8/18/2019 Brownian Motion - A Laboratory Experiment
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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.
8/18/2019 Brownian Motion - A Laboratory Experiment
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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 .
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r
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309