Fundamental Theory of Gases, Liquids, Philip Empedocles
Univers~tyof Michigan Ann Arbor, 48104
and Solids by Computer Simulation Use in the introductory course
The substance of the introductory course in chemistry a t many prominent universities has been subjected, during the past decade, to some far reaching changes. In particular, fundamental concepts of atomic science, quantum chemistry, and statistics often precede and structure a phenomenological account of chemistry. Fundamentals are appealing, hoth because they give a more honest account of what we believe to he the truth, and because i t is only possible to cover large amounts of material by first establishing a theoretical framework. The teacher often finds it essential to his purpose to describe the molecular orbital theory or the Boltzmann distribution of energies. However, he is seldom able to give a strictly logical account in an introductory course, both hecause the time available is limited, and because the student cannot he presumed to possess the necessary experience with mathematics. The modern, "fundamental" course thus often leaves the student with a n assurance that the concepts are sound, but no evidence that they proceed from a few well established postulates via a strictly logical, mathematically precise argument. Computer Simulation of Atomic Motions
The author bas adapted the technique of computer simulated, Monte Carlo studies of the motion of atoms in gases, liquids, and solids to the teaching of freshman chemistry, hoth in lectures and in the "laboratory." The theoretical technique was originally introduced by Alder and Wainwright.' They modelled an infinite gas, liquid, or solid as an array of identical cubical volumes, each containing 100 hard-sphere atoms. A computer was instructed to follow the trajectories of the atoms in one cube over a finite time interval, by repetitively solving Newton's second law. Then as any atom passes out of one side of the cube another identical atom (its "ghost") moves in from the opposite side. The number of atoms in the volume remains constant. Alder and Wainwright first studied the increase in entropy of a gas of atoms with a n initially non-equilibrium distribution of velocities, as it relaxes to a Maxwell distribution. Subsequent refinements allowed them to represent atoms by more realistic squarewell potentials, and to make sophisticated, accurate predictions of the structure and dynamics of fluids for model ~ y s t e r n s .Such ~ calculations form the basis for the definition of an "ideal liquid" of hard spheres, for the discussion of the perturbation of this ideal liquid resulting from more realistic atomic forces, for comparison with the results of analytical theories of the liquid state, and even have been used to suggest models of liquid structure and dynamics upon which new theories can be based. Recently several workers have used more realistic two- and threebody potentiah3 Similar techniques meanwhile have been used by theoretical astronomers to study the condensation of matter in space, and by plasma physicists. A computer program of this type has been used by the author in the second semester of the freshman-oriented general chemistry course a t the University of Michigan.4 The program differs from its research counterparts in four ways.
(1) The computer drives a display of the atoms on a cathode-ray tube as they move on a uniformly slowed-down time-scale. The atoms are represented as circles, to which the velocity vectors and other characteristicsof the motion can be attached. (2) Most of the information necessary to set up any demonstration can be in-putted directly into the machine through the screen, by means of a light pen. Setting up can be dane almost instantly. Velocities, positions, and masses can even be changed during the execution. (3) The atoms move in two dimensions only. This was dane for clarity, and eliminates a complicated problem in computer graphics which arises from the need for perspective. The atoms are programmed to interact by a Lennard Jones 6/12 potential, with an adjustable well depth, atomic radius and mass. (4) The demonstration may he run live for small groups of students. For large general chemistry classes the sequence of positions are preserved on film, and may he speeded up or slowed down over a considerable range by the use of the time lapse photography." The aim of this work has been to eliminate some of the abstraction from the principles of statistical mechanics as applied to chemistry, and to remove the need for experience with purely mathematical skills on the part of the student. Indeed the comuuter has lone been used bv scientists