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Teachable Unit:Brownian Motion
Created by:
Claudia De Grandi and Katherine Zodrow
May 2013
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
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• 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• 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:
- R. Brown(1827): botanist observing motion of pollen grains
- Einstein’s theory and connection to Avogadro’s number (1905)
- Perrin’s experiment (1908)
Binomial Distribution
probability of k successes in n
trials
p = probability of one success
average number of successes :
variance :
reminder
Learning goals:
•extract the diffusion coefficient D
• understand how the variance depends on time
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 gaslaw
Volume
Pressureideal gas constant
Temperature
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
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
Einstein’s theory
Einstein’s theory
Einstein’s theory
Einstein’s theory
Diffusion coefficientFriction coefficient
Einstein’s theory
Diffusion coefficientFriction coefficient
measurable!in Brownian
motion
viscosity
Radius of
green particle
s
Einstein’s theory
known quantitiestime
Diffusion coefficientFriction coefficient
measurable!in Brownian
motion
viscosity
Radius of
green particle
s
Today’s Recap
Diffusion coeff. of 2D brownian particles
Diffusion coefficient of1D random process
Avogadro’s number!
Einstein’s theory
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
- 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 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
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
volume of a mole
Estimate of molecular radius
at room Temp. (T=300)and 1 atm.(100kPa)
estimate of particlesradius
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
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).