stance. These substances are to be taken hy the student on an imaginary trip. As slhe nroeresses throueh the imaeinarv . " . lauds. s h e is confronted with situations that involve imaginary dangers. The student is to escane the danger hv huruine one substance or by mixing twq of thkm. In either case, s h e must specify the amounts to he used. If slhe chooses a solution that is ineffective, then slhe remains in that situation and tries again. In each situation the optimum solution earns the most points. Ineffective solutions, wrong answers and careless errors will cause points to he lost. The program is written in standard BASIC and has approximately 400 statements. The program was originally written for the DEC system PDP-20 where it requires about 20 blocks of memory (2984 8 hit words). The program listing comes with an explanation of how to expand it by adding more lands and situations, if desired. The oroeram as written would also function on the TRS-80(16K). To obtain a program listing, along with a line hy line commentarv and an exnlanation on expanding the program, send your name and address along witha checkor money order for $10 Cmade out to Richard Carman) to R. Carman a t the address ahove.
Non-Linear Least-Squares Optimization of Parameters for First-Order Kinetics on Programmable Calculators Dornenico De Marco, Athos Bellorno, and Alessandro De Robertis Instilute of Analytical Chemistry University of Messina Messina, Italy In first-order or pseudo-first-order kinetics the value of an experimental variable y that is linearly related to concentration is given by y = y-
+ Aye-k'
where y- = limy,
Ay = y o - y - ,
y o = limy
I-m
i -0
and h is the rate constant, and t is time. We describe here a non-linear least-squares program for optimization of k , Ay, and y, on jy, t J experimental data. Two equivalent listings of the program are available from the authors. One is for the Hewlett Packard HP 9825 desk computer equipped with 9821112 Matrix ROM. The other is for the Texas Instruments TI59 Programmable Calculator equipped with 1014976-1 Master Library Module 1. The least-squares sections are identical for hoth machines, but there are differences in data input and in calculation of starting values. On the basis of exoerimental values. the proerams selfdefine y , Ay and k siarting values and kini&ze"the sum of the squares of the differences between calculated and experimental values of y by Newton's method ( 1 3 ) .At the end of each iteration the user must decide whether to stop or continue the fitting procedure. When convergence is reached (h, y - , and Ay the same in successive iterations), the user must transfer program control to a second section where standard deviations in h, y - , and Ay, as well as other measures of goodness of fit are calculated and output. Program listings and directions for use are available from the authors at the address given ahove. 696
Journal of Chemical Education
Commentary on Countercurrent Distribution Program Distribution Curves (Chromatograms) for Many Stages George Vogel Boston College Chestnut Hill, MA 02167 David Holdsworth's note on countercurrent distrihution (14) in Bits and Pieces, 2, invites several comments. First, no
separate expression (and no corresponding, much longer program) is required when the volumes of moving and stationary phases are different. Instead, one simply replaces K , the distribution coefficient, by K', the distrihution ratio KVJV,. (It is clear that, e.g., doubling the volume of the mobile nhase must have the same auantitative effect as doubling K:) Mv second comment is that, althoueh the ex~ressioneiven for calculating the amounts in each fkction isberfect~icorrect. in wactice it cannot be used when the numher of stages (n) kxceeds 69, because 70! already overflows virtually any calculator. Even though Holdsworth's second expression avoids explicit calculatkn of factorials, for n > 69 overflow will occur euentuallv (unless the calculation is stopped .~ nre. niaturel! or #~111lti{~li~~~t1;o1i I.< alternattd with d ~ ~ i . ~ i 10loOin the hinomial. This occurs after 329 transfers (330 tubes). Manual inspection can allow further transfers, though this is slow with the calculator operating in semiautomatic mode. Students heing introduced to countercurrent extraction appreciate the distrihution is hinomial by first choosing K = 1and nlottine distrihution curves re ore sent in^ a few solvent transfkrs. A &aph representing the Heparation of p-hydroxyhenzoic acid and gallic acid in an ether-water system after fifteen transfers is sufficient to show complete separation (14). For such a program using a simple calculator, without a factorial key, the data required for the sixteen plots is displayed
