Norman C. Craia. -. David D. Sherertz, Terry S. Corlton, and Martin N. Ackermann
Oberlin College Oberlin, Ohio 44074
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Computer Experiments Some principles a n d examples
For several years we have been developing programs for computer experiments for use in undergraduate instruction in chemistry, primarily at the first and second year levels. I n most cases computer programs mith similar computational algorithms have already been described in the literature. We therefore do not attempt to give details of the mathematics or the computer programs for these experiments in this article. Rather, emphasis is placed on some principles of design and on the uses of such experiments, areas which, we believe, have received less attention to dat,e. Of particular interest to us have been computer experiments that extend a student's experience beyond what is feasible or even possible in the laboratory a t a given level. Such experiments fall roughly into two categories. The first category consists of the direct use of mathematical models such as quantum mechanical ones. It is this model manipulation category that has no simple laboratory counterpart. Even where related laboratory experiments are feasible, such as the analysis of the Balmer spectrum of atomic hydrogen as an example of applied quantum mechanics, the impact on t,he student is blunted by the long chain of inference involved. I n contrast, the computer can provide direct and rapid access to the solutions of quantum mechanical models in the form, for example, of electron density distrihutioris for atomic or molecular orbitals. By working directly with the mathematical models the student gains a feeling for the behavior of such systems. Comparable experience usually is not attained until an advanced level when he is in a position to appreciate the equations themselves. The other category of computer experiments consists of the simulation of possible, though impractical, laboratory experiments. Although equivalent experiments could he conducted in the laboratory, they are not pract,ical due to limitations in equipment, special reagents, or the skill of students a t a given level. For exampk, a student may perform an acid-base titration with a pH-meter in a general chemistry laboratory. However, the scope of such an experiment is usually narrow for it is unlikely that the student has sufficient skill to obtain data for an interesting system such as a multiprotic amino acid. The chances are that the instructor will have settled for a simple monoprotic acid. Not only does computer simulation permit t,he introduction of experiments that extend the laboratory experience hut computer simulation also makes it possible to provide the student with a richer experience 310
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in data interpretation and hypothesis making. Because of the constraints on the design of instructional experiments, instructors commonly write experiments in a form that leaves little maneuver mom for the student. I n this situation the emphasis is on techniques and minimizing time-consuming missteps. As a consequence the student often misses the opportunity to do "real science" in the sense of performing an initial, survey type experiment and then choosing refinement steps himself in order to reach desired conclusions. I n a computer simulated experiment, on the other hand, a series of useful experiments can be performed so rapidly that the student can afford to make a number of seemingly unprofitable tries. Emphasis is therefore on the inductive reasoning process rather than on the details of laboratory technique. I n his column in the Scientific American Martin Gardner has recently dralvn attention to new games that highlight inductive reasoning ( I ) . Our use of the ORBITAL program, which is described later in this paper, bears a resemblance to the game of "Patterns." I n order to make possible this emphasis on the inductive reasoning process, the computer programs for the experiments provide for easy and repeated interaction between the student and the computer. Some of the experiments are cast in the format of the problem of identifying an unknown, such as an amino acid from its titration curve. As a first step in these experiments in the unknown fomzat the student receives some computer-generated, preliminary data from the instructor. This step also reduces the extent of computer-fright for students who are using the computer for the first time. After examining the preliminary data the student formulates questions which he then asks of the computer. Responses to these questions and subsequent ones, based on increasingly refined knowledge of the system, lead the student to a convincing explanation of his data and to the identity of his unknown. I n other experiments which are not in the unknown format the student has control of the various parameters and has as his goal the exploration of the range of validity of certain approximations, such as the steady state approximation for the MichaelisMenten enzyme mechanism. Alternatively, his goal may be the generation of sufficient experimental data to allow him to characterize a system, such as a phase diagram constructed from cooling curve data. These are examples of what we call the explorationformat. Experience with student use of the computer experiments has led us to introduce an accounting subroutine called PEAR that keeps track of the number of runs
each student makes. This subroutine denies a student access to the computer program if he exceeds the run limit specified by the instructor. In the absence of this feature some students circumvent the inductive reasoning process by making large numbers of computer runs involving slight variations in the parameters. The accounting procedure disciplines a student to analyze carefully the results of each step in the experiment and to incorporate these results into the planning of the next step. In the next two sections we describe some computer experiments which we have developed or have under development. The computer programs are all written in Fortran IV G. We would be happy to supply listings of the programs and instructions for use upon request. Computer Experiments Involving Mothematical Models
T v o programs have been developed for calculating electron probability density maps. One called ORBITAL is for atomic (hydrogen atom) orbitals; the other called MOLORBIT is for molecular orbitals based on the simple LCAO approximation. We consider ORBITAL first.
