Brownian Motion - A Laboratory Experiment

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    Phys

    Educ 23

    119881 Prlnted In

    the

    UK

    Brownian motion a laboratory

    exper iment

    Haym K rug lak

    Robe rt Brow n, a Bri t ish botanist , reported in 1828

    that plant pollen particles n water viewed through a

    microscope were in constant , hel ter-skel ter motion.

    The pheno meno n is known as Brownian mot ion and

    can be observ ed with various microscopic particles

    in fluids.

    In 1877, Delsaux.aFrenchphysicist ,suggested

    that the Brow nian motion was caused by the bom -

    bardment of particles by the molecules of the fluid.

    However, this was only a quali tat ive explanation at a

    t imewhensomescientistshad eservat ionsabout

    the a tomic-molecular theory.

    Between 1905 and 1908, AEinstein 1956)de-

    velopedanelegantandcomprehensive heory of

    Brownian mot ion.

    J

    Perrin. a French physicist, con -

    firmed by e xperim ent the formu las derived. One of

    these was for calculat ing Avog adro's number

    N

    R T t i 3 q r s '

    where

    N

    is A vogad ro's num ber, R is the universal

    gas con stan t , T is the Kelvin tempe rature of

    HzO

    is a con stan t, obs erva tion perio d, is the viscosity of

    H 2 0 ,

    is the radius

    of

    part icle and

    S

    is the root mean

    square displacement . The theory a l so predic ts that

    Haym Krug lak is Professor Emeritus Western

    Michegan University where he was from

    954-

    78

    Professor of Physics. He is also their

    consultant on lecture demonstrations andas

    been a Visit ing Scholar at the Universityof

    Arizona since 1982. A Wisconsin graduate he

    obtained hisPhD from the Univers ity of

    Minnesota andhas been Visiting Assistant

    Professor of Physics atPrinceton University and

    Assistant Professor of Physics Astronomy and

    Natural Science at the University of Minnesota.

    He has published widely in the fields of the

    teaching of physics mathematics astronomy

    and general science.

    0031-9120 88 050306 04502 5C

    Q 1988 OP Puu l6h lng

    Ltd

    I

    M

    Figure

    1 Schematic of the experimental set-up. C, black

    and white television camera;M. monitor. with 48cm

    (diagonal) screen;

    L.

    collimated light source:

    F.

    heat

    filter;

    S. microscope

    s2 =2 Dt . wh ere D is the diffusion coefficient.This

    equation is called the Einstein-Smoluchowski Diffu-

    sion Law (Atkins 1982).

    The aboratory project on Brownian motion de-

    veloped by J le

    P

    Webb (1980) st imulatedme o

    devise an experimen t which could be com pleted in a

    three-hourperiod. Theexp erim ent was tried with

    studen ts in a third-sem ester physics co urse for ma-

    jor s in physics and engine ering at W estern Michigan

    Univers it y (W M U) .

    Apparatus

    The main components

    of

    the experimental arrange-

    mentaresho wn in figure 1 . The atexspheres in

    dilute queoususpen sion re rojected with a

    microscope and television came ra onto a mon itor .

    With heeyepiece emoved,a

    20X

    objectivepro-

    vides adequate magnificat ion. The

    TV

    cam era lens is

    also emovedand eplace d with

    a

    IO-1Scm tube

    which helps to align the cam era an d to exclud e som e

    of thestray elections.Aheat filter betwe en he

    i l luminatorand hemicroscopemirror is used t o

    maintain a constant suspension temp erature.

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    The cov er glass for the depression sl ides is sealed

    withnail polish.Latex pheres

    of

    0.93,1.10and

    1 .35ymdiamete rwe re ested with good esul ts .

    Wh en not in use the slides were rotated slowly in a

    special rotator as suggested by Web b. With a 60cm

    distance between the bottom

    of

    the

    TV

    camera and

    themicroscopestage, he magediame ters of the

    1.1 pm spheres on the monitor screen were on the

    ord er of 1c m .

