Solution Equilibrium

We viewed solubility from a big picture perspective and very briefly touched on the idea that it is governed by the same equilibrium principles as any other reaction. Let us now apply what we have already learned about equilibrium to explore the more technical aspects of solutions and solubility.

Equilibria Review

As a quick recap chemical reactions eventually reach a state of equilibrium in which the forward and the reverse reaction occur at the same rate. At this point, the reaction reaches equilibrium, a stable state in which the amount of products and reactants is unchanging.

When it comes to solutions the same thing occurs and in this case equilibrium represents the point at which precipitation and dissolution occur at the same rate. Basically, your lump of sugar sits at the bottom of the glass undissolved.

Getting Specific: K_sp

Instead of describing this point with a generic equilibrium constant, we use the solubility product constant (K_sp). The reality as it will be shown time and time again is that all K values are all the same, product concentrations at equilibrium over reactant concentrations at equilibrium. However, we must also remember that we never include pure solids or liquids in our equilibrium constants.

[latexpage]
\[
MgCl_{2(s)} + H_2O_{(l)} \to Mg^{2+}_{(aq)} + 2Cl^-_{(aq)}
\]

This point is especially important when dealing with solutions because as we can see from the reaction above everything on the reactants side won’t be included in our equilibrium constant equation. Therefore all K_sp coefficients end up being equal to the product concentrations at equilibrium. Since K_sp follows the form of K_eq any stoichiometric coefficients end up as exponents as per usual.

Thus the Ksp expression for the reaction shown above would be:

\[ K_{sp}=[Mg^{2+}][Cl^-]^2
\]

Q is IP

As we have seen before not all reactions are at equilibrium but are actively moving towards equilibrium. Previously this was termed the reaction quotient (Q) and gave us an indicator of where the reaction was headed. Basically, it is a snapshot of a reaction at a particular point in time not just at equilibrium as is the case for K values.

Chemists decided they didn’t want to use Q for everything so they defined a new term the ion product (IP), which is just Q for solutions. Here all of the same tenets of equilibria also hold true. Meaning the comparisons between K_sp and IP follow the same direction as K_eq and Q.

Ksp > IP	-ΔG	Unsaturated
Ksp = IP	ΔG = 0	Saturated
Ksp < IP	+ΔG	Supersaturated

For solutions, we also add an additional descriptive term that describes the state of the solution. These terms are unsaturated, saturated, and supersaturated. An unsaturated solution can continue to dissolve additional solid since it has room for more ions.

By comparison, saturated and supersaturated solutions are out of room for more dissolved solutes so any additional solid added will simply sink to the bottom and sit there.

Where supersaturated solutions differ is that somehow, usually by heating, we were able to cram in more dissolved ions than can actually fit. As a result supersaturated solutions actively precipitate their dissolved ions back into their solid form if they can.

Examples: IP and K_sp in Action

This is what happens when you harvest salt from seawater. The water in seawater is slowly evaporated which increases the concentration of the dissolved ions. Once K_sp < IP, salt crystals start to form which are then further dried, packaged, shipped, and sold so you can season your lunch. In this case, the evaporation leads to a supersaturated solution, precipitation, and a tastier lunch.

This is also why people with kidney stones are advised to drink more water. As they increase the amount of water in their kidneys they dilute the ion concentration of their urine causing K_sp > IP resulting in the dissolution of any painful kidney stones that might be hanging around. In this scenario drinking more water lead to an unsaturated solution, dissolution of the kidney stone, and a lot less pain.

ICE, ICE Baby

Any discussion of equilibria wouldn’t be complete without a discussion of ICE tables. In the context of solubilities, ICE tables allow us to determine the specific amount of dissolved ions present in solutions. We will use the same consistent setup for our ICE tables then discuss one of the unique pieces about Ksp that allows us to use a neat shortcut.

Step 0: Setup the ICE Table Template

The ICE table template gives us a nice consistent way of setting up our ICE tables. It ensure that we don’t miss any pieces of information and also helps us remember what steps to take when solving.

To see how this template works for solubilities we will consider the following question.

CaCl₂ has a Ksp value of 9×10^-6 if we added enough CaCl₂ to water to create a 4M CaCl₂ solution what will the concentration of Cl- ions be at equilibrium?

Step 1: Write Out The Balanced Chemical Reaction

First, we will need to write out the properly balanced chemical reaction. Since this is a solubility question you will want to look for the distinct components the given reactant will break into. In this case, CaCl₂ will produce a calcium ion and chloride ions.

\[CaCl_2_{(s)} \to Ca^{2+}_{(aq)} + Cl^{-}_{(aq)}
\]

then we will balance the equation

\[CaCl_2_{(s)} \to Ca^{2+}_{(aq)} + 2\; Cl^{-}_{(aq)}
\]

Place the balanced reaction at the top of the template above ICE.

Step 2: Determine the K_sp Equation

After you have filled in the balanced chemical equation fill in the K_sp equation with the chemical species concentrations and exponents on the basis of the balanced chemical equation as you would when writing out any Keq value. Just remember we don’t include solids in our solubility expressions so CaCl₂ will be excluded.

Step 3: Fill in the I, C, and E

Now write in initial concentrations of each in the I column.

Step 4: Plug E into the Ksp Equation and Solve