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RUSSELLH. ad,t6d B ~byn Gambler, Kenyon OHCollege 43022
Solving Equilibria Problems with a Graphing Calculator A Robust Method, Free of Algebra and Calculus David K. Ruch and T. G. Chasteen Sam Houston State University Huntsville, TX 77341
Solving equilibrium problems in chemistry involves solvine mathematical eauations.. freauentlv of an intimidatine nature. Because many of these equations are extremely difficult to solve using purely algebraic methods, other approaches are oRen used. For example, the method of successive approximations (MSA) is frequently introduced in freshman chemistry as a means for solving these equations. Although MSA is a powerful technique, it will not work in some problems ( I ) , and the explanation for why it works requires some calculus background (2).Moreover, the equation being solved usually requires some algebraic manipulations before MSA can be applied. Analternative approach is to solie the problem graphically. This method frees the student from algebraic manipulations and allows the teacher to select from a wider range of chemical equations because the algebraic complexity of the problem is no longer an issue. The graphical method works as follows. To solve an equation of the form Ax) = 0, simply graph the equationy =fix).The intersection point of the curve y = f (x) and the x axis gives the solution to the problem. The procedure described is easy to carry out on a graphing calculator or spreadsheet, and students quickly become proficient. Graphing calculators have the advantage of being portable. so they can be used during in-class exams. They are also becoming increasingly inGpensive--about the brim of a freshman chemistrv text. Tbev are also becomine more user-friendly and moreeommon as their use in prec&ulus and calculus courses increases (3). To illustrate the ease and robustness of this method, consider the K., problem involving the solubility of lead chloride, PbC12, in the presence of a common ion. For instance, what is the molar solubility of lead chloride in a solution of 2.00 x lo-' M sodium chloride? The K, for PbC12is 1.70 x lo6. Thus, we have the following.
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P~CIZ
Initially
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A
+
solid
pb2+ + 0M
2cr
Fiaure 1. The araoh of v = lx1/2x+2.00 x 1 0 ~-)1.70 ~ x lo4. as it apbears on a'graphing calculator screen forthe range of values shown. The xand , vmordinates are - - ~ - ~ ~shown - - --~- those -..d . the . .cursor. ....,which can lrace''aong the curve. The solution to tne eq~ationis the xrntercept ofthe cme. Because the t~dcmar- are 1 x to4 units apart it is c ear tnat tne so.utlon of x 0.0103 is correct to 3 slgniflcant figures. ~
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Finding reasonable range values can be done using trial and error, but students with a little physical insight will quickly get a good graph displaying the solution. Any graphing calculator will do the actual graphing in a matter of seconds. The 'trace" feature on the calculator can then be used to move the cursor along the graph of the function, with the coordinates of the cursor displayed at the bottom of the screen. Most graphing calculators have a "zoom" feature that can be used to zoom in on the intersection point to any desired accuracy. After zooming in on the intersection point, the solution to eq 1is found to be 0.0103. By examining the scale tick marks on the x axis, the student can be sure that this solution is correct to 3 significant figures (see Fig. 1). Another aDDroach to solvine this ~roblemis to neelect the initial N ~ C concentration I in eq 1. This resultsln a s i m ~ l cubic e whose solution is 0.0162. This value is not adeq&e because it has a 57% relative error, but it makes a reasonable first approldmation for the MSA method. Now to solve this problem using MSA, most students would first rewrite eq 1as
Using 0.0162 as a first guess, Table 1shows the suecessive approximations from MSA, which are slowly oscillating away from the true solution ofx = 0.0103. The fault is not with the initial guess: It can be shown that any non-
..
Iteration To solve eq 1 on a graphing calculator, a student would type in the equation I O - ~ )-~1.70
lo6
(2)
and then set reasonable range values for the viewing screen. A184
Journal of Chemical Education
~
Table 1. Results of MSA for solving eq 1. Each Is substituted Into the riaht-hand side a~~roximation of eq 3, and the result Is the next app~oximation. The true solution is 0.0103.
