Teachable Unit:
Brownian Motion
Created by:
Claudia De Grandi and Katherine Zodrow
May 2013
claudia.degrandi@yale.edu
katherine.zodrow@yale.edu
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Unit Summary
This unit contains materials for 2 or 3 class periods. Parts of
this unit can stand alone.
Teaching Materials Include
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Powerpoint slides detailing unit
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Powerpoint slides to be used in the classroom
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2 Matlab modules (for beginners) to help explain
Brownian motion
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In class quiz/questions
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Homework
Part 1 (75 min)
1.Introduce learning goals
2. Perform a 1D random walk as a class
3. History of Brownian motion (lecture)
4.Reflection and discussion
Part 2 (75 min)
1. Present and discuss the solutions to the
homework (from Part 1)
2.Computer lab group activity with Matlab
(Module I)
- Students follow instructions on handout to simulate
and analyze 1D random walks
- Students hand in a final lab report
- Students are given final answer key sheet
- TAs and Instructor available for in-class help
Part 3 (75 min)
1.Introduction to data analysis: How do researchers
use Brownian motion?
2.Computer lab group activity with Matlab on 2D
random walks (Module II)
3. Final question: Estimate the radius of an atom
4.Discuss solution of the question
5.Reflection, final comments on initial learning
goals.
Assessment
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Initial reflection on learning goals questions (see slide 9)
Initial multiple choice quiz about binomial distribution (see
slide 11)
Homework (end of Part 1) on diffusion in different
viscosities
2 Matlab Modules, to be turned in as a short lab report
In-class final problem questions about the size of an atom
Final homework/report: revised and detailed answers to
learning goals questions
Materials needed
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coin to flip for each student
a relatively spacious room to implement the random walk
activity or a large white board and Post-it stickers
a computer and screen to project slides
Device for students to use Learning Catalytics (formative
assessment questions and class random walk activity)
Computer with Matlab for each student group
Handout for students with a copy of the slides
Classroom Slides
Brownian motion, Atoms and
Avogadro’s Number
•How do we know atoms exist?
•What is the size of an atom?
•How would you observe an individual atom?
Brownian motion, Atoms and
Avogadro’s Number
•How do we know atoms exist?
•What is the size of an atom?
•How would you observe an individual atom?
Suggestion: make a ‘diary’ to keep track of your learning
process
At the end of the 3 lectures your homework will be to summarize
what you have learned and give your best answers to those
questions.
Today in class, we will
1. Review the binomial distribution: quiz
2. Perform a 1D random walk as a class
and extract our diffusion coefficient
3. Review the history of Brownian motion:
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R. Brown(1827): botanist observing motion
of pollen grains
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Einstein’s theory and connection to
Avogadro’s number (1905)
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Perrin’s experiment (1908)
Binomial Distribution
reminder
p = probability of one success
probability of k
successes in n
trials
average number of
successes :
variance :
Learning goals:
• understand how the variance
depends on time
•extract the diffusion coefficient D
Learning goal: • understand how to extract the diffusion
coefficient from 2D images
• relate the diffusion coefficient to the
Avogadro’s number
QuickTime™ and a
decompressor
are needed to see this picture.
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ideal gas constant
Ideal gas
law
Pressure
Temperature
Volume
ideal gas constant
Pressure
Ideal gas
law
Temperature
Volume
In our case:
water
# of moles
mole=as many molecules as in 12 grams of 12C
mole= Avogadro’s number(NA) of molecules
total number of molecules
Boltzmann constant
ideal gas constant
Pressure
Ideal gas
law
Temperature
Volume
In our case:
water
historically
# of moles
Einstein’s theory
of Brownian motion!
mole=as many molecules as in 12 grams of 12C
mole= Avogadro’s number(NA) of molecules
total number of molecules
Boltzmann constant
Einstein’s
theory
Einstein’s
theory
Einstein’s
theory
Einstein’s
theory
Diffusion coefficient
Friction coefficient
Radius
of
green
particles
viscosity
Einstein’s
theory
Diffusion coefficient
measurable!
in Brownian
motion
Friction coefficient
Radius
of
green
particles
viscosity
Einstein’s
theory
Diffusion coefficient
measurable!
in Brownian
motion
Friction coefficient
time
known quantities
Today’s Recap
Diffusion coefficient of
1D random process
Diffusion coeff. of 2D
brownian particles
Einstein’s theory
Avogadro’s number!
Brownian motion, Atoms and
Avogadro’s Number
•How do we know atoms exist?
•What is the size of an atom?
•How would you observe an individual atom?
Homework
1) At 20 °C, the dynamics viscosity η of water is
10-3 Pa*s. Glycerol is 1 Pa*s. We place particles
in these two solutions, holding everything else
constant. Give a quantitative relationship for
the diffusion of these particles in these two
solutions.
2) Sketch a plot that compares <x2>vs. time for
particles in each of these solutions.
Today in class, we will
1. Review solutions to the homework
2. Use Matlab software to simulate and
analyze in details 1D random walks
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Work in groups of 2/3 people
Follow instruction on handout
Hand in a report by the end of the lecture
Today in class, we will
1.Use Matlab software to simulate and analyze 2D
Brownian motion (like reproducing Perrin’s exp. images)
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Work in groups as Module I, hand in final report
- You will extract the Avogadro’s number from your data
2.Group problem: estimate the size of a molecule from
Avogadro’s number
3.Final discussion on learning goals and final homework
Estimate of molecular radius
Assume Avogadro’s number NA= 6 X1023
NA is the total number of molecules in a mole
Reminder Ideal gas law:
Work in groups to find an estimate of the size of
a molecule in a mole
Estimate of molecular radius
volume of a mole
at room Temp. (T=300)
and 1 atm.(100kPa)
estimate of particles
radius
Brownian motion, Atoms and
Avogadro’s Number
•How do we know atoms exist?
•What is the size of an atom?
•How would you observe an individual atom?
Final homework/report/reflection:
write down your best answers, compare with your
initial guess, discuss what are the most important
things you have or have not learned during this
teaching unit
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Additional reading
Haw, M D. (2002) Colloidal suspensions, Brownian motion,
molecular reality: a short history. J. Phys. Condens. Matter
14:7769.
Philip Nelson’s book: Biological physics: Energy,
Information, Life (Chap. 4).
Random Walks in Biology, Howard Berg
Investigation on the theory of The Brownian Movement,
Albert Einstein, Dover Publications (1956) (original
Einstein’s paper on Brownian motion).