The initial stages of the animation provide the observer with important information prior to viewing the PES. In particular the main features of molecular potential energy curves for diatomic molecules (using Hz and H F as examples) are reviewed and the coordinate system for plotting the three-dimensional PES is firmly established. The remainder of the animation concentrates on visually exploring the PES for reaction 2 by moving the ohserver's eye over the surface and providing, for example, views down the entrance valley for the reactants and exit valley for the products. T o aid the eye the PES, which is plotted against a dark background, is "sited" within the outline of a cube and is itself plotted in color ranging from hlue for low energy to red for high enerE V . ~The use of color is essential t o the animation: it not onlv enhances the visual impact of the PES hut also gives thk observer a real sense of examinine a solid obiect. When the observer's eye is moved to look down upon the surface, the minimum-energy path is immediately recognizable as a blue band, and the change in shading from light to dark hlue reflects the exothermic nature of the reaction. A articular advantage of the computer animation technique isthat various sections of the PES can he easily (and reversibly) removed. This property is used "to travel up" the reactant entrance valley and so reveal the energy barrier to reaction at the transition state. We believe the present animation represents a very good e x a m ~ l eof the Dower of comouter eraohic . techniaues linked with \;idea rero;dingasan aid in the teaching ofhkhly \,isual chemical conceots. It would also be of ronsiderahle interest to extend the sequence to include molecular reaction dynamics and display trajectories over the energy surface. Other areas of application could be, for example, atomic and molecular orbitals or the shape-selective properties of zeolite catalysts. Upon request, a more detailed description of the computer animationfvideo system used in the work can be supplied along with a videotape (state PAL or NTSC) of the computer animation in VHS format. In order to cover handling, copying, and postage charges, please send a check, payable to the Open University, to MM for f20. Additional information regarding the television program (see footnote 3) in which the animation is used is also available.
Numerical Solutions of Kinetic Equations on a Spreadsheet Douglas A. Coe Montana College of Mineral Science and Technology Butte. MT 59701 The purpose of this article is to illustrate how the EulerCauchy method (8)can be used with a spreadsheet to solve numerically first-order differential equations that frequently occur in chemical kinetics. The spreadsheet will be set up to analyze the following system of coupled elementary reactions: 1
k,
A-B-C k.,
k.~
(1)
For this system two coupled first-order differential equations can be written:
* A linear relationship between the magnitude of the potential energy and the "color wheel" on the computer graphics terminal was used.
496
Journal of Chemical Education
In the Euler-Cauchy numerical solution of these differential equations, the derivatives are written as ratios of finite differences. For example, eq 2 would be expressed as
In this equation [A], and [B], are the concentrations of species A and B a t time t and [A],+D,is the concentration of species A a t time t Dt. Dt is a small finite increment of time that should be chosen so that the relative changes in the concentration of any species during a single time step is small. Equation 4 can he rearranged to allow the concentration of species A a t the advanced time step, t Dt, to he calculated from the concentrations of species A and B a t the previous time step, t:
+
+
A similar equation can be derived for species B, [B]~+D, = [Bit + k,[AItDt
- k - , I B l P - kJBltDt + k-z[CltDt (6)
The time dependence of the concentration of species C could be obtained in the same way, but it is simpler to determine the time dependence of species C in this constantvolume system from a mass balance relation [Clt = [A10 + [Blo + IClo - 14,- [Bit
(7)
Here [Ajo, [BIo, and [CIo are the initial concentrations of species A, B, and C. The spreadsheet used in these calculations was SuperCalc 3. Other spreadsheets, for example, LOTUS-123, would work as well; however, a definite advantage exists for spreadsheets that3can plot X-Y graphs. These calculations were done on a DEC Rainbow 100 microcomputer with 256 K of RAM memory. SuperCalc 3 occupied 199 K of this memory. The kinetics calculations required an additional 15 k of memory. Calculation of the spreadsheet takes less than 10 seconds of computer time. Input for the spreadsheet calculation includes the initial concentrations, the rate constants, and the amount by which the time will he incremented. The choice for the unit of time used in defining the rate constants is arbitrary, provided all rate constants are defined using the same time unit. The amount by which the time will be incremented should have the same time units as the rate constants and he less than or equal to one-tenth the inverse of the largest rate constant. This restriction on the time increment ensures that the Euler-Cauchy method will provide acceptable accuracy for most instructional applications of this spreadsheet, For short reaction times acceptable accuracy may require choosing an even smaller time increment. A portion of the spreadsheet used to analvze this svstem of counled elementarv retlctiuns and containing these input parameters is shourn in Firurt I . In this examole. onlv svecies .4 is nrrsent initiallv. and its initial concentiation is aihitrarily set a t 1.0000 M cell F8. The particular set of rate constants used in this example are contained in cells F12 throughF15. The amount by which the time is incremented, 0.100, is contained in cell F20 and is one-tenth of the inverse of k-1 = 1.000, which is the largest rate constant. The actual spreadsheet calculations are contained in the rectangular block of cells defined by columns A through D and rows 25 through 75. For the example just described the concentrations of the species A, B, and C are plotted versus time in Figure 2. In this particular example the rate constant k-1 = 1.000 was chosen to he substantially larger than the rate constant k2 = 0.0500, illustrating the rate-determining step approximation with the second step rate determining. How well does the Euler-Cauchy numerical method model the actual kinetics? T o answer this question let k , = 1.000,
i
A
I
I
B
~
I
C
I
I
D
I
I
E
I
I
I IF
TiliS SPREAOSHEET CALCULATES THE CONCENTRATIONS OF CHEMICAL SPECIES A , 8 . AN0 C AS A FUNCTION OF TlME FOR l H E SYSTEM:
[Alo
INVUT ME INITIAL CONCENTRATIONS:
= 1 = 0 LC10 = 0
[BID
D . O00
INPUT THE RATE CONSThNTS: (niESE RATE CONSTANTS SHOULO BE EXPRESSED I N THE SAME UNITS OF TIME) AMOUNT BY WHICH THE TIME WILL BE INCREMENTED: ( I N i i i E SAME UNITS AS THE RATE AS A RULE OF THUHB O f SHOULD BE LESS THAN OR EQUAL TO ONE-TENTH OF THE IHYERSE OF THE LARGEST
CONSTANTS.
