# Accounting for the Presence of Complexing Ligands

**c) Now you realize that, for the solution in part (b), lead can form soluble hydroxide complexes. Incorporate these into the expression. **

*What complexes can form? What is the modified expression for S for the lead species?*

By looking up *K _{f}* values, students should realize that lead can form three soluble hydroxide complexes in water. They should recall from part (b) that they need to incorporate the total lead into the S expression and that knowing value of αPb

^{2+}will allow them to solve for S.

*What is the value of αPb ^{2+}? What does the value of αPb^{2+} tell us about the formation of hydroxide complexes?*

As with αPO_{4}^{3-}, once they have written the correct expression, provide them with the exact numerical value. It should take them no more than five minutes to write the correct expression. The students should realize that with an α-value of approximately 1, there is essentially no complexation of lead with hydroxide.

*What would happen at a high pH? What does a plot of the solubility of lead(II)phosphate versus pH look like? What would happen to the solubility at highly acidic, intermediate, and highly basic pHs. *

Where is the solubility minimized? Spend about ten minutes discussing what can occur at extreme pH values and what factors determine the pH at which solubility is minimized. Also discuss methods that are used to remove metals as their insoluble hydroxide species (a common industrial strategy to remove toxic metals from waste streams) from solutions.

**d) Revisit part (a). What is the actual solubility of lead phosphate in unbuffered water given that other equilibria will simultaneously occur?**

Allow the students ten minutes to think about how to approach this problem. Students may initially be tempted to find α-values for the ions at a pH of 7 forgetting that α-values are only valid for a constant pH. Allow them to calculate αPO_{4}^{3-} in order to demonstrate that the pH will change. Talk about how in order for the pH to remain constant, we would have to make unrealistic assumptions such as no H_{3}O^{+} or OH^{-} reacting or both sets of competing reactions consuming exactly the same amount of H_{3}O^{+} and OH^{-}. Discuss how this is an example where there are no simplifying assumptions that we can make and that the only way to solve this problem is to solve a set of simultaneous equations.

*How many unknown variables are there in this system? What are the equations?*

Students may have a surprising amount of trouble counting the unknowns. Once they have all agreed on a number, challenge them to come up with the same number of equations to relate those unknown variables. They may need some assistance figuring out what sorts of equations will be helpful. Remind them that something like *K _{sp}* is an equation that relates some of the unknown variables. Allow them about five minutes to come up with as many equations as they can. They may be tempted to list the S expressions as equations, but remind them that doing so would introduce S as a new variable. They will most likely come up with eight or nine on their own but not know about a mass and charge balance. Point out that equating the two S expressions is the mass balance for the system.

*Ask them whether there is something else besides mass needs to be balanced in a solution?*

Groups usually realize that the solution needs to be neutral so that the change must be balanced.

*What is the expression for the charge balance for the system?*

Instruct them to write all of the cations on one side and all of the anions on the other. Make sure that they correctly account for ions with charges other than one. Many groups put the coefficient in the wrong place in their initial attempt at writing a charge balance – they want to take a 3- ion and divide its concentration by three rather than multiplying it by three. A good example is to ask them to consider Na3PO4 dissolving in water and to write a charge balance for only that species, keeping in mind the relationship between the concentrations of the ion.