in about 2l12 minutes. The data required to plot the purification of henzylpenicillin in 70 tubes using an HP-67 (or HP-97) instrument, with a factorial key, is displayed or printed in ahout two minutes for each solute. Each point on the graph can he readily checked if necessary. This time factor is reduced in Vogel's continuous plot program, but the mass in each tube, beginning from tuhe 1, must he displayed in turn, the display series cannot easily he initiated a t any selected tuhe, and the mass in a particular tuhe cannot he directly examined or checked if the student makes a graphical error. Consequently the student may not appreciate that the mass in any selected tuhe can he readily calculated by a hinomial expression. Vogel's statistical expression is an interesting one. However, it proved to be inaccurate for the class uses I mentioned in illustrating the principles of countercurrent distrihution. I t has the disadvantage that for most students the calculator hecomes a number-disolavine . " "blackbox." The derivation of the expression programmed, the program, and its application could confuse manv and daunt all hut the bravest teachers and students. I prefer to have the calculator evaluate expressions that students can derive and use themselves. Literature Cited (1) Wuzniak, S., "The AppleII," Hyts.215). 34 (May 19771. (2) Sauitiky, A, and Onlay,M. J. E.,Anal. Chem., 36, 1627 (18641. (a) criffith3.rcter R.. " ~ ~ ~ ~ ~~e /~ ho nri pmu in a schemistry." plenum PW, 1978. (4) Murgieh. J., J. CHEM.EDUC.,SW!. 421 (1877). .I. P.."Qu~ntumChemi~tiv."Academic Reas. New York. 1978. Aooendix 10. 15) LOWP. (6) Fleser. L. F: Williamson. K, I.'"organic Erperimmt.?4th ed., ~ e & , i e r i n g t o n ,
Maaarchu~etts,Experiment 54. , New York, 1969, Chapter 14. (7) Sukal, R. 8 . andRohlT, F.J.. R m m ~ i r yFreeman, (81 Day, R. A . Jr.. and Underwood, A. L.. "Quantitative Analysis: 2nd ed., Prentice-HnU, Englewood Cliffs, New .leraey. 1971. (9) Kerni~han,B. W., and Piauger, P. J., "Software Tools..' Addison-Wesley. Readinp. Masachoaetta, 1978. (10) Munn, R. I.. and Stewart. I. M.. "RATMAC: Kernighan and Plaueor's Structured FORTRAN Prnerammine Laneuaee."TR-675. Camouter Science Center. Universitv M ~ ~c & ~ pa&, ~I ~~1 9 i s~- ~ , 111) Munn. R. J.. and Stewart. d. M.. "RATMAC Primer: TR-804. C o m o u t e ~Science Center. University of Maryland, College Park, 1979. (12) ~ o b i n s ~ A. n . I . . , s c ~ PZ~ O~ ~T . , 119no). ~ ~ 113) Demidavlch, B. 1'. and Mamn. I. A . "Computational Mathematics? MIR Pub..
Computer Series Reprint Volume Available The Division of Chemical Education is pleased to announce the publication of "Iteratiuns: Computing in the Journal Computer Series. of Chemical Education" edited by John W. Moore, the current editor of the JOUHNAL'S The first 16 articles from the Computer Series together with a dozen more full-lengthdescriptions of computer applications from the 1979through April 1981issues of the JOURNALare contained in this volume. It begins with an introduction to amputers, educational applications,and sources of materials. Also included are a thorough treatment of computer-assisted instruction, computer graphics, microcomputers and desktop computers, simulation and data analysis, computers and testing, and information storsee and retrieval. Aoolications of oocket calculators. use of com~utersin ohvsieal and organic chemistrv me comprehensive reference. This paperback is available postpaid for $9.75. Address prepaid orders to Journal of Chemical Education, 238 Kent Road, Springfield,PA 19064.
Volume 58 Number 9
Seotember 1981
697
