All things bright & beautiful, all creatures great and small. Remember that hymn? We often talk about remembering the little things, well here’s a lesson in remembering the big things as well!
This post started life with an entirely different title. It was initially called, Should I ignore ‘-x’ in my ICE table? I started writing it a LONG time ago, and it had an entirely different thrust at that time. Then, the 2016 AP Chemistry statistics came into the public domain, and the national results for Q6 found their way on to my radar.
In short, 8% of kids failed to write anything meaningful at all for question 6 and they scored 0/4. Another 65% tried to answer the question and they scored 0/4 – yes, that’s 73% of the population scoring 0/4 on this question – staggering! 12% scored 1/4, and another 12% scored 2/4. Finally, 1% scored 3/4, and 1% scored 4/4. From a purely statistical viewpoint, this is a disastrous question.
Having said that, I actually think the question is fair! I know, it’s crazy that I should side with the CB/TDC on a matter such as this, but I think the fault could easily lie with teachers.
There are two separate issues here, one for part (a) and one for part (b).
Let’s split this into two ‘sections’. Part (a) first. I’ll deal with part (a) by circling back to the original thrust of this post and the addressing it’s connection to 2016, 6(a) at the end of the first section.
Part (a)
Should I ignore ‘-x’ in my ICE table? It’s a commonly asked question that has a pretty simple answer for AP chemistry, and that answer is usually ‘yes’.
What do we mean when we say ‘ignore -x’? What we are really asking is, ‘Is ‘x’ small enough to be considered negligible?’ For example, lets take a look at the dissociation of a solution of the weak acid, ethanoic acid, with a concentration 0.100 M of in water.
CH_{3}COOH_{(aq)} | + | H_{2}O_{(l)} | ⇌ | H_{3}O^{+}_{(aq)} | + | CH_{3}COO^{–}_{(aq)} | |
Initial | 0.100 | 0 | 0 | ||||
Change | – x | + x | + x | ||||
Equilibrium | 0.100 – x | 0 + x | 0 + x |
In this case the degree of dissociation of ethanoic acid is extremely small, so subtracting a very tiny number from 0.100 is considered to be approx. equal to 0.100, i.e., 0.100 – x ≈ 0.100. (You may have heard of the ‘5% rule’, but there really no need to apply this approximation in an AP context unless an AP question was written in such a way that insists upon it, and that’s a situation that I cannot imagine ever occurring).
In the calculation of Kas and Kbs for weak acids and bases, the assumption of ‘x’ being negligible is both reasonable and indeed ‘necessary’, since the understanding is that quadratics will not be examined/need to be solved on the AP exam. Leaving ‘x’ in the problem causes the creation of difficult quadratic equation that means time is spent on math, and not on chemistry.
You can also safely make the ‘x is negligible’ assumption when dealing with common ion problems, i.e., that any ion concentration resulting from the dissociation of the ‘insoluble’ salt will be sufficiently small when compared to the ion concentration generated by a completely soluble salt, to make it meaningless. For example, take a look at the dissociation of the essentially ‘insoluble’ zinc hydroxide in water. With a Ksp of approx. 1 x 10^{-17}, making x = 1.4 x 10^{-6}.
Zn(OH)_{2(s)} | ⇌ | Zn^{2+}_{(aq)} | + | 2OH^{–}_{(aq)} | |
Initial | 0 | 0 | |||
Change | + x | + 2x | |||
Equilibrium | 0 + x | 0 + 2x |
So, if this equilibrium is established in a solution with a pH of 12, where the [OH^{–}] = 1 x 10^{-2 }(i.e., is a factor of 10,000 times bigger than that that comes from the zinc hydroxide), then
Zn(OH)_{2(s)} | ⇌ | Zn^{2+}_{(aq)} | + | 2OH^{–}_{(aq)} | |
Initial | 0 | 0 | |||
Change | + x | + 2x | |||
Equilibrium | 0 + x | 0 + 2x + 1 x 10^{-2} |
and the ‘2x’ can be ignored, since it is completely negligible when compared to the 1 x 10^{-2}.
