Brownian Motion and Diffusion
Learning Objectives
Students will be able to:
- Explain Brownian motion as the result of molecular collisions with suspended particles
- Apply the Stokes-Einstein equation to calculate diffusion coefficients
- Derive the relationship between diffusion distance and time (random walk)
- Calculate the mean free path and relate it to particle behavior
- Predict how particle size affects diffusion-dominated transport
The Big Question
"Why do small particles behave so differently from large ones, and how does random molecular motion affect particle transport in air?"
Historical Context: Einstein and Brownian Motion
In 1905, Albert Einstein published a paper explaining the erratic motion of pollen grains observed by Robert Brown in 1827. Einstein showed that Brownian motion provides direct evidence for the existence of molecules and allows calculation of Avogadro's number.
Einstein's Key Insight
A particle suspended in fluid is continuously bombarded by fluid molecules from all directions. These collisions are random and unbalanced, causing the particle to move in an erratic path. The statistical properties of this motion can be precisely predicted from the kinetic theory of gases.
The Stokes-Einstein Equation
The diffusion coefficient D quantifies how fast particles spread due to Brownian motion:
D = kT / (6 pi eta r)
| Variable | Meaning | Value/Units |
|---|---|---|
| D | Diffusion coefficient | m2/s or cm2/s |
| k | Boltzmann constant | 1.38 x 10-23 J/K |
| T | Temperature | Kelvin |
| eta | Dynamic viscosity of air | 1.81 x 10-5 Pa-s (at 20 C) |
| r | Particle radius | meters |
The Random Walk
Because Brownian motion is random, a particle does not travel in a straight line. Instead, its displacement follows a "random walk" characterized by:
Mean Square Displacement
x2 = 2Dt (1-D)
r2 = 6Dt (3-D)
Displacement grows with the square root of time, not linearly
Root Mean Square Distance
xrms = sqrt(2Dt)
The typical distance traveled in time t
Diffusion Coefficients for Aerosol Particles
| Particle Diameter | D (cm2/s) | xrms in 1 second | Example Particles |
|---|---|---|---|
| 0.01 um (10 nm) | 5.2 x 10-4 | 0.32 cm | Nucleation mode, viruses |
| 0.1 um (100 nm) | 6.8 x 10-6 | 0.037 cm | Accumulation mode, bacteria |
| 1.0 um | 2.7 x 10-7 | 0.007 cm | Fine PM, some bacteria |
| 10 um | 2.4 x 10-8 | 0.002 cm | Coarse PM, pollen |
Note: For very small particles, the Cunningham slip correction factor must be applied as particles become smaller than the mean free path of air (~65 nm).
The Mean Free Path
The mean free path (lambda) is the average distance a gas molecule travels between collisions:
lambda = kT / (sqrt(2) * pi * d2 * P)
For air at standard conditions: lambda ~ 65 nm
Key insight: When particle diameter approaches lambda, the assumption of continuous fluid breaks down and particles "slip" between gas molecules, moving faster than predicted by Stokes' law.
Activity: Diffusion Calculations
Problem Set
- Calculate D: Determine the diffusion coefficient for a 50 nm particle at 25 degrees C (298 K) using the Stokes-Einstein equation.
- Random walk: How far (rms) will this particle travel in 1 minute due to diffusion alone? Compare to a 5 um particle.
- Temperature effect: By what factor does D increase if temperature rises from 20 C to 35 C?
- Virus transport: A respiratory virus (diameter ~100 nm) is exhaled into still air. Estimate how far it could travel by diffusion alone in 10 minutes. Why is this distance misleading for understanding disease transmission?
Key Takeaway
Brownian motion, arising from random molecular collisions, dominates the transport of ultrafine particles (<100 nm). The Stokes-Einstein equation provides a quantitative framework for predicting diffusion rates, which depend inversely on particle size. This diffusion-dominated regime is critical for understanding the behavior of viruses, nanoparticles, and the smallest components of air pollution. In the next lesson, we will explore how particles deposit on surfaces through multiple mechanisms.