6.021/Notes/2006-09-08

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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}

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