Home work 2.3: Carefully read the hand out material regarding the constant gradient approximation. 1)Derive DA , 1 1 Ac(0) = Ac(0) exp(-쓰(+x (÷+-)t) 2)Assume the molecules are ions. Using Fick's Law, Particle flux j=cv, Terminal speed v=µE, E=-dVldx (V: voltage), and Stokes-Einstein relation to derive the voltage difference across the membrane. KT -In(그) AV =V, -V, Assuming ideal diffusive behavior, for a sufficiently low diffusivity sample and sufficiently large vessels, the concentration profile across the specimen should become practically linear after some initial induction period. At this point, the flux of iodide would be constant across the sample, and the corresponding concentration gradient would also be constant. This behavior is referred to here as the constant gradient approximation (CGA), and has been used elsewhere to analyze diffusion data. Figure 1: Schematic of the constant gradient approximation (CGA) for a sample with thickness I. Vessels 1 and 2 contain an ionic species with concentrations ci and c2, and have volumes vị and v2, respectively. The schematic shown in Fig. 1 depicts a system in the constant gradient state. The thin line depicts the concentration of iodide throughout the system; constant in each vessel and a straight line with constant slope across the sample. The specimen apparent diffusivity is D, the thickness is 1, the area is A, and the volume of each vessel is vi and v2, each with iodide concentration ci and c2, respectively. Under the CGA, the flux is constant and the rate of change in iodide concentration in each vessel is also a constant: ôg DA G -c _½ ôc y ôt (1) %3D Upon making the following substitution for the concentration difference, Ac(t)=c1(t)- c2(t), the time dependent behavior for Ac(t) can be expressed as an exponential: DA,1 1 sc(1) = Ac(0)exp(-24E+-») (2)

Modern Physics

3rd Edition

ISBN:9781111794378

Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Chapter11: Molecular Structure

Section: Chapter Questions

Problem 12P

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Transcribed Image Text:Home work 2.3: Carefully read the hand out material regarding the constant gradient approximation. 1)Derive DA ,1 1 q