lec05 - In response solute particle is equally likely to...

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HST.410J/6.0 7 J Lecture 5 February 22, 2007 Concentration at a point in space and time volume V amount of substance in V concentration c(x,t) = lim V 0 V Flux at a point in space and time window area A φ (x,t) amount of substance flowing flux φ (x,t) = lim A 0 through test window A in Δ t A Δ t Δ t 0 Fick's First Law Harvard-MIT Division of Health Sciences and Technology HST.410J: Projects in Microscale Engineering for the Life Sciences, Spring 2007 Course Directors: Prof. Dennis Freeman, Prof. Martha Gray, and Prof. Alexander Aranyosi Figure from Weiss, T. F. Cellular Biophysics, Vol. I. Cambridge, MA: MIT Press, 1996. Courtesy of MIT Press. Used with permission.
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x 0 l 2l 3l 1 1 1 1 2 1 3 3 1 1 0 1 2 3 +3 +2 +1 0 - 1 - 2 - 3 m n F V C Random Walk Model • number of solute particles << number of solvent particles • motion of solute determined by collisions with solvent (ignore solute-solute interactions) • focus on 1 solute particle, assume motions of others are statistically identical Every τ seconds, solute particle gets hit by solvent particle.
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Unformatted text preview: In response, solute particle is equally likely to move + l or -l . τ = mean free time ; l = mean free path 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1-7-5-3-1 1 3 5 7 1 2 3 4 5 6 7 W(m,n) m n igures from Weiss, T. F. Cellular Biophysics, ol. I. Cambridge, MA: MIT Press, 1996. ourtesy of MIT Press. Used with permission. Fick's First Law How long till half the solute diffuses to | x |> x 1/2 x 1/2 2Dt 2 3 ~ < 2Dt 2 3 ~ > x 1/2 4 9 ~ > 2 x 1/2 2Dt 2 x 1/2 D ~ > ≡ t 1/2 t Importance of Scale 2 x 1/2 cm 2 = ; D = 10 − 5 for small solutes (e.g., Na + ) t 1/2 D s t 1/2 x 1/2 1 membrane sized 10 nm µ sec 10 cell sized 10 µ m 1 sec 10 dime sized 10 mm 10 5 sec ≈ 1 day Figure from Weiss, T. F. Cellular Biophysics, Vol. I. Cambridge, MA: MIT Press, 1996. Courtesy of MIT Press. Used with permission. –3 –2 –1 3 2 1 50% 68% 95% 99% c(x,t) = n o 4 π Dt e-x 2 /(4Dt) x 2Dt...
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lec05 - In response solute particle is equally likely to...

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