In the both the case of Ka’s/Kb’s and Ksp calculations, I would strongly encourage students to make a very brief note of the fact that they are making such an assumption about the insignificance of x.
There have been AP questions in the past, where the degree of dissociation has been given, i.e., the ‘x’ value was quoted in the question. One such example s 2002B, 1. Under these circumstances, obviously one can (and should) include the ‘x’ value, but it may be that after the consideration of significant figures that one finds that it does not matter if one does include the x value or not.
So what the heck has all of this go to do with 2016, 6(a)? Well, it appears that a TON of students failed to understand that there is another circumstance when x can, and should, be basically ignored. That situation is the precise opposite of what I have written about above, i.e., it is when the equilibrium, rather than being skewed so far to the reactants side that x is infinitesimally small, it is skewed so far to the product side that x is so massive, that we effectively no longer have an equilibrium at all.
The clue was GIVEN in the question in 2016, 6 via the words, “Considering the value of K for the reaction“. The value was 7.7 x 10^{7}, i.e., MASSIVE! This means that effectively, the limiting reactant is reduced to concentration of zero, and the product concentration will be related to it (after consideration of the stoichiometric ratio). Essentially, there’s no equilibrium at ALL.
Lots of teachers will have talked about this, but it may be that tying it to the ‘x being so tiny so ignore it’ mantra, they can make the point more clearly.
Part (b)
Part (b) really has no relationship to the ‘x is big/small’ discussion above, but it offers teachers a really very important lesson – VERY.
I believe that part (b) is easy, IF you apply Q versus K. Prior to the new course beginning I was never a proponent of Q versus K, rather I was (and to some extent still am), a Le Châtelier guy. However, if one has been paying attention to the CB/TDC/CED, it was easy to predict that an ‘obscure’ question like this one was going to come up, and Q versus K would be the simplest answer.
The same thing happened last year when I heard a casual, off-the-cuff remark about microwaves. BOOM, there’s a question that referenced microwaves! Then there was a reference to non-uniform samples being used in PES plots. BOOM, there it is referenced in a MCQ.
The lesson here is simple. If you want to maximize your student’s scores, you’ve got to get a really good handle on what the CB/TDC is thinking whether you agree with it or not. Know thine enemy!
What should a teacher do to prepare if they are not as wired in as you and some others are?
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Of course, another answer is they need to be tuned in as much as humanly possible to the extent that each of their individual circumstances will allow. If they are not, then they’ll miss stuff.
I wonder how many students were confused that “considering the value of K” actually meant “don’t use K at all in calculation.”
It’s impossible to know, but I find it hard to believe that would be the case very often.
1. Requiring analysis of the quadratic equation is fine. We are not talking tensor calculus here! We used the QE all the time when I did chem in the 80s. It IS NORMAL.
My 5th edition Chemical Principles (Masterton, Slowinksi, Stanitski, p. 1981) has the quadratic as the normal procedure for problems requiring it (not simplifiable because of large or small K). In light blue pen on the margin, they say you can use trial and error also. But quadratic is the normal method expected for a LOT of end of chapter homeworks. Is explained a few pages into the first chapter on equilibrium as normal part of ICE problem solving as an example.
2. The applied algebra aspect of chemistry is GOOD. Kids need to handle a hard science like this with a lot of units, equations, conversions, etc. The may not even remember sources of the elements in descriptive chemistry, but they can really use a toughening up on applied algebra (especially engineers and medical types). For that matter if descriptive chemistry is important why the heck did the nuke solubility rules (the core of qualitative analysis!)
3. The MSS text (1981) also clearly discusses the implication of K being very small, large, or intermediate. In terms of how you make an assumption for which side of the equation is at completion (and then X solved for the small side). This explanation is also a few pages into the first chapter on equilibrium. Is it no so in texts nowadays?
N.B. I do think the AP instructions designed to help (perhaps) ended up confusing the students more than if the problem were just stated as a simple, here’s the info, find the unknown. I don’t even think it was intentional. But the whole wordiness, requiring parsing, seems to arise from or relate to the flawed AP emphasis on conceptual understanding.
