Russell H. Batt
Kenyon College Gambier, OH 43022
A Piagetian Learning Cycle for Introductory Chemical Kinetics
I wish to describe a Piagetian learning cycle based on Monte Carlo modeling of several simple reaction mechanisms. The objective of this learning cycle is to help general chemistry students better understand the relationship between the stochastic microscopic and the deterministic macroscopic aspects of chemical reactions. There is growing interest among science educators in the use of the Piagetian learning cycle for teaching abstract scientific concepts and formal operational thought (J-3). Very briefly, a learning cycle organizes the student's learning experience into three phases: Exploration, Concept Invention or Development, and Discovery or Concept Application (4). During the Exploration phase the student investigates experimentally a particular phenomenon, posing questions and making observations largely unguided by any detailed experimental protocol provided by the instructor. In a well-designed learning cycle this free form experimentation will provide the student with concrete experiences which he/ she cannot fully understand utilizing only his/her accustomed concepts and reasoning processes. This stage puts the student into a state of "disequilibration." During the Invention phase the student, aided by discussion with the instructor, begins to develop the concepts and reasoning necessary to under stand the Exploration phase experiences, thus beginning "self-rE)gulation" or "equilibration," the process in which the learner begins to restructure his/her concepts so as to make better sense of new experiences. The Discovery phase involves the application of these newly developed intellectual capa bilities to related phenomena and experiences. Those general chemistry students who have not yet com pleted the transition from concrete operational to formal operational thought usually experience considerable difficulty when expected to think at the high level of abstraction re quired to understand stochastic molecular behavior. Several authors (.5-8) have described ingenious mechanical devices intended to help such students conceptualize molecular pro cesses by providing visual analogs to random molecular be havior in systems undergoing reaction or in dynamic equi librium. However, these devices generally give only semi quantitative results at best and are not amenable to simple mathematical analyses. Other authors have presented sta tistical mechanical (9--JJ) and Monte Carlo techniques (12,13) intended to help students understand the stochastic and deterministic aspects of chemical reactions, but the mathe matical arguments used are beyond the abilities of most general chemistry students. This learning cycle utilizes the particularly simple Monte Carlo method for modeling chemically reacting systems first described by Schaad (14), who represented systems of chemically reacting molecules by large arrays of digits in a computer. Others (15,16) subsequently applied this method manually on a smaller scale, representing individual molecules by digits in 10 X 10 matrix arrays drawn on paper. For ex ample, the irreversible first order process 1 ._ 0 is simulated as foUows. All positions in the array (reaction vessel) are filled initially with ls, representing initiaJ unit concentration. An array position is selected randomly, and if that position con tains a 1 (reactant), the 1 is converted to a O (product) and another array position is selected randomly. If the array po sition selected contains a 0, no reaction event occurs and an� other position is selected. Time is represented by the se quential random selection and testing of array positions, that is, the random selection of ten array positions might represent the passage of one_unit of time. The concentration of reactants or products is represented by the total number of reactant or 634 I Journal of Chemical Education
product digits present in the array. At the end of eaqh time unit the product and/or reactant digits present in the array are counted thus providing time-concentration data for the simulated reaction. For a more detailed description of the procedure, see reference (15). For the particular purposes of this learning cycle, the pedagogic advantages of Schaad's method are that it simulates both microscopic and macroscopic aspects of reacting systems and provides the student with a visual analog to random mo lecular behavior. Also, it is applicable to a wide variety of mechanisms and is amenable to a simple but accurate math ematical analysis which directly illustrates how the stochastic and deterministic behaviors are related. The Learning Cycle1
Since the students are encouraged to play a quite active role in the learning process, the successful implementation of a learning cycle requires more class time than is generally re quired to present the same material via the usual lecture format. The major role of the instructor is to facilitate learning rather than simply to dispense information and the correct answers. This particular learning cycle requires from two, to two and one half, 3-hr laboratory periods, plus two homework assignments. The prior experiences, abilities, interests, and responses of the students will significantly affect how any learning cycle develops in practice and thus the following description is purposely quite general. Additional details and copies of the written materials given to the students can be obtained from the author. Exploration Phase
To begin this phase, the students generate experimental data by using Schaad's method to simulate by hand the pro cess 1 ._ 0. Each pair of students is provided with a 100 posi tion paper simulation board, a list of random numbers in the range 0-99, and a set of directions for doing the simulation such as outlined above. Student interest can be increased by approaching the simulation as a game, played on a game board according to a set of rules. The students are asked as a group to design, execute, and report on investigations of the effects of varying several simulation parameters, for example, the initial number of ls (reactant digits) on the simulation board and the number of positions selected per time unit. This game provides concrete experience with the behavior of the model system. By striking an appropriate balance between direction by the instructor and student-initiated activity, this initial gaming exercise can be completed in about 1 hr. After the students have completed a few manual experi ments, they are giv�n data generated by a computer simula tion of this same process employing a 4096-position simulation board and various values of the simulation parameters. Group discussion of the manual and computer data follows, during which the students are encouraged to compare the manual and computer data, and to search for patterns in the computer data. Then the students are asked to consider how this sim ulation and the resulting data might be described mathema tically. Generally a majority of the students quickly recognize their inability to develop such a mathematical description. They are now in the state· of "disequilibration." They are 1 Information about a network for sharing chemistry learning cycles may be obtained by writing: CHEMCYCLES, c/o Professor Sharon K. Hahs, Dept. of Chemistry, Box 52, Metropolitan State College, 1006 11th Street, Denver, CO 80204.