Figure 1. Sample output fmm program ORBITAL 3howing electron probobility density in the yr plane of a 4d,' atomic orbital. Contour liner have been sketched in. The plane displayed is perpendicular to the x-axis above the yr plane. (To rove space only the upper left quadrant and 1 of the mop is shown.)
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As shown in Figure 1 the output from ORBITAL is a two-dimensional map of numerical values of the electron probability density ($%)expressed as a decimal fraction of the maximum value (2). Representative orbitals from 2p, to 4fa are coded in the present version of the program. The student user has control of the size of the grid (in Angstroms) and the effective nuclear charge as well as the two coordinate axes to which the displayed plane is parallel and its intersection with the third axis. This program is used for an experiment in the unknoml format. A student is given a code number and is told that it corresponds to one of the orbitalsL from 3s to 5gn. By investigating the electron density distribution of the unknown orbital he is expected to identify it as to quantum numbers. Along with his
code number the student receives the map of the most revealing plane as a starter. The student is encouraged to sketch in contour lines of constant probability density as shown in Figure 1 and, after he has formed an hypothesis about the identity of his orbital, to examine the electron density distribution in other critical planes to confirm his identification or to revise Before attempting this experiment students are made amare of the relationship between quautum numbers and nodal features of atomic orbitals (8). The program RIOLORBIT, which produces electron density maps for molecular orbitals, has an output format similar to that of ORBITAL. However, the length of the grid in the dimension in which the paper is fed through the printer is adjusted to the internuclear distance specified by the student user. In addition to the internuclear distance the user designates the effective nuclear charge of each nucleus, the hybrid mix of 2s and 21, orbitals at each nucleus (or simply a 1s orbital), and the mixing coefficient for the molecular orbital. Thus, the student experimenter can construct density maps for a variety of molecular orbitals at different internuclear distances for both homonuclear and heteronuclear diatomic molecules. We have used this program as an experiment in the exploration format with students constructing series of maps to represent the formation of several molecular orbitals as functions of internuclear distance. Chemical kinetics is another area in which computer experiments with mathematical models have proved to be helpful. An experiment of this type is one based on the simple Michaelis-Menten mechanism for enzyme kinetics
By numerical integration of the appropriat,e set of rate equations program ENZYME calculates the concent,rations of S, E, ES, and P as functions of time. As shown in Figure 2, these results are printed out in graphical form vith provision for scale expansion for small concentrations. The experiment is used in the exploration format. The student's goal is the det,ermination of the range of validity of the steady state approximation by varying the rate constants and the initial concentrations of S and E. To date we have used this program only with physical chemistry students to extend their understanding of the steady state approximation. As background they know from a full mathemat,ical t r e a h e n t the conditions which the rate constants must meet if the kinetic scheme C,
k2
A-B-C
is to be approximated by a steady state. They also know that ES/S = Eo/(K&l S) for the RIichaelisRIenten scheme in the steady state approximation. In
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It is dmirnble that the range of possibilities far orbitals be greater than the number nvnilable in the program. Otherwise, il single density map would be sufficient to identify the orbitals with the highest n and I qnarltom numbers, which are 41 in the present case. Program ORBITAL is also capable of producing eloud4ype electron donsity maps by a technique of overprinting. These mans are osefnl for checkine student resnh and for class discussions. Volume 48, Number 5, May 1971
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ahow i.'zooo CONCENTRATION
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Figure 2. Somple output from program ENZYME l o w i n g coneentrationl of rubstrote, S, enzyme, E, enzyme-rubstrote complex, ESICI, and product, P, as a function of time. So = 2.0 M, Eo = 0.1 M, lRKIFlkl = 2.0 M-I ~ec-', IRKlB1kLl = 1.0 sec-', and (RK2F)kn = 0.1 set-'. M is the number of rubintervdr into whLh each interval is divided for calculation purpo9es. Scole factor for S and P ir 1 ; for E and C, 1 0 . The axes have been redrawn here for clarity. There lobelr are printed b y the computer in the oetval output.
