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Unformatted text preview: Fick’s laws (1855)
1. J … concentration (net) flux, number of molecules
that move across an area A per unit time
(units: #/area/time)
c … concentration, here: number of molecules per
volume
(units: #/volume)
x … average distance traveled by each molecule
average distance traveled by each molecule
in xdirection (positive or negative) in the time
interval t c x, t 2 c x, t D
t
x 2
diffusion equation 2.
where D x 2
2t one particular solution: x2 C1
c x, t exp C2 4 Dt t J … concentration (net) flux, number of molecules
that move across an area A per unit time
(units: #/area/time) Diffusion + Convection
c x, t 2 c x, t D
t
x 2 c … concentration, here: number of molecules per
volume
(units: #/volume)
x … average distance traveled by each molecule
average distance traveled by each molecule
in xdirection (positive or negative) in the time
interval t diffusion equation Adding drift (at velocity v(x)) to Fick’s first law:
drift (at velocity
to Fick
law: J x D dc x dx v xc x Drift is caused by force f acting on molecules, where c x, t t D 2 c x, t x 2 f x v x f x
c x, t x At equilibrium, c is stationary (doesn’t change with time), and is given by the
Boltzmann distribution:
distribution: Concentration Probability
number of molecules per volume
c x, t … number of molecules per volume
at position x at time t p x, t … probability of finding a molecule
at position
at position x at time t
time c p, so we may replace the “concentration flux” J by the “probability flux” j p x, t 2 p x, t f x D p x, t t
x x 2...
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This note was uploaded on 07/12/2011 for the course BIM 162 taught by Professor Heinrich during the Spring '11 term at UC Davis.
 Spring '11
 Heinrich

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