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![Page 1: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/1.jpg)
Teachable Unit:Brownian Motion
Created by:
Claudia De Grandi and Katherine Zodrow
May 2013
![Page 2: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/2.jpg)
Unit Summary• This unit contains materials for 2 or 3 class periods. Parts of
this unit can stand alone.
• Teaching Materials Include
Powerpoint slides detailing unit
Powerpoint slides to be used in the classroom
2 Matlab modules (for beginners) to help explain Brownian motion
In class quiz/questions
Homework
![Page 3: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/3.jpg)
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
![Page 4: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/4.jpg)
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
![Page 5: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/5.jpg)
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.
![Page 6: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/6.jpg)
Assessment• 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
![Page 7: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/7.jpg)
Materials needed• 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
![Page 8: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/8.jpg)
Classroom Slides
![Page 9: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/9.jpg)
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?
![Page 10: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/10.jpg)
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.
![Page 11: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/11.jpg)
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:
- R. Brown(1827): botanist observing motion of pollen grains
- Einstein’s theory and connection to Avogadro’s number (1905)
- Perrin’s experiment (1908)
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Binomial Distribution
probability of k successes in n
trials
p = probability of one success
average number of successes :
variance :
reminder
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Learning goals:
•extract the diffusion coefficient D
• understand how the variance depends on time
![Page 14: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/14.jpg)
Learning goal:• understand how to extract the diffusion coefficient from 2D images • relate the diffusion coefficient to the Avogadro’s number
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QuickTime™ and a decompressor
are needed to see this picture.
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290 295 300 305 310 315 320
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Ideal gaslaw
Volume
Pressureideal gas constant
Temperature
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Ideal gaslaw
mole=as many molecules as in 12 grams of 12Cmole= Avogadro’s number(NA) of molecules
Volume
Pressureideal gas constant
Temperature
# of moles
total number of molecules
In our case: water
Boltzmann constant
![Page 21: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/21.jpg)
Ideal gaslaw
mole=as many molecules as in 12 grams of 12Cmole= Avogadro’s number(NA) of molecules
Volume
Pressureideal gas constant
Temperature
# of moles
historically
Einstein’s theory of Brownian motion!
total number of molecules
In our case: water
Boltzmann constant
![Page 22: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/22.jpg)
Einstein’s theory
![Page 23: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/23.jpg)
Einstein’s theory
![Page 24: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/24.jpg)
Einstein’s theory
![Page 25: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/25.jpg)
Einstein’s theory
Diffusion coefficientFriction coefficient
![Page 26: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/26.jpg)
Einstein’s theory
Diffusion coefficientFriction coefficient
measurable!in Brownian
motion
viscosity
Radius of
green particle
s
![Page 27: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/27.jpg)
Einstein’s theory
known quantitiestime
Diffusion coefficientFriction coefficient
measurable!in Brownian
motion
viscosity
Radius of
green particle
s
![Page 28: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/28.jpg)
Today’s Recap
Diffusion coeff. of 2D brownian particles
Diffusion coefficient of1D random process
Avogadro’s number!
Einstein’s theory
![Page 29: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/29.jpg)
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?
![Page 30: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/30.jpg)
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.
![Page 31: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/31.jpg)
Today in class, we will
1. Review solutions to the homework
2. Use Matlab software to simulate and analyze in details 1D random walks
- Work in groups of 2/3 people- Follow instruction on handout- Hand in a report by the end of the lecture
![Page 32: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/32.jpg)
Today in class, we will1.Use Matlab software to simulate and analyze 2D Brownian motion (like
reproducing Perrin’s exp. images)- 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
![Page 33: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/33.jpg)
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
![Page 34: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/34.jpg)
volume of a mole
Estimate of molecular radius
at room Temp. (T=300)and 1 atm.(100kPa)
estimate of particlesradius
![Page 35: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/35.jpg)
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
![Page 36: Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 claudia.degrandi@yale.educlaudia.degrandi@yale.edu katherine.zodrow@yale.edu.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649da95503460f94a95fa3/html5/thumbnails/36.jpg)
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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).