addition to the use of ENZYME in the physical chemistry course we see the possibility of using this experiment in our honors level introductory course. Laboratory-Equivalenl Computer Experiments
A program called TITRATE has been developed for use in an experiment in the unknown format on the titration of a polyprotic acid (4). As an unknown the student receives the code number of an amino acid (in hydrochloride form). He knows that i t is one of eighteen naturally occurring amino acids for which he has the pK.'s and that it is diprotic or triprotic. Along with the code number he receives a table of preliminary titration data generated on the computer. This data consists of pH versus volume of base added at 10-ml intervals and thus can be used to sketch a rough titration curve.3 From the rough curve the experimenter can locate probable endpoint regions for which pH values at small volume increments are needed. In a single run the student can then calculate a series of ten consecutive points, which, if well planned, define an a For the instructor's use TITRATE is capable of producing a graph of the preliminary data on the high speed printer. This feature of the program could be used to provide preliminary graph for student users.
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Journal of Chemical Education
endpoint region. By several such refinement steps he attempts to define all of the important features of the titration curve and thereby determine the pKds. From the observed pK:s the student identifies his amino acid insofar as possible. In some cases a unique solution is not possible. He also calculates the concentration of his acid from the titration curve and the known concentration of the base. A second program of this laboratory-equivalent type is in the development stage. It is intended to simulate a thermal analysis experiment in which cooling curves are used to establish phase diagrams for solid-liquid equilibria. Undergraduate experiments in this area are usually limited to uninteresting systems chosen because they can be performed with simple equipment in the 0-100°C temperature range. Also these experiments are usually restricted to simple eutectic systems because of the tedious and time consuming process of collecting cooling curve data for more complex systems. With a computer-simulated experiment it seems possible to give a student considerable freedom of choice in the system he studies. High temperature systems of alloys or fused salts and low temperature systems with compound formation due to hydrogen bonding or chargetransfer would be accessible. Systems exhibiting solid solution formation might well be included. Some computer experiments fall on the borderline between direct simulation of laboratory experiments and the exploration of mathematical models. One such experiment is a nuclear magnetic resonance program ( 5 , 6 ) , called SPIN3, which permits the student user to explore the relationship between the appearance of an nmr spectrum and the critical parameters for a threespin (At to AI\fX) system. The user controls the chemical shifts, coupling constants, and the radio frequency and obtains an output in the format of a typical nmr spectrum. This program is intended for use as an experiment in the exploration format by students a t the introductory or organic level. It serves as a bridge between the empiricism of correlation rules and the detailed quantum theory they encounter later. Another such program, which is in the planning stage, is concerned with the normal vibration frequencies of a three-atom system. I n this case the student user would explore the range of validity of the group frequency approximation and the effect of geometry and force constants (bond strength) on vibration frequencies, e.g., the sensitivity of the carhonyl stretching frequency to substituents such as fluorine, hydrogen, and methyl. He vould supply force constants, bond lengths, the bond angle, and atomic masses and receive a printergenerated output in the format of 6n infrared spectrum. Computer Environment
Oberlin College has an IBM 360/44 computer with a core of 128K bytes. The system has a high-speed printer and is disc oriented. The programs described in this paper reside in object form on a user disc that is continuously accessible to student users. The programs make no use of tapes. Chemistry students use only a few job control and data cards in each run. For short programs (15-30 sec) of the sort described in this paper, turn-around times are generally 10 min or less in our batch processing environment. Thus, it is quite feasible for students to conduct experiments on the
computer which require several runs. Remote terminals are not available. One should recognize that several of the programs require the high speed printer and thus would not be useful on remote typewriter terminals. Programs have the following lengths in (decimal) bytes: ORBITAL (46,500 K), MOLORBIT (14,000 K), ENZYME (8,500 I
