Daniel Hinds, David D. Parrish, a n d Michael A. Wartell Metropolitan State College Denver. Colorado 80204
Modeling of Molecular Velocity Distributions A physical chemistty experiment
An area of ohvsical chemistw for which i t is particularly h a r d to find app;opriate experiments is t h e kinetic theory i f eases. Exoeriments are zenerallv limited to measuring iransport properties of gases w h e r e t h e fundamental physical properties such a s molecular velocity distributions a r e n o t clearly related to the highly averaged macroscopic quantities measured experimentally. Direct measurement of velocity distributions are excluded h v t h e exoense and so~histication of t h e required apparatus. T o ease this orohlem. we have develooed a comoutermodeled kinetic ihevry ex'periment for t h e physical chemistry laboratory. T h e goals for t h e experiment include ~~~~
+
where dn, = number of particles with speeds between c and c dc, c = speed, N = total number of particles in the system, rn = massof one article, k = Baltzman constant, and T = temperature. The average, most probable, and root mean square speeds can be obtained
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1) To illustrate kinetic theory thmugh experimental data in a clear and direct manner 2) To teach elementary computer programming 3) To give experience in handling distribution functions 4) To teach statistical data analvsis 5) To point out the potentials add limitations of models in general
All of the,c rcsulrsnre well known, but less fnmilar are the distrihutim functions and wnespmding speeds for a two-dimmuionalgw. The distribution function is dnC(2d)= N
(6)
ee-mr21zkT dc
(5)
and the average, most probable, and root mean square speeds are
Theory The kinetic theory of gases which is presented in its simplest form in most physical chemistry courses, is based on the following assumptions 1) A gas is composed of a large number of minute particles. 2) The particles move in straight-line paths. 3) All collisions, whether particle-particle or particle-wall are elastic. Using these assumptions and classical mechanics, the empirical pressure-volume relationship, P V = constant, for a gas can be derived. Comparison of this equation with the ideal gas lawallows the temperature dependence of the average molecular kinetic energy to be
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i. n..f-. o r.r d . -.
A further important result is the distribution of particle speeds which is shown to be given by the three-dimensional Maxwell distri-
The computer model reported here is restricted to two dimensions. Not onlv does this restriction reduce the model com~lexitv. , .meed . mlculnt~on,nnd ensp \,iwali,ation of results, but it also allmvs the ;tudent Comcwe fully under-land t h r chdracter ufdirtrihutitm funr~ two and three dimenciom. t i m i 11). wmparmg the p r e d i r t ~ mfor
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Experimental The computer model consists of a two-dimensional box in which are placed twenty-five circular neon atoms. (Box size, number, and type of atom can be varied, but are held constant for each laboratory section.)At the beeinnine of the emeriment (time = 0). all atoms have an initial meed ar5.00 2 10' cm/kc~.(variable within the oroeraml. I~uthave mndom d~rertimqnndpositivns within the t ~ The x iizr of t he I11n is adjusted as a funrtam ot the atomic diameter sv that the atoms occupy 10%of the total area. This allows a reasonable number of collisions per unit time. For neon, the atomic diameter was set a t 2.4 A.This main program has been written for the students in Basic language and runs on a Hewlett-Paekard 9830 mini-com~uterwith 8 ~ m e m o r and y platter. Once the "experiment" is begun, the program allows the atoms to collide according to their initial directions and follows further eollisions between particles and with the walls. The result, after many collisions, is a distribution of speeds which can be compared with the theoretical function. The data provided by the computer program include the following ~~
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~
~
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1) Initial particle posmons and directions 2) I'article positrons, d~rectwns,and sprrd after ten time rtcps
3, I'article poiitims, directions, and speed airrr twenty t:me steps 4) Final particle positions, directions, and speeds after "infinite
time"
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I
Figure 1. Sample particietrajectories.Thecircles indicate the positionsofthe 25 particles after 20 time steps for one sample calculation. The lines indicate the approximatetrajectories followed by the centers of each particle. 670 1 Journal of Chemical Education
The data is presented in tabular form as well as in the pictorial form as shown in Fimre 1. The progr& operates by allowing the particles to move in straight line trajectories for a small time step (1 X 10-l3 sec is a convenient increment). If any particle is found to overlap the edge of the hoxor another particle a t the end of this time, its position and velocity are corrected to those appropriate for the elastic collision. After each
particle is checked, another time step is allowed. About 500 time steps &err generally allowed turnsure complete randomimtionof wloritir\. A ~ypicnlp n i r l e d l haw undergone appruximately d0mllisimr with vthrr pnrtirlcs and 20 colliGnx with the walls.
