Teachable Unit:Brownian Motion

Created by:

Claudia De Grandi and Katherine Zodrow

May 2013

claudia.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

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.

50 100 150 200 250 300 350 400 450 500 550

50

100

150

200

250

300

350

400

50 100 150 200 250 300 350 400 450 500 550

50

100

150

200

250

300

350

400

290 295 300 305 310 315 320

365

370

375

380

385

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).