# Lab 3, part C: Brownian motion

##### Motivation

This week you will take another round of bead displacement data using a smaller bead (either 1 or 2 micron diameter). Hopefully this will reveal that – as your intuition should tell you – smaller objects are more affected by Brownian motion. For comparison: mammalian cells are typically much larger than 1 micron; small bacteria are typically about a couple of micron in size; cell organelles are smaller than a micron; individual proteins are much smaller than a micron.

##### Lab writeup

Use this blank template as a starting point for your writeup.

###### Questions

For each of two beads that you tracked

1. Is the bead’s mean and mean-squared motion consistent with Brownian motion? Analyze x- and y-behavior separately and also together.
2. What is the diffusion constant for your bead?
###### Hints

There are several ways to demonstrate that Brownian motion is occurring, but one possibility is to

1. Calculate the mean displacements in x and y, as a function of time. There are a couple of ways to do do this, but the most straightforward is, for each bead, to subtract off the initial position (so that Δx(t) = x(t) – x(0)), and then average together all 50-some bead displacements at time t to get the average <Δx(t)>.
2. Plot <Δx(t)> and <Δy(t)> versus time to show that the average displacement is zero.
3. Calculate the mean squared displacements in x, and y, as a function of time. Plot <Δx2(t)> and <Δy2(t)> versus time and fit each to a straight line (with zero intercept) to show that <Δx2(t)> = 2Dt in x and y. The diffusion coefficient is half the slope of the fit line.

These two plots, along with the value for D, will convey all the quantitative information in your lab. Be sure to explain exactly how you obtained your numbers; for instance, how you calculated mean and mean-squared displacements (using words and/or equations), how you calibrated space and time in your video, how you obtained a value for D from the graph(s), etc. Watch your units!

When evaluating your value for D, compare it to the Stokes-Einstein prediction for the diffusion constant of a sphere.