2.00 x 10-'M
at equilibrium solid
= ( z H ?+~2.00
~
Approximation
negative first guess other than the exact solution w i l l lead to successive approximations moving away from the true solution (4). If eq 1is instead rewritten as
then MSA will work, but very slowly. With the first guess of 0.0162, MSA with eq 4 will take over 400 iterations to converge to the correct solution. Moreover, it is unlikely that most students would realize in advance that eq 4 is preferable to eq 3 when using MSA. The graphical approach is also useful with equations of the following form, which arises in monoprotic acid-base equilibrium problems.
One approach is to convert eq 6 to a quadratic equation, but this must be handled with care due to numerical precision considerations in the quadratic formula (5, 6). Another approach is to use MSA, stopping the iterations according to the "5% rule" (7,8) or an ionization ratio (9). If the graphical approach is used, there is no need to wony about numerical precision in the quadratic formula or rules for stopping the MSA iteration process. Another point to consider when solving mathematical equations is the possibility of multiple solutions, some of which may not be physically sensible. How does the student using MSA know-thereis only one physically sensible solution to the equation given? Agraphical representation of the equation will easily conv*e the skeptical student that only one such solution exists because the curve will cross the x axis only once over the range of physically sensible x values. Finally, we are not advocating the eradication of the quadratic formula or MSA as methods for equation solving. The MSAapproachis an important concept, useful for solving mmplex, multivariable problems beyond the scope of the graphical methods presented here. However, depending on the course and student background, the graphical approach provides a simple alternative or supplement to MSA.
Wave Functions for Hydrogen Atomic Orbitals Using Mathcad Dean E. Turner Murray State University Murray, KY 42071
Several notes have appeared recently in this Journal showing how Mathcad can be applied to problems in chemical education (10, 111, including one by Rioux that implemented a Taylor series expansion for radial wave functions for the hydrogen atom (12). A similar application using Theorist by Prescience on the Macintosh has also been described (13). Reading Rioux's paper, I was inspired to use Mathcad to derive exact wave functions for hydrogen and to prepare a doeument that plots them. The user need only enter the desired quantum numbers to see vari-
ous graphs describing orbitals up to 4p, making it useful for classroom demonstrations or students' exploration Times for recalculationand replotting vary from 1-2 min using a 33-MHz 80386-DX computer with no coprocessor. Most of this time is required to prepare a three-dimensional plot of electron density. (Much less time is needed if the radial function alone is desired.) The complete document appears in Figures 1and 2 on the following pages. I can offer my personal testimony to the power and ease of using Mathcad. Although I had not studied quantum mechanics since my undergraduate days about 15 years ago, I was able to develop this document in a single weekend while also taking care of young children. I believe that programs like Mathcad make it possible to reasonably ask a typical undergraduate to perform tasks like this that have heretofore been too difiicult. The program's symbolic processor does calculus and algebra, and its array and graphics tools perform calculations and display the results. Thus, students should be able to apply the principles they have learned even to mathematically complex pmblems.
Interfacing Atomic Absorption Instruments to a PC for Student Laboratories John M. ~okosa' and Keith M. Dery GMI Engineering and Management Institute Flint, MI 48504
Several models of Perkin-Elmer atomic absorption (AA) instruments were designed to be interfaced with the Perkin-Elmer 3600 data station, a computer used with PE instruments since the late 1970's. Because we wanted to use the PE-5000 (both airlacetylene flame and graphite furnace modes) for a new course in environmental chemistry, we also needed a suitable computer interface for the AA, in order to handle several laboratories of 20 plus students. An easy to use program incorporating data collection, analysis, and graphics was the basic need for the students. Unfortunately, the PE-3600 software and Perkin-Elmer's recently introduced Wata Management S o h a r e " for the PC (functionally identical to the 3600 software)were more demanding than required for student use.2 Experimental Examination of the microprocessor documentation lead to the realization that the PE-5000 communicates bidirectionally with an external computer usiw ASCII codes. allowing not only downloading if raw data, but also full Antrol of the instrument. We therefore decided to write a program for two-way communication with the instrument in Basic (Microsoft QuickBasic 4.5) computer lanrmaw. Basic was chosen because it is an easy m&hiie language to understand, program in, and change - by. anyone . familiar with a microcomputer.
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'~uthorto whom correspondence should be addressed. 2Availablefrom Pehin-Elmer Corporation, Norwalk, CT. (Continued on nertpage)
Volume 70 Number 7 July 1993
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