to free them from mathematical drudgery, and to afford them more time for imaainative research. To the inexperienced student the application of equations to the study of matter can present an insuperable barrier to understanding. This technique allows the lecturer to effortlessly by-pass the equations, so that their structure and purpose are clear before the student is asked to attend to the formal aspects. The computer provides an extremely versatile microscope with which to view the atoms. The atomic characteristics can he adjusted to represent a wide variety of pure substances and mixtures. The motion can he stopped, and the structure of a solid, liquid, or pa, analyzed a t leisure. The current values of the mathematical variables can be displayed. The motion can he replayed a dozen times, or run backwards. An almost uhlimited range of general chemistry demonstrations can he set up on short notice. A crystal forming, melting, dissolving, or evaporating can be watched. It is possible to study how much energy is lost by a hot molecule on collision with a d
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'Alder, B. J., and Wainwright, T. E., J. Chem. Phys., 31, 459 (1959).(see also referencescited therein). ZAlder,B. J., Young, D. A., and Mark, M. A,, J. Chem. Phys., 56.3013 (1972). 3Hansen, J., and Verlet, L., Phys. Reu.. 184, 151 (1969);Babetic, M. V., and Barker, J.A., Phys. Reu., B2.4169 (1970). T h e program was devised and coded by J. Blinn. SA Beaulieu R 16 animation camera was used for this work. It has a speed range 2-64 f.p.5. For use in scenes with large numbers of atoms the range is extended with a "Pulsar" animator from American General Products, Lnc., with a variable filming rate down to 1 frame/lO s. Flicker on the screen, generated by phase differences between the rate of picture refreshment and the filming rate, may he eliminated by the use of long exposures. The computer is a PDP9. Volume 51, Number 9. September 1974 1 593
Typical Computer Demonstrations for General Chemistry
wall, or demonstrate the molecular origin of gas viscosity, and show whether "persistence of velocities" is liable to influence the nredictions of the kinetic theow. Exneriments can he performed to determine how the characteristics of the atoms involved influence the distribution of velocities in a gas, and so on. The table gives some typical demonstrations and the figure shows the screen in operation. Some Caveats
There are three sources of difficulty in using a demonstration like this. (1) The demonstration is two-dimensional. In qualitative discussions this presents very little problem. However, some statistical results do depend on the dimensionality. Thus, for example, the functional form of the Maxwell distribution of speeds is different in two and in three dimensions. Again in a random Brownian walk the long time probability of returning to the starting point (persistence) is different. Furthermore, the range of solid polymorphs is restricted in two dimensions. (2) The demonstration is classical rather than quantal. It is not entirely clear today whether this difficulty is very important; however, for liquids and gases classical results are widely accepted at temperatures significantly above absolute zero, and for particles heavier than He or Hz.At an elementary level the greatest difficulty comes in the discussion of the quantization of vibration (rotation) in single molecules and in solids, and in the reflection of this quantization in the statement of the principle of equipartition of energy. One should, of course, also be aware of the Gibbs anomaly in the entropy, which is of an essentially quantum mechanical origin. (3) There is a practical limit to the number of atoms which can be followed, (we seldom use more than 60). There are thus liable to be fluctuations in the properties and distributions. For singlephase demonstrations these have not proved troublesome, although it is sometimes necessary to average results over s long time. For two-phase systems, particularly the solid/liquid equilibrium, results are quantitatively unsatisfactory and perhaps misleading. It is generally agreed in the literature that well over ten times as many atoms would be needed for a satisfactory quantitative demonstration of a two-phase system. There are however methods for obtaining quantitative estimates of twophase systems, which involve studying the two phases separately over the conditions in which they would be expected to be in equilibri~m.~ The discussion is not suitable far an elementary class. Students do not seem to find it hard to accept that the instantaneous averages over a small sample are not reliable.