    Experimental procedure

    As is com mo n in the USA , students work in pairs

    (occasionally in groups of three).Withplastic

    t ransparency taped

    o

    the moni tor screen, one f the

    partners gives the imesignal .Theotherpar tner

    mark s with a fine felt ippen hecentre of the

    sphere’s posi t ions and abels hem 1, 2 , 3 . . . If the

    sphere drif ts off th e scree n, ano ther is selected and

    fol lowed.Wh en otal of

    25 displacements

    are

    obtained, he ransparency is remo ved.Then he

    second par tner repeats the procedure . The tempera-

    tur e is mea sured by placing the bulb

    of

    a t he rmo-

    meter betwe en he op of theslidecoverand he

    microscope object ive. Each transparency is label led

    with the student’s name, date, sphere diameter and

    temperature . The displacement cal ibra t ion is deter -

    mine d by replacing he lide with a ransmission

    grat ing and measu ring the spacing between i ts l ines

    on the m oni tor screen.

    The transparen cy with the recorded sphere posi-

    t ions is taped onto a sheet

    f

    graph paper and .using

    an arbitrary origin, the

    x

    and

    y

    posi t ion coordinates

    read off andrecorded.Alternat ively,

    if

    acopying

    machine is avai lable, the t ransparency record can be

    t ransferred onto graph paper and copies made for

    the par tners and the lab ins t ructor . This as don e a t

    W M U .The wo ranspare ncies yield 50 displace-

    ments in each

    of

    the two direct ions. Thus, 100 data

    pointsareavailable oranalysis.Tim e is usually

    adequ ate in a three-hour laboratory per iod to repeat

    the observat ions with

    a

    different time interval.

    A hand calculato r with a basic statistics program

    was made available

    i n

    the laboratory. Each s tudent

    computed he mean, var iance and s tandard devia-

    t ion for the displacements before leaving the labora-

    tory.

    I t

    is possible for a single p erson o perform he

    exper imentprovided n udible , utomat ic ime

    signal is availab le. The circuit for such a timer was

    designed by

    R

    Hiltbrand and is given in the Appen-

    d ix

    I t

    was used by all the stu de nts

    a t

    W M U .

    Data analysis and results

    Seven WM U students per formed the exper iment a t

    15 and 30s intervals. Three students used somewh at

    different imesbecause

    of

    anaccidental misalign-

    ment of the t imer cont rol knob. A ‘ randomwalk’ of

    a phereat 15 s intervalsplotted by on e of the

    observers is shown in figure 2. A histogram of the

    data by this student and his partner ( team no 1) is

    displayed in figure

    3,

    which also shows a Gaussian

    curveittedoheidpoin ts of theroups

    (Snede cor and Coch ran 198 0). A lot on probabil i ty

    graphpaper wasused

    to

    test hegoo dne ss of fit.

    The gra ph (figure 4) sho ws that with only

    100

    da ta

    points hedistribution

    of

    displacements is agood

    approximation

    to

    a normal or Gaussian distr ibution.

    The probabil i ty graph serves another useful pur-

    pose: checking the calculations for the mean and the

    standard deviation of the displace men ts. The mea n

    can be read

    off

    at the 50th percenti le: two standard

    deviat ions are ncluded between he 16th and 84th

    percentiles.

    Themean nd tandarddeviat ion

    for

    t he

    1 0 0

    monitorscreendisplacementsat

    1 5 s

    intervals or

    Figure

    2

    A

    rando m walk of a l 10

    pm

    latex sphere at 15

    t ime intervals

    I

    Figure

    Histogram of 100 apparent displacem ents with a

    fitted Gaussian curve

    I

    l l

    4

    2 0 2

    Apparent dtsplacement / c m -

    307

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    9

     