Analysis of Data and Results First, the student is asked to verify that representative pnniclep a r r i ~ l ~ a nparride-wall d collisions are elartir. Thm 1s accomplished by ramparing tcjtal energy wlli\ion partners before and otipr LC& lision, "ring thcdata in r l , end 12, ahwe. Sectmd, I ha studmt is required 10 n m p a r e qualitatively the distrihution of final weeds n n h the two dmrnriunnl theory. To dl, this, the student writes a computer program which plots a histogram of nurnher of oarticles versus oartide smed. The task not onlv teaches Rnaic pwgmmming but nlro yirlda a gwd to.4 fur compnrrmn of results. T o rumpare the rhecmtiral and ~ c t u a distril,utions, l t h dis~ tributions must he normalized according t o ~~~
~
.
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Jdn,=N and plotted on the same graph. The theoretical normalization can be done as a continuous integral while the "experimental" normalization must be calculated as a discrete sum. For contrast the student is also asked to compare the two-dimensional actual results to the threedimensional theoretical results, so the three-dimensional theoretical result must also he normalized and plotted on the same graph. In general the statistical fluctuations in the experimental histogram derived from the student's 25 speeds are too great to evaluate the agreement between theory and experiment. Therefore the student also compares all the particle speeds accumulated from the total class. Figure 2 illustrates the comparison for the data accumulated in a recent semester. From this extended data set the student can see that the data does more closely fit the two-dimensional curve. During this comparison, the student must realize an important point. The temperatures corresponding to two and three dimensions are different. Since all collisions are elastic, the total kinetic energy of the system cannot change. Therefore, the root mean square speed of the system remains constant a t the initial speed assigned to all the particles. This root mean square speed of 5.00 X lod cmlsec used in the experiment above, when substituted intoeqns. (4) and (81, gives temperatures of 202 K and 303 K for three and two dimensions, respectively. Third, the student is required to compare quantitatively the experimental results with the theoretical predictions. The center of the top of each histogram interval is taken to he a single data point. A second computer program is written by thestudent tocarry out a chi-square goodness-of-fit calculation' comparing the experimental results to bath the two- and three-dimensional theoretical predictions. The calculat~ansare done for both the student's own data and the complete set of class data.
SPEED (macmen)
Figure 2. Experimentaland theoretical speed disbibutions.The slid and dashed CUNBS are U w k e t i c a l oredinionsfatwa srd mree dimensions.. resoectivalv. The histogram gives the results of 375 particle speeds. The symbols indicate the data points used in me statistical "goodness-f-fit" calculations.
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Far the data given in Figure 2, the comparison with the two- and three-dimensional theoretical curves gave reduced chi-square values of 0.52 and 1.80, respectively. If the data were consistent witha theoretical curve and statistical fluctuations accounted for all deviations. a value of 1.0 w u l d be expccrcd. Thus, the experiment agrees with the tuo-dimensional c u n e a* expected, nlthmgh fortuituudly well tthersrsmlva IOD~prvhabilityof ubtaininosuch .+good tit~.Also,the three-dimensional curve may be rejected with greater than 95% confidence. Finally, the student derives, calculates, and compares values for C ,, C,,,, and ,C for the two- and three-dimensional theoretical results and for the experimental data. The computer program with documentation as described here is available upon request. ~
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'See, for example, Bevingon, Philip R., "Data Reduction and Error Analysis for the Physical Sciences," McCraw-Hill, New York, 1969.
Volume 55, Number 10. October 1978 1 671
