sconic shanes in the context of pattern-forming phenomena of chemical interest. The ~ o s s i b i l iof t ~runningthe program under different concentration and sticking conditions, apart from favoring the discussion on the cen&al role played by concentration and potential on electrochemical processes, facilitates the c o n t k t of such students with r e d experiments on electrodeposition recently performed (9).In addition, a diversity of aspects that can be of interest t o a wide range of chemistry students may be examined under the spirIt of such simulations. F r o m a formal or mathematical point of view the question itself of the definition of fractaiitv. its ouantitative determination in terms of fractal dim;nsioni, or even the practical implementation of the different nrocedures to evaluate those dimensions, depending on thlgeometry and characteristics of the aggregate, are worth examining carefullv. Apart from that, other questions with more phys&al c h e m & j incidence, like ~rownian-motion-simulated diffusion, electrochemical processes on electrode surfaces, or even more generically diffusion-controlled reactions taking place on surfaces or lines could he approached in thecontext of such simulations. Acknowledgment
We thank M. Vilarrasa for helping us in preparing some preliminary versions of the simulation routines, and F. Mas for a critical reading of the manuscript.
Figure 1. Number of m e r s on each aquare (random placement).
Literature Cited L Mandelbwl. H H I b r F m d o l G e o m e r r y o / N o t u n : P r r c m a n San Frdncluo. 1882. 2. Witrcn T A In c h o m p ond bforrrr. Prcc c d the K A n J AS1 Lcs Hovrhes Seeawn xI.vI. 19bG. 5twlet.e. .I.; Vann~mcnus.J Stom. K FA- Pion1~-HollandAmaur-
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dam. 1987. Budemki. E. B.;Dapie, A. R In C c m p ~ e h h h i iTm~fi*i* i i*/Ele~L~hhmi*tty; CCCCCY, B.E.;Bochia, J.O'M.'Yeager,E.;Khan,$. U.M.; Whi&,R.E,Eds.;Plenum:New York, 1983: Vol. 7, Chapter 7, Pa* A end B. Coafa, J. M. Fvndomntoa de Ehcfrodico;Alhambra: Madrid, 1981.Southampton EieNochemiatry Gmup.lhtrumpntol Method8 inEleclmchemi.lry;Chiehe8ter, 1985. Wit&n,T.A.;Sander.L.M.Phya.R~~.L~tt. 1981.47.14W-1403:Phy8.R~~.B 1985.27, 56865697. Vass. R F.; Tomkieviee, M . J. Electrockm. Soe. 1985,132,371-375. Ju1lien.R. Comments Cond. Mot. Phya. 1987.23,177-205. Meskin. P. Phya.Re". A 1988.27.26162623: Phya. ROY. B 1984.30.4207-4214. Meakin. P. Phys.Reu. A 1983.27.60PM17: J. Chom. Phys. 1984.80.2115-2122. Mstsushita, M.; Ssno, M.; Heysbwa, Y.: Honjo, H.; Sawads. Y. Phys. Rsu. Lett. 1984,
53,286:Sawads.Y.:Dougherty,A.:Golluh.J.P.Phys.Reu.Lelt.1986,56.12M),GRer, D.; Ben-Jamb, E.; Clarke, R: Sander, L. M. Phya. Re". Lstf. 1986.56 1264:Arwd. F.: Ameodo, A,: Grasseau, G.; Sluinney, H.L. Phys. Re". Lett. 1988.61.2558.
Counters on Grids Ben Sellnger and Ralph Sutherland
The Australian National University GPO BOX4 Canberra, ACT.
Ausbalia 2601 Statistical thermodynamics is often illustrated with the use of colored balls1, cards, or dice games2. We have chosen the latter approach with an example generally used to illustrate the distribution of energy (as well as molecules) as in an Einstein solid and exploited i t further and linked the results to more familiar experiences. We have called our game Counters on Grids3. You proceed as follows: Set u p a grid, say 6 X 6, and on i t place any number of counters, say 108 (to give a ratio of 3 counters per square). You can place them on squares a t random or on squares evenly, or all the counters on one square. Alternatively you can choose your own unique scheme. You then plot out a histogram of the number of squares with0,1,2, etc., counters versus the number of counters 0,1,2, etc. For placement of the counters on squares a t random, one set of results is seen in Figure 1. Now you use a pair of dice (6-sided for a 6 X 6 grid, or 4-, 8-, 10-sided etc. for other-sized grids), to specify a particular square on the grid, like in a city map reference. One die 508
Journal of Chemical Education
Figure 2. Numbers of squares with 0. 1.2,
. . . countem.
defmes the horizontal coordinate and the other the vertical coordinate. On the first throw of the dice, you pick up a counter, if there is one, from the square selected by the throw. If you hit a blank square you throw again. On the next throw you place the counter down again, onto the square selected by the throw. You repeat this hundreds of thousands of times and note the changing shape of the histogram along the way. We programmed the game for the Macintosh4, but i t can he run on the tiniest of computers. What happens is very interesting. No matter how you start, you end un with the same result. a fluctuatine. -. ao~roximatelv .. exponentially falling distribution. Let us start with 108 counters placed randomlv on a 6 X 6 arid. You usuallv obtain a histogram with a humpjn the middie, Figure 2 (approaches normal for a high ratio of counters to squares). The random experiment of dice throwing (say 1000 times),
' Pwter. George. "The Laws of Disorder"; BBC Films.
School Sci. Rev. 1976. 57. 854. Henderson. C. N.; selkge;. 6. K. "Microprocessors in Chemir try". in Bringing Computers info College and University Teaching; HERSA, Teniary Education Centre UNSW: 1981; pp 33-42; paper and videotape. 'Program on disk available from the aurhors for $25 lncl postage.