    99. 0

    -

    -

    95

    -

    t

    80

    6 0 -

    4 0 -

    E

    20-

    z

    =

    5

    x

    U

    W

    +

    -

    -

    1 -

    0 2

    -

    0 5

    -

    -

    0 .011 ' ' ' ' ' '

    - 4 2 0

    2 4

    Apparentdlsplacernent l

    c m -

    Figure

    4 A

    plot of the data

    i n

    figure

    3

    on probability

    graph paper

    Figure A computer-generated histogram and Gaussian

    of

    350

    displacements at 15 S intervals observe d by seven

    students

    Frequency

    H s togr im

    44

    , , j , , I , , ~ , , , , ,

    I

    I

    e

    DELXY15

    Figure A computer readout of

    a

    normal probability plot

    for the datao f figure 5

    ~ .'?,

    P

    1

    ?

    Table 1 Summary of an experiment on B rownian motion.

    Western Michigan University, autu mn

    1986

    Students 1 0 15-1X.6 500

    6.3

    O

    3 G 3 5 . 4

    250 7 . 1

    Faculty

    4

    14,6

    150

    6 0

    the wo tudents

    of

    t eam

    no 1

    were1.23cmand

    0.065

    cm respectively.Thescreencalibration was

    2 .77( rmcm- ' .Wi th= 2 9 7 K andhephere

    diameter of

    1.

    l o p , th e c alc ula te d v alu e

    of

    Avo-

    gadro's number was N = 5 . 5 x 10'hkgmol".

    The two par tners and another team recorded 100

    displacements

    on

    two ransparenciesat 30s inter-

    vals. Theresult ingstandarddeviat ion was 1.68cm

    a n dN = 6 . 3 x 1 O 2 " k g m o l - ' .The a t io of thestan-

    darddeviations

    for

    the

    30

    and

    1 5 s

    displacements,

    S3,,/SI5=

    .68/1.23= 1.37. This experimental rat io i s

    in goodagreem ent with thepredicted 1.41 value

    from the Einstein-Smoluchowski law.

    O n e

    of

    the studen ts with a lot

    of

    computer exper i -

    ence used a eni thmicrocomputernd

    S I AT

    G R A P H I C S software?

    to

    make hecalculat ionsand

    plot the graph s

    of

    this data shown

    i n

    figure 5 and 6 .

    The value of N was 5 . 4 ~ kgmol". Then seven

    of the students also recorded 150 displacements at

    30 s. obtaining a value of

    N = 6 . 9 ~

    02hkgmol" .

    Two faculty mem bers and he aborato ry super-

    visor each plot ted a random walk unde r he same

    conditions as the students. Their v alues

    of

    N for the

    15

    interval were 5.7,

    6.5

    an d 5 . 7 ~ kgmol".

    for

    anaverage of

    6 . 0 X IO'

    kg

    mol- .

    Was hisa

    matter

    of

    greater observat ional experience or luck?

    Probablyboth Theresults

    of

    all theobservat ions

    are summarised in table I

    Error analysis

    Th e largest uncertainty is associated with th e tar get-

    ing

    of

    the sphere on the mo ni tor screen:

    * 1

    mm at

    best or approx imate ly0 .3pm

    i n

    theslidesuspen-

    sion. Thus he uncertainty in the mean square dis-

    placement for the

    15

    S time interval could be as high

    as

    15%

    and for the 30 s interval .

    1 0 .

    The d i amete r

    o f

    the sphere used

    i n o u r

    experiment was given by

    the manufacturer o 1 . The emperature has he

    least uncerta intly, perhap s

    0 . 3 ' .

    Thus. the overal l

    uncertainty i n the Av ogadro n umber i s o n the order

    -I- Statistical Graphics Corporation, S ? A I - I ~ K A P ~ ~ I

    Software Publishing Corporation.21 IS East Jefferwn St .

    Rockville. M D

    20x52.

    USA.

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    of

    5%

    for the

    30s

    in tervals and

    f 2 0 %

    for the

    15

    measurements.Theexperimentalvalues of

    N

    fall

    within these limits.