unable to analyze and understand these new experiences using only their accustomed concepts and reasoning processes. Invention Phase
The objective of this phase is to begin the process of"selfregulation" by guiding the students through a simple probabilistic analysis of the simulation, with attention initially focused on a derivation of the deterministic equation describing the change in the population ls, t:.[1], for any one time unit, t:.t.This analysis initially poses difficulties for many students, but discussion of other statistical phenomena with which they have had experience-coin flipping, for example, leads most to accept, at least tentatively, the following equation: ~[1] = _ (number of positions selected)
~t
~t X (
[I]
total number of positions
)=
- k
[I]
(1)
where [IJis the average number of ls present on the simulation board during time unit t:.t.This first class period concludes with a brief discussion by the instructor of simple graphical methods by which a linear equation such as eqn. (1) can be tested against experimental data . As homework each student is asked to use the experimental data from the computer simulations for testing the accuracy of eqn . (1) and of any alternative equation he/she believes might apply. One of the pedagogic advantages of Schaad's method is the quite good statistical precision obtained with populations of only a few thousand digits, thus giving rigor and credibility to the testing of the theoretical equations . For example , eqn. (1) predicts that a plot of t:.[1]/t:.tagainst [IJwill be linear with slope -k and that the quantity (t:.[1]/t:.t)/[I] will also equal -k. For the case of 4096 array positions and 410 random position selections per t:.t,the theoretical value of k is 0.1. A typical experimental run, beginning with 4096 ls (reactant digits) and extendinK for ten time units, gave a good linear plot of t:.[1]/t:.tversus (1] (with a least squares slope of 0.105 ± 0.004) and experimental values of (~[1]/~t)
[1]
which scatter from the theoretical value by a few percent at most, their average being 0.099. This excellent agreement between experiment and theory gives the students confidence in eqn. (1) and in the reasoning employed in its derivation. Discovery Ph~se
In this phase the students apply what they have learned in the Exploration and Invention phases to two additional processes: 1 + 1 -----0 ·+ 0 and 1 + 2 -----0 + 0. As part of their homework assignment , they are asked to formulate appropriate game rules for these processes, by extension of the rules by which the y played the 1 -----0 game. The second laboratory period begins with a discussion of their homework results and then moves on to develop equations, analogous to eqn. (1) , predicted to describe the macroscopic behavior of the two second order processes. During these discussions some kinetic terminology can be introduced in preparation for what follows, for example, macroscopic, microscopic, mechanism, order, rate, rate constant, and rate equation . Thus far discussion has purposely been in terms of the random conversion of digits, with no explicit reference to molecules or chemical reactions. The intent is that the students discover for themselves that these stochastic games are analogous to chemical reactions , rather than to assume as much a priori as is done in references (9-16) . At this point the 2 These programs are sufficientl y simple as to be implemented readil y on a minicomputer or on most microcomputers after conversio n to the par ticular local BASIC diale ct. Copies of th e list ings, documenta tion and sample outpu t may be obtained from the auth or. Th e au thor will also suppl y anyone who wishes to try thi s learnin g cycle, but who does not have access to a suita ble compu te r, with the necessarv simulation results . Please include a self-addr essed, stamp ed 9 X 12 envelope.
