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Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 [email protected] [email protected]

Teachable Unit: Brownian Motion Created by: Claudia De Grandi and Katherine Zodrow May 2013 [email protected]@yale.edu [email protected]

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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

Teachable Unit:Brownian Motion

Created by:

Claudia De Grandi and Katherine Zodrow

May 2013

[email protected] [email protected]

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

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

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

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

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

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

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

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

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

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

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)

Page 12: 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

Binomial Distribution

probability of k successes in n

trials

p = probability of one success

average number of successes :

variance :

reminder

Page 13: 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

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

Learning goal:• understand how to extract the diffusion coefficient from 2D images • relate the diffusion coefficient to the Avogadro’s number

Page 15: 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

QuickTime™ and a decompressor

are needed to see this picture.

Page 16: 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

50 100 150 200 250 300 350 400 450 500 550

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Page 17: 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

50 100 150 200 250 300 350 400 450 500 550

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Page 18: 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

290 295 300 305 310 315 320

365

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385

Page 19: 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

Ideal gaslaw

Volume

Pressureideal gas constant

Temperature

Page 20: 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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
Page 37: 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

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).