RATE CONITANT. 1
Ot
=
,100
h., = 10.000. and hLr = h-? = 0. The snreadsheet is now modeling a pair of coupled, irreversible first-order reactions. As before.. onlv. snecies A is nresent initiallv a t a concentra. tion ot 1.000 M. 'l'he annlytic olutions of the rate equations 191 can also br calculated on the spreadsheet (these calvulae plots of the tions are not shown in Fig. 1).~ i ~ u 3r shows concentration of the intermediate species B versus time for !loth the analytic 3ol11tion.indfor 111; Euler-Cauchg numerical solution for different choices of the time increment. Time increments of 0.250 and 0.150, which are 25% and 15%, respectively, of the inverse of hl, do not model the analytic solution very well. As the time increment decreases the nu. mrrical solution approaches the analytic a~lution.When the time increment is U.100 or 10ac of the in\,erse of the lareest rate constant, the numerical solution represents the analytic solution fairly well. Once the spreadsheet is set up, the effects of other combinations of rate constants are easily explored. With the spreadsheet, students can "discover" the particular conditions that lead to the rate-determinine-step. auoroximation. the steady state approximation, or a number oibther kinetic situations that can arise with the model described here. I have used this approach in a second-semester physical chemistry course. Students in this course were asked to develop t h e required spreadsheet and then to use this spreadsheet to demonstrate that, if the second step in a twostep mechanism was rate determining, then this implies that the first step in the mechanism for the reverse reaction is rate determining. At the freshman level the already developed spreadsheet could be provided on a disk to allow students to "explore" some simple kinetics. ~~
.. . .. .. .. .
Figure 1. A portion of the spreadsheet that uses the Euler-Cauchy method to solve numerically a system of linear differential equations describing lhe kinetics of the pair of coupled elementary reaclions represented by eqs 1 and 2.
~
~~
~
Huckel MO Theory and Electron Spin Resonance in the Spectroscopy Course Ronald D. McKelvey University of Wisconsin-La Crosse La Crosse, WI 54601
TlME Figure 2.A plot ofthe concentrations of the species A, B, and Cversus time for the coupled elementary reactions described by eqs 1 and 2.
Figure 3. Plots of the concenhation of me intermediate species B versus time for a pair of coupled ineversible first-order reactions for the analytic solution (-) and for the Euler-Cauchy numerical solutions for time increments of 0.100(. . . .),0.150(---),and0.250(-.-.).
In the last 20 years organic textbooks have nearly doubled in size, but most schools still teach the course in the same format of three lectures a week for twosemesters. One wav to deal with the ever-increasing amount of material is to defer some of these t o ~ i c sto more advanced courses. This aDproach makes a l i t of sense, since many of these topics s e e k more appropriate for advanced students, especially those who are going on to graduate school in chemistry. The number of advanced organic electives that can be offered, however, is limited. In this paper I will describe the integration of Huckel MO theory into our spectroscopy class. In the process, a unit on electron spin resonance was included. The course includes both lecture and lab. Where equipment is not available computer simulations are used.sTiward the end of the course, Huckel MO theory is discussed, and students are taueht to calculate small svstems bv hand. including the factoring of the secular determinant using svmmetrv. At this point thev can solve the 6 X 6 for benzene. only then are the; shown Low to do the calculations on the computer. The MO energies are used to introduce UV-VIS spectroscopy. For many, this is their first exposure to antihonding orbitals. Although excellent match of theory with absorption spectra is not possible, students can see the correlation between conjugation and absorption maxima. Some of these include: Proton NMR simulation, "RACCOON" (runs on IBM-PC), by P. Schatz, available from Project SERAPHIM; "Principles of FT-NMR" (Apple IN+), available from John W. Blunt. Department of Chemistry. Unlv. of Canterbury, Christchurch 1, New Zealand: Dynamic exchange, "TWOSITE" (Apple II+), by Richard A. Newmark, available through Project SERAPHIM. Volume 64
Number 6 June 1987
497
