Brownian motion: An undergraduate laboratory experiment - Journal

Mar 1, 1982 - Displacements of Brownian Particles in Terms of Marian von Smoluchowski's Heuristic Model. Hermann Klein , Dietrich Woermann. Journal of...
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Brownian Motion An Undergraduate Laboratory Experiment G. P. Matthews University of Oxford Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3 0 2 Few chemistry students realize that Brownian Motion may be studied quantitatively to yield a reliable estimate of Avogadro's constant. A simple and very successful experiment demonstrating this has just been introduced into the first year Physical Chemistry course at Oxford University. I t was originally developed as a project for students by J. le P. Webb a t Sussex University physics department ( 1 ) ;the Oxford apparatus is essentiallv the same. but the theoretical a ~ o r o a c hhas the value of the experiment in a physiEal chemistry teaching course has been gauged from discussions with undergraduates. Theory After Brown first observed the phenomenon in 1827 (21, there were many years of debate as to the causes of Brownian Motion, and indeed a discussion of some of the blind allies which were followed can be quite informative ( 3 ) .Nearly a century elapsed before Einstein, quite unaware of previous work on the subject, made conclusive predictions of a diffusion effect arising from the random thermal motions of solvent particles (4). The complexities of his original considerations led Einstein to describe a simple derivation in terms of osmotic pressure (4). Smoluchowski derived essentially the same formula as Einstein by a different method, and the formula may also be ohtained by rigorous and highly complex considerations of the vector analysis type (5).However, the derivation of Langevin ( 6 ) is probably the most suitable for undergraduates since it leads most quickly to the Einstein formula, although without yielding the deeper insights of the Einstein and Smoluchowski approaches. The Einstein formula ohtained hy each of the different methods may he written in the form: -

Ax2 =

RTI -3mp-L

where R = gas constant, T = absolute temperature, r = time interval between observations, 7 = viscosity of water at temperature T, r = radius of latex particles, and L = Avogadro's constant.


Journal of Chemical Education

The quantity &? has the following meaning. A particle is observed at time 0 and time 7.Durineu this time interval it has undergone a displacement As, whose projection onto the xaxis is Ax. The same oarticle is observed at later times. alwavs separated by the same time interval r , i.e., at times 2 r , 3 ~ ,.. . and Ax is determined for each interval. These v&es are squared, their mean is computed, and the result is Ax2. The displacements thus ohserved are in no sense the actual path of the particle since it undergoes millions of collisions in different directions between time 0 and r , nor. for the same reason, is A X / Tthe x-component of its velocity. I t can be seen from the Einstein formula that if is measured in a suitable experiment, and T, r , 7 and r are known, then the Avogadro constant L may be obtained. Experimental Apparatus Quantitative measurements of Brownian Motion based on the Einstein formula have been facilitated by the availability of latex particles of an accurately known diameter (f 1%).Their motion in aqueous suspension is observed with a standard microscope fitted with an X40 objective and a widefield XI5 Huygen's eyepiece. Into the eyepiece is mounted a gratieule in the form of an image on reversal film of an original diagram, photographed d a m so that its line spacing is approximately twice the diameter of the particles when viewed through the microscope. A separate, adjustable light source is used, defocussed to give a wide, even illumination without danger of convection effects. On the microseooe staee is mounted as~iritlevel.and

collar to prevent removal of the eyepiece and a shim t o prevent over~zealousracking-dom. The only other item of apparatus required is a "tumbler"-a platform rotating once per minute on which can be mounted a microscope slide, edge on, and a boiling tube of stock solution.


A 2.5-ml bottle of 1.09 pm diameter latex particles suspended in water may be purchased for the experiment.' This is made up into the


DOW Uniform Latex Particles, available in U.K. from Unisclence Ltd.. 8, Jesus Lane. Cambridge, CB5 8BA.

Figure 3. Cumulative Normal probability curve.


Figure 1. Central region of the eyepiece graticule.

Figure 2. Normal probability cutve.

stock solution bv dissolvininn one small droo in a boiline" tube of uurified water. The .- ~ - small ~ ~ bottle o f rusnrnsion must, he krnt in a refrieerator., while the stock solution, made up freshly by staff every fortnight, is kept in the tumbler. The tumbling motion inhibits the sticking of the latex particles to the glass tube and extends the life of the prepared slides from hours to days. The two students take a microscope slide with a cancave depression, fill it with stock solution, slide over a cover slip and seal it. So that they do not have to call upon the services of a demonstrator, clear instructions are given on how to focus the microscope and identify a Brownian particle. The students then practice taking accurate readings on the graticule. If the particle is not touching a line, its position is taken as 0.5. Thus, in Figure 1,which shows t,he central region of the graticule, the co-ordinates of the particle are (-4.0, +7.5). Each student then observes a Brownian particle far 13 min and reports its x and y coordinates at 30~secintervals. Some slight refocussing is often necessary to keep the particle in view. If a particle is lost, the experiment may be continued with another. In all, (25 P)/2 minutes of observation time are required, where P is the number of different particles tracked. This yields 50 values of Ax and Ay, the change in graticule position after a 30-sec interval. Since the r direction is arbitrary, the Ax and Ay values can be combined to give 100 values of Ar in all. The temperature is noted during the readings, and the appropriate viseositv of water interuolated from a table of valuer at 1d e ~ r e ein~




.~~ ~






Statistical A n a l y s i s T h e valnes of Ax from t h e e x p e r i m e n t may simply be sauared a n d averaged t o give x 2 . However. t h e Oxford experiment imposes-two fGrther constraintson t h e results. Firstly, since t h e movement is random, t h e values of Ax


0014 .



