

Simplifying Approximations:
There are a couple approximations or assumptions that can make many of our equilibrium calculations a little easier to deal with.  These assumptions are usually valid for problems where the equilibrium is either quite strongly reactant-favored or quite strongly product-favored.  For very reactant-favored equilibria, we can assume that the change in concentration of reactants is small enough to be negligible.  For strongly product-favored equilibria, we can often treat the problem more like a limiting reactant/theoretical yield problem.  These approximations can be the only way to solve some of the higher order polynomials that come up fairly often in equilibrium calculations.

Correction from class:
As I said a few days ago, equilibrium problems are the places I am most like to get myself in trouble when I make things up during class.  I had a little "oops" today in class with the initial concentrations of the lead and sulfate stock solutions.  The set-up should have been 50.0mL of 1.0M solutions.  All the concentrations (and moles) in the equilibrium tables were fine if this stock concentration is used.