Regarding quadratics and algebra, I largely agree with you but the oft stated reason behind avoiding quadratics on the exam is this. The time spent manipulating and solving a quadratic is time spent ‘not doing chemistry’. Now, there is an argument that such math is such an integral part of chemistry anyway, that it ought not to matter.
I think a lot of teachers DO discuss the idea of K being large and therefore essentially destroying any practical aspect of ‘equilibrium’, but my point above is that by pairing it directly with the K is small conversation, maybe the point can be made even more clearly. It sounds like MSS does that. BTW, in my limited experience of MSS I HATED that book – HATED it!
As for confusing kids, quite possibly. The AP exam is now wordier that ever, with tons of absolutely unnecessary language tied to many questions. It must be a nightmare for students whose 1st language is not English, but are otherwise excellent students of chemistry. The exam actually disadvantages them now.
It’s probably the student and the teacher more than the book but I learned pretty well from MSS. I loved MSS but it could just be a bias from familiarity. However, I read the chapters and worked the homework problems. In a lot of cases self studying. (Meaning the book was clear enough to teach without a teacher and the problems had an appropriate range of difficulty and matched text well.)
I do remember late 90s as a TA and seeing texts at a university. They used Zumdahl for chem 101 and Oxtaby/Nachtrieb for advanced/accelerated. Zumdahl was very “standard” and very similar to MSS. In contrast ON, assumed some pre-existing knowledge and didn’t teach in the standard order (stoichiometry delayed until after bonding). It was the sort of book that professors liked, but then they already know the topic and are more research oriented so a book like that appeals to them. Even though it is worse pedagogically.
MSS is really good at calculational problems: equilibrium handled in three different chapters (gases, acid-base, solubility product). Same concept practically, but handled 3 times–helps you get good at it. Reaction rates (similar math algorithms to equilibrium excepting the temperature dependant calcs). thermo. Stoich is hit hard at the beginning.
Bonding is OK–no complaints.
Chapter on solutions (colligative properties, solubility rules) is quite good and thoughtfully staged right before the Ksp chapter.
The special topics (quant titrations, coordination compounds, nuclear, baby orgo, etc.) are well done. Just a question of how much of that you get to in a year. I mean there is a chapter on polymers, could probably skip that unless a very fast, advanced class. And maybe skip the baby orgo.
The one area that caused students issues were the two or three chapters on descriptive chemistry (mostly sources of the elements, some industrial processes) mixed into the book. Don’t really seem to fit with all the ICE and stoich and delta G calc chapters. The chapters themselves are pretty enjoyable (learning how to make steel or separate air by fractional refrigeration) and written at appropriate level–easier than Greenwood. But it is a little harder as a student to kind of shift gears and make some flashcards to memorize reactions, to being able to write paragraph answer on the Bessemer process, versus the standard chemistry calculations.
I like the idea that you are applying one thing from a class (QE) and using it in another. Feature, not bug. QE is very common in a lot of follow-on math and science courses. It is such a core equation.* Not some oddity like binomial theorem, but core core. And easier than some math topics like derivatives (touched on briefly in frosh chem thermo explanations).
*Not relevant to AP chem, but a similar example is the second order constant coefficient diff eq (whole damped and underdamped thingie). You just see that damned equation over and over in EE, mechanical controls, quantum mech, etc.
Part of the confusion from their instructions is that it is unclear if the extra information is an added requirement or just helpful explanation.
P.s. Of course for cubics, doing estimations by a few repeated trials is normal. But I remember the cubic problems being pretty rare.
Google search on quadratic equation and chemical equilibrium:
https://www.google.com/search?q=quadratic+equation+equilibrium+problems&sourceid=ie7&rls=com.microsoft:en-US:IE-SearchBox&ie=&oe=
Normal, normal, normal college chemistry. Look at the links to Perdue, Texas A&M, and UC Davis.
Well, the extent to which the new AP exam is failing to ‘match’ college 101 chemistry in several respects, is an ongoing conversation.