You are a chemist involved in developing a new product for a textile company. As part of the new process, a suspension of the compound lead phosphate will be used to treat the surface of the textile. The lead phosphate will end up in the waste effluent from your plant. This effluent will be discharged to the local municipal waste water treatment plant. Unfortunately, from your standpoint (fortunately, from the standpoint of an environmentalist) the waste water treatment plant faces strict requirements on the amount of lead that is permitted in their end products. (A waste water treatment plant ends up with "clean" water and a solid sludge. Most lead ends up in the sludge, and the Environmental Protection Agency has set a limit on how much lead is permitted in the sludge.) Most municipalities will require you to enter into a pre-treatment agreement, under which you will need to remove the lead before discharging to the plant. For example, the City of Lewiston will require you to discharge a material that contains no more than 0.50 mg of total lead per liter.
Lead phosphate is a sparingly soluble material so most of it will actually be a solid in your waste, thereby allowing you to filter it out before discharge to the treatment plant.
What is the concentration of total dissolved lead in the discharge?
I use this problem on the first day of class to show the types of problems we will be addressing by the end of the equilibrium unit and to set the stage for the different processes (acid-base reactions, formation of water-soluble metal complexes, solubility of a sparingly soluble salt) that we will be using over the term. It allows me to introduce the different tables of equilibrium constants that we will use over the term and to show how these different processes usually occur simultaneously in real systems (e.g., environment, living organisms) and that we need to develop the expertise to handle these real systems.
The in-class problems on chemical equilibrium are intended for a student who has taken general chemistry and had a previous introduction to the topics of chemical equilibrium, acid-base chemistry, solubility equilibria, and complex-formation equilibria. I usually find that, while students have been introduced to these topics, their understanding is still marginal and they need a refresher on the concepts that we had hoped they would learn in the general chemistry course.
Before passing out the first problem set to the class, I spend a relatively brief period of time discussing with them some background information on chemical equilibrium. Using the generalized equation shown below, I ask them what things we might say about the equilibrium state of this reaction.
aA + bB = cC + dD
Usually it does not take too long before students in the class indicate that one characteristic of the equilibrium state is that the concentrations of the chemicals remain fixed. I point out that viewing the system at the macroscopic level (i.e., concentration), we would define equilibrium as a static state. Students also offer up that another thing that characterizes the equilibrium state is that the rate of the forward reaction is equal to the rate of the reverse reaction. Therefore, at the microscopic level of individual species, we find that the system is dynamic and species are actually changing their identities.
I then ask if someone would provide me with the equilibrium constant expression, and many of the students can usually give the answer I am looking for.
Equilibrium constant expression
I then ask the person who provided the answer (and also throw this out more generally to the class) whether they are so certain that this is the correct expression that they would be willing to stake their entire grade in the course on it – if it’s correct, they receive an A for the course and do not have to attend; if it’s incorrect, they receive an F for the course and also do not have to attend (one bright side (?) to taking the offer no matter whether the answer is correct or not is that they don’t need to attend the class). The students immediately sense that the answer is likely not correct (why the offer if it is correct?), but rarely do they come up with why (we either do not cover the concept of activity in general chemistry or if we do, since we then do all the calculations using terms for concentration, they forget this).
This allows me to introduce the concept of activity, and how equilibrium constant expressions are correctly written in terms of activity and not in terms of concentration. I write something to the effect of the following on the board, and ask them if they could identify an A species in the picture that might be regarded as “inactive”. They readily identify the one shown in bold as an “inactive” form of A.
A B B A A A B B A B
I then ask them why they think equilibrium constant expressions and calculations in general chemistry were always done using concentrations rather than activities. They are usually stumped by this and I do not let them spend too long thinking about it. I point out that chemists rarely shy away from difficult calculations, so it is likely something different than that, and indicate that in many instances we simply do not know how to accurately express the activity of a chemical, so that using concentration as a approximation for activity is the best we can do. I also point out that since any equilibrium calculation that uses concentrations is at best an approximation of the system, that it will reasonably allow us to make other approximations when doing equilibrium calculations that will not compromise the outcomes. Furthermore, I point that from the perspective of an analytical chemist, what we usually care about is whether a reaction goes to completion or not (how large is K) or whether a reaction hardly occurs at all (how small is K), and that many analytical procedures are predicated on using systems that have either exceptionally large or exceptionally small values of K so that we can be assured that we are measuring all of what we desire or that no other substance is interfering in the measurement.
Realizing that we will use concentration as an approximation of concentration, I then ask them to consider whether this approximation is more valid at high or low concentration. I encourage them to talk to their neighbors (they are not yet in assigned groups) and then take a poll of the class. Usually it seems that most students think the approximation will be better at high concentration. I go back to my example of As and Bs that is still on the board and ask them to put in more A species and tell me whether it will lead to more or fewer inactive forms. The students then realize that the concentration is a better approximation of activity at lower values. I also point that those methods we do have for rigorously determining activity also break down at higher concentrations, so that they rarely help us in the cases where we would most benefit by using activity instead of concentration. I also point out that while we could get more accurate answers for systems at low concentrations, the use of concentrations is valid enough that all the extra work to use activities is often not warranted.
With this background, I indicate that throughout the remainder of the course we will always use concentration as an approximation of activity, that with our use of concentration we must always keep in mind that we are obtaining “ballpark” figures for the amounts of species in a solution that is at equilibrium, and that because we are only obtaining ballpark values, it will allow us to use other approximations to simplify many of the calculations.
Occasionally throughout the unit I like to ask them to just generally assess the extent to which the utilization of concentrations as an approximation of activities is valid for the system described in a particular problem.
With this background the students are ready to divide into their groups and start on the first problem on the in-class set. When I have passed out the first set, and asked them to read the first question, I point out that the concentration provided in this (0.155 M) and all subsequent problems refers to the initial concentration of species in solution (rigorously referred to as the Formality of the chemical) rather than the equilibrium concentration – that the is prepared with a certain concentration but then proceeds to equilibrium such that the final concentration will be different than the initial concentration.