students are encouraged to note , for example, the formal similarity between macroscopic stoichiometric chemical equations and the symbolic descriptions of the digit conversion processes, and that on the microscopic level both processes involve transformations of one or more discrete reactant entitie s (digits, molecules) int o discrete product entities. Discussion then moves to the possibility that these games might be analogous to chemical reactions in several important ways. One is that the equations developed previously might describe the time-concentration behavior of chemical reactions as well as that of the digit reactions. As homework, the students perform computer simulations of the processes 1 + 1 -----0 + 0 and 1 + 2 -----0 + 0 and use the resulting data to test the accuracy of the two rate equations derived earlier. Also they are provided with some real pseudo first-order and second-order kinetic data and are asked to determine if these data also can be described accurately by the previously derived game rate equations. Approximately 1 hr of a third laboratory period is devoted to a summary discussion of the previous work. Terms such as stochastic, deterministic, population half-life, digit mean life, molecularity, and mechanism are explained by reference to the model systems and computer data. The analogy between chemical reactions and the model systems is further developed and implications of the analogy explored, particularly with reference to microscopic and macroscopic interpretations of chemical rate constants. The Computer Slmulatlons2
The Monte Carlo simulations are performed with four short programs , written in BASIC-PLUS , running .on a DEC 11/70 minicomputer. Each molecule is represented by one bit and the reaction vessel by a core array of 256 or 512 sixteen bit words. Each bit is addressed by two random integers , one specifying the array word and the other the bit position in that word. Each bit can be tested for 1 (reactant) or O (product), converted from 1 to O or the reverse by use of the logical operators AND or XOR. For example, if the Nth bit of the Kth word is to be tested, the logical expression W(K) AND 2**N is evaluated using integer mode arithmetic. If the bit is 1, this expression is true; if 0, it is false . Simulation of the process 1 -----0 uses a 4096 bit reaction vessel. Both the number of ls initially present (initial reactant concentration) and the number of bits tested per time unit can be varied within preset limits. The larger the latter quantity, the faster the reaction proceeds, i.e., the smaller the mean life of reactant bits. Simulation of the process 1 + 1 ----0 + 0 uses a 8192 bit reaction vessel and the number of pairs of bits tested per time unit can be varied . Only if both randomly selected bits are 1 does a reaction occur, both then being converted to 0 + 0 is quite similar ex0. Simulation of the process 1 + 2 ----cept that the reactant 1 is represented by bits in one 256-word reaction vessel and the reactant 2 by bits in a second 256-word reaction vessel. The initial number of 1 molecules is fixed at 4096, while the initial number of 2 molecules can be varied between 2048 and 4096. Also, the number of pairs of bits tested per time unit can be changed to vary the rate of the reaction. Literature Cited ( 1) Grot z, Leonard C., J. CHEM. ED UC., 56, 7 (1979). (2) Good, R. , Mell on, E. K., and Kromh out , R. A., J. CHEM. ED UC., 55, 688 (1978). (3) Good, R. , Kromhout , R. A., and Mell on, E. K.,J . CHEM. ED UC., 56,426 ( 1979). Thi s reference is an annotated reso urce paper citing 131 papers dealing with appli cat ions of Piaget 's work to scien ce teac hin g. (4) Ka rplu s, R. , J . Res. Sci. Teaching, 14, 169 (1977). (5) Dave np ort, D. A., J. CHEM. EDU C., 52, 379 (1975). (6) Birk , J . P. , a nd Gunt er, S. K., J . CHEM . ED UC., 54, 557 (1977) . (7) Alden, R. T ., and Schmu ckler, J. S., J. CH EM. EDUC ., 49, 509 ( 1972). (8) Haupt ma nn , S., and Menger , E., J. CHEM . EDUC. 55, 578 (1979). (Thi s paper conta ins many references to earlier work.) (9) Bouche r, E . A., J . CHEM. ED UC., 51, 580 (1974). (10) Sta rzak, M. E., J . CHEM . EDUC., 51, 717 (1974). (11) Dixon , D. D., and Shafer , R. H ., J. CH EM. EDUC ., 50, 648 (1973) a nd see Soltz berg, L. J. , and Waber, F. G., J . CH EM. ED UC., 51, 576 (1974). ( 12) Pa ra, A. F ., and Lazza rini , E., J . CHEM . EDUC., 5 1, 336 (1974). (13) Moebs, W. D., and Haglund , E. A., J . CH EM . ED UC., 53, 506 (1976) . (14) Sc haad , L. J .. J. Am er. Chem. Soc., 85, 3588 ( 1963). (15) Rab inovitch, B., J . CH EM. ED UC., 46, 262 (1969). (16) Mun son, J . W., an d Connors, K. A., Ame r. J . Pharm . Ed uc., 35. (197 1).
Volume 57, Number 9, September 1980 I 635
