# 6.021/Notes/2006-09-08

## Diffusion

• Most fundamental transport process and the one we understand the best
• Process by which solute is transported from regions of high concentration to low concentration
• Graham made some observations related to diffusion
• Quantity transported is proportional to the initial concentration
• Transport rate slows with time
• Transport of gas is greater than 1000x faster than liquids

### Definitions

• Concentration: $c(x,t)=\lim_{V\rightarrow 0} \frac{amount}{volume}$
• But matter isn't discrete, so limit doesn't make sense
• Practically, cells have about $\frac{1}{6}\frac{mol}{L}$ NaCl which is about $10^8\frac{molecules}{\mu m^3}$
• Cells are greater than m3 so there are many molecules in a typical cell
• Thus, we'll assume matter is continuous to simplify the math
• Flux: $\phi(x,t) = \lim_{A\rightarrow 0 ; \Delta t\rightarrow 0}\frac{amount}{A\Delta t}$

## Fick's First Law

• Adolf Fick (1855) at age 25, came up with Fick's first law by analogy to Fourier's law for heat flow
• $\phi(x,t)=-D\frac{\partial c(x,t)}{\partial x}$
• We define flux to be positive in same direction as increasing x
• Diffusivitiy units: $D=\frac{m^2}{s}$
• This is a macroscopic law

## Microscopic basis for diffusion

• 1828: Robert Brown (Brownian motion)
• Even dead things moved
• Albert Einstein with random walk model
• Assumptions
• Number of solute much less than number of solvent
• Only collisions between solute and solvent
• Focus on 1 solute molecule and assume others are statistically identical
• Every τ seconds, molecule equally likely to move + l and l
• l = 2pm for small molecule (really small length scale)
• From random walk model, easy to derive Fick's first law

$\phi(x,t)=-\frac{l^2}{2\tau}\frac{\partial c}{\partial x}$ so $D=-\frac{l^2}{2\tau}$