Film Production as a Teaching Medium in General Chemistry The production of films for use in general chemistry is accomplished in a four-stage sequence, each stage being used as a separate teaching medium. 594
1 Journal of Chemical Education
Conservation of energy and momentum. Separation of molecular motion into translation, rotation, and the normal modes of vibration. Why a solid is a regular array. Structure of the liquid and the radial distribution function. Melting. Langevin criterion of melting. Establishment of the Maxwell distribution of speeds. Boltzmann's distribution of energy. Entropy and diffusion. Pressure of a pas. Brownian motion and the random walk. Brownian motion and the Langevin force. Raoult's law. Increase of entropy and the approach to equilibrium Differential cross-section of an atom. Reactive cross-section in an exchange reaction. Reactive cross-section in a fragmentation reaction. The R.R.K.M. theorv of unimolecular reaction rates. i'he accamodation eaeficient. Equipartition of energy. Molecular beams. Surface tension. Fluidity of a liquid. Ideal gas laws and the deviations from ideality. Viscositv and the kinetic theorv. of eases. ~ q u i l i b h u mconstant. Principle of microscopic reversihility. Live Experiments A senarate course is eiven outside the chemistw - deoart. ment for non-science majors about "gases, liquids, and solids." Aporoximatelv half the class time is spent in the computer iiboratory, &here the students acquire hands-on experience with molecular motions. Each week's work culminates with a n experiment on the computer. For example, in a recent examination students measured the reactive cross-section for an atom/diatomic molecule exchange reduction. Sometimes live data is captured on film so that the experiment can be recreated for a larger class in the "laboratory" with a projector and a film loop cassette. For example, students may he asked to determine the corrections to the ideal gas law resulting from increase in pressure, by examining molecular motions in gases with different volumes/atom. Single Concept Film The experiments give some experience in setting up the computer demonstration to clarify a concept in the most effective way. A single concept film running 3-10 rnin is then scripted and shot. The result is transferred to audiovisual tape and the sound track added. A class of 250 students reaularlv breaks down into sections of about 25 students for "recitation," under the direction of a teaching fellow. The films are introduced into recitation as follows: (a) 5 min quiz, (b) film, (dl discussion and review of parts of the film, (d) 5 min quiz. In this way we try to focus on the problems before introducing the film, bring them out for discussion, and finally review whether the exercise has affected the student's understanding. The teaching fellow summarizes the comparative results of the two quizzes, and notes on a log any significant points of difficulty raised in discussion. Evaluation During the course of a week the film is reviewed by ten freshman sections. The quizzes and the script may be changed two or three times to reflect our increased understanding of the difficulties students have in mastering the concepts involved. Also a data sheet is periodically circulated to the students. It includes on the left-hand side summaries of concepts and laws, statements, simple questions upon which the students can test themselves, and
difficult questions analyzed step by step. The students evaluate their own performance on the right-hand side of the data sheet. The right-hand side is detached, a statistical summary is prepared and examined for significant areas of confusion. Rescripting After a week's operations research on the appreciation of a new concept by the class the whole film is re-scripted; additional scenes may be shot and some of the original scenes may he edited out. A permanent copy of the film is produced for inclusion with the lectures in subsequent courses. Pedagogical Advantages ~, A computer program such as the one described here has four clear advantages over hooks and lectures in the mesentation of the mechanical and statistical foundations of chemistry. (1) Time is an explicit observable. The student can see a sequence of events which simply cannot be described. Removing the abstraction often eliminates an insuperable barrier to the novice. (2) The foeus can be placed in the mathematical variables governing the motion, such as the velocities. This allows the student torelate the formulas to the phenomenon. (3) By observing natural phenomena in this detail the student can gain a preliminary grasp of s whole sequence of concepts. A second point which baffles the student of general chemistry seems to be appreciation of the whole structure of what he is learning, rather than of the individual links. Once the structure is clear he
is better mepared to deee the uhvsical laws. models, and the mathematicai summa6 together; ihe purpose ofeach is clear. (4) The fourth advantage is psychological. Each event is unique. Even the lecturer or teachine fellow watchine molecular motion in class freouentlv . , .oerceives new facets to moblems he mav have roneidrrrd in the abstract over a long pmod Thus i r is possible to rrnnsmir to the rtudrnt in a large scale academrc rnrll, something of the novelty, enrhuainsm,and individuality of a seminar. Other Applications with Similar Potential in the Classroom There are many applications of modern theoretical techniques allied to computer graphics technology which could readily he adapted to use in the class room for en masse instruction. Two examples which could be applied almost without change are the computer organized technique for searching for synthetic pathways in organic chemistry, pioneered by E. J. Corey a t Harvard6 and the technique developed by Fromm, e t al., for analyzing hydrodynamic flow.' Acknowledgment This work was assisted by an award for innovative teaching from the College of Literature, Science and the Arts of the University of Michigan. Corey, E. J., and Wipke, W. T., Science, 166,178 (1969). 7Harlow, F. H., and Fromm, J. E., Sci. Amer., p 104 (March 1965).
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