    Summary and conclusions

    The availabi l ityof latex microspheres. compact tele-

    vision camerasandelectroniccalculatorsmake i t

    possible toperform n xper iment

    on

    Brownian

    movement in one laboratory per iod. A more accu-

    ratealue of

    N

    caneetermined by othe r

    methods . However , the exper iment descr ibed above

    has several valuable pedagogical outcomes. Under-

    graduates tudentsgetexp erien ce with several ex-

    perimentalechniques:i)ecordingrandom

    walk' of a micro sphere ; (ii) plotting a histogram of

    displace men ts; (iii) itting a Gau ssian curve o he

    histogra m: (iv) check ing the goo dness of fit analyti-

    cally

    or

    with probability graph paper; (v) calibrating

    screen displace men ts with a diffraction grating; (vi)

    calculat ingvogadro'sumberromhex-

    per ime ntal data : (vii) verifying data validity w ith the

    Einstein-Smoluchow ski Law.

    The exper iment a l so provides valuable pract icen

    unit conversion and error analysis. Another instruc-

    t ive feature: he xperimentmakes he tudents

    aware of Einstein's work other than relativi ty. The

    students ' react ions to the experiment w ere posi t ive:

    'interesting', challenging'. fun'.

    Acknowledgments

    I would ike o xpress my appreciat ion or he

    encouragement and he lp

    of

    the fol lowing: D Dona-

    hue. D Kangas . J Kessler . W Merrow. L Oppliger

    and H Tarkow.

    Appendix

    A

    compact, variable imer-beeper

    In some laboratory exercises a t imerwith an audible

    signal is useful.

    For

    exam ple , in the exper iment

    on

    Brownian motion described above, a single student

    can pinpoint the moving sphere at the beep of the

    t imer preset for the desired t ime interval . The t imer

    is also useful in such experiments as the iming of

    radioact ive decay and charge-discharge of a capaci-

    tor.

    Thecircuitdiagram hown

    i n

    figure

    1

    was de-

    signed by

    R

    Hiltbrand (Western Michigan Universi-

    t y . US A) . Th e t im er is cal ibrated with a stop watch

    and he ime ntervalsmarkedaround heknob-

    pointer of the potentiometer . The dimensions ' of

    the t imer are

    IOXX

    cm.

    TCG 123

    Figure A I Schematic circuit

    for

    the timer-beeper

    References

    Atkins P 1982

    Physical Chemistry

    2nd edn (San Francisco:

    Einstein

    A

    1956 lnvestigarions o [he Theoryo [he

    Freeman)

    Brownian

    M ovemen t

    R

    Furth ed. ,A Cowper

    tr .

    ( New

    York: Dover)

    Snedecor G

    and

    Cochran W 1980Statistical meth ods

    7th

    edn (Ames,

    10:

    Iowa State University Press) (The

    normal curvefitting

    is found

    i n most elementary

    statistics

    texts).

    Educ. 15 116

    Webb J le P 1980 'Einstein

    an d

    Brownian motion' Phys .

    Hertfordshire Science

    Teaching Scholarship

    A scholarship s offered to science teachers

    f

    at

    leas t three years exper ience, workingn secondary

    schools, middle schools

    r

    colleges of furthe r

    educa t ion , t o enab l e t hem to work fo rix mon ths,

    starting in April or July next year , on a project

    f

    thei r choosing. The project must , however ,e one

    which is intended to mak e the teachingf science

    more effect ive by producing and test ing new ideas,

    ra ther than conduct ing academic research into a

    science subject .

    replacement for one term and the remainder (about

    f250

    for mater ia l s , equipment and t ravel expenses .

    The scholarshipwill pay for the teacher's

    For full

    detai l s and appl icat ion forms wr i te to the

    Science Adviser , CountyHall, Her t fo rd , Her t s

    SG13 8DF.

    Closing date for appl icat ionss

    November 1s t .

    309