Several articles on the use of graphing calculators to solve chemical problems appeared in the May 1999 issue of this Journal, including Henry Donato's paper on solving chemical equilibrium problems (1). Although I was intrigued to learn that it is so easy to obtain chemically reasonable roots to complex polynomial equations using modern graphing calculators, I found myself asking "So what?" Is this really useful for our students? Will they learn important chemistry by using this technique?
Donato teaches his introductory chemistry students to use their graphing calculators to solve for exact roots of polynomial equations that stem from equilibrium constant expressions. He gives three types of examples: gas phase, solubility, and acid-base equilibria. With Keq's of 10-5 and 10-4, standard approximation methods lead to errors of ca. 3 and 16%, respectively. Donato believes that solving for exact polynomial roots allows students to appreciate "the limitations of approximations and hence obtain a better understanding of the ways chemists describe [equilibria]". And further, that "students can feel confident that they have completely specified the solution to the problem.
I would disagree on two counts. First, by solving for the exact root of a polynomial equation, one has in no way "completely specified" what actually happens in an equilibrium chemical system. Since 1966, authors have repeatedly reported that, owing to the effects of pressure, temperature, and ionic strength, one can obtain only the vaguest idea of a salt's actual solubility from the value of its Ksp (2-5). This point was confirmed by a laboratory project in the very same issue of this Journal as Donato's article, which showed that Kf for the FeSCN2+ formation reaction decreased by over 50% as ionic strength increased from 0.05 to 0.3 M (6). Owing to the effects of both non-ideality and competing side reactions, calculations based on known equilibrium constants may give answers that are off by an order of magnitude or more compared with experimental results. It therefore does not seem particularly useful to ask students to solve a polynomial in order to get an "exact" solution that may still be wrong by an order of magnitude.
Second, even if one ignores the above problem and pretends that Keq's accurately describe what happens in real reactions, I'm still not convinced of the usefulness of this technique in the introductory chemistry classroom. As Stephen J. Hawkes has stressed repeatedly (3, 7), when deciding what to teach, we should always ask ourselves "Why should they know that?" If the standard approximation algorithm yields a solution that is correct within 4%, is it really worth the extra effort to teach students to use a graphing calculator to solve a polynomial in order to get the "exact" solution? Especially when the "exact" solution is probably not really accurate anyway? I believe that the time in class would be better spent explaining the chemical intuition that lies behind the approximation method, and in engaging in a more rigorous qualitative discussion of equilibrium systems (3, 5), leaving the solving for exact polynomial roots to math courses.
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