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

## 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
• $\displaystyle{ \phi(x,t)=-D\frac{\partial c(x,t)}{\partial x} }$
• We define flux to be positive in same direction as increasing $\displaystyle{ x }$
• Diffusivitiy units: $\displaystyle{ D=\frac{m^2}{s} }$
• This is a macroscopic law

## Microscopic basis for diffusion

• 1828: Robert Brown (Brownian motion)
• Every $\displaystyle{ \tau }$ seconds, molecule equally likely to move $\displaystyle{ +l }$ and $\displaystyle{ -l }$
• $\displaystyle{ l=2 }$pm for small molecule (really small length scale)
$\displaystyle{ \phi(x,t)=-\frac{l^2}{2\tau}\frac{\partial c}{\partial x} }$ so $\displaystyle{ D=-\frac{l^2}{2\tau} }$