2 Ax (graticule units1

Figure 4. Set of undergraduate results. -best-finingstraight line lines.


should follow a normal (Gaussian) distribution, and secondly t h e distribution should be svmmetrical about t h e mean dis-

.- .

probability curve has t h e familiar shape shown in Figure 2, where ri is the standard deviation. T o avoid the wroblem of the infinitely narrow class widths implied by thiscurve, the cumulative probability function may be plotted, as shown in Figure 3. This indicates, for example, t h a t if 100 measurements are made, 84 of them will occur below t h e mean and u p to 1standard deviation above t h e mean (assuming t h a t they follow a normal distribution). Probability graph paper has its ordinates plotted so t h a t this cumulative probability curve becomes a straight line. Figure 4 shows some typical undergraduate results plotted in this way. Since only 100 measurements are made, t h e cumulative number of readinas is onlv s ~ e c i f i e dt o &%. Similarly, as t h e position of the'particle isbnly measured t o t h e nearest half a graticule unit, these readinps . . arc only defined e.g., Chartwell Graph Paper ref. 5571, available in U.K. from agents of H. W. Peel and Co. Ltd., Jeyner Drive, Greenford, Middlesex. Volume


Number 3

March 1982


to *'I4 unit. The students are, therefore, encouraged to plot their points as rectangles illustrating this uncertainty; for example, the hundredth point on the graph is shown as a rectangle of width 3.25 to 3.75 graticule unit and height 99.5 upward. If the data are not so represented, they have a misleading scatter at each end of the graph which results in unduly pessimistic error bounds, and the hundredth point cannot be plotted. Three lines are shown on the graph-a best-fitting straight line and two "bounding" lines to give an estimate of the statistical uncertainty of the result. The fitting of straight lines to the points is equivalent to the fitting of normal distributions to the randomly scattered values of Ax. Each of the lines passes through the point (-%,50), which is the condition that the distributions are svmmetrical about the mean disnlacement of 0. The central point is not (0,501 as might be exdected, because the point at 0 on the abscissa includes the number of readings of Ax which are negative or zero plus the number of displacements up to +'I4 graticnle unit, i.e., the cumulative frequency at this point should represent more than half the total distribution. Any drift of the articles due . systematic . to convection currents or the microscope not being vertical shows up markedly when the points are plotted on the graph, and i t proves impossible to draw a best-fitting straight line through the point (-'/4,50). On the full size graph paper, each standard deviation measures 3.15 cm on the ordinate. The standard deviations of the distributions are, therefore, easily measured by finding the gradients of the lines in cmlgraticnle unit, and multiplying their reciprocals by 3.15 cm. The best-fitting straight line in Figure 4 yields a standard deviation u of 1.38 graticnle units. Calibration with the stage graticule shows that a movement of 1 eyepiece graticnle division corresponds to 3.56 X 10Wm. Therefore, u = 4.91 X 10F6m.u is the root mean square deviation of x from the mean. Thus, since the mean is zero, u = Rearranging the Einstein formula:


L =-- RTr 3a7rr2

and substituting the experimental values and constants in S. I. units:

The bounding lines suggest that the range of statistical uncertainty is 5.5 - 10.0 X loz3,and since this is an order of magnitude greater than uncertainties from other sources, it largely determines the error limits for the experiment as a whole. These data are typical of those obtained by students and are satisfactory results from such a simple and brief experiment. Analysis of Student Project Results The statistical uncertainty of the result could he decreased by increasing either the number of observations made or their precision. However, because the observations are near the limit of clear visibility, either course of action would lead to undue fatigue in the students. Alternatively, one could comhine the results from several experiments. This has not been attempted in the Oxford undergraduate laboratory hecause of the administrative difficulty and the clumsiness of having to correct for different temperatures. However, in the Sussex Universitv from different students were indeed combined, thLove'ral1 temperature variation being only 1.5 K (1).Fortv-three students contributed a total of 458 measurements and a normal curve was fitted by inspection to a ~


Journal of Chemical Education




histogram of the results. The heightihreadth profile of the curve was measured and yielded L = 6.53 X loz3,with f12% error limits largely attributed to the fitting procedure. A chi-square analysis of the goodness of fit of the histogram to this normal curve gives x2(18) = 82. If, however, the results are analyzed using probability graph paper, a better fit is obtained directly (x2(18) = 42), and the bounding lines give a meaningful estimate of the uncertainty limits, which again happen to be f12%.The value of L, however, is not improved, heing 5.25 X loz3. The -13% deviation from the accepted value of 6.02 X loz3is readily explainable in terms of a :12% statistical uncertainty with a 1%error from other sources. For the Oxford experiment, the deviation from the accepted value could always he explained in terms of the statistical uncertainty alone, although because of the small number of observations, the limits for any single set were invariably wider than f 12%. Value of the Experiment .

completion. Many have not seen Brownian motion previously, or have been shown only an unclear glimpse of dancing smoke particles. Their first response is therefore one of enthusiasm at having seen "proper" Brownian motion and the movements which form the basis of the kinetic theory of gases. The experiment also gives an introduction to random processes, of which the chemist is all too often unaware. Frequent misconceptions about them are revealed in such comments as: "We observed one particle which had all negative displacements, another which went the other way all the time, and another which hardly moved at all. Wasn't that wrong?" An inspection of the graph in this case showed immediately that the movements of all three narticles were Darts of the same statistical distribution. Similarly, some students expect to he able to ohserve systematic effects on the random movements before the analysis is carried out: "Our particle didn't seem to move very far in 30 sec. Does i t really make any difference hmv lajr1: ~ U I lmk I Jar i r or'''' ' l h w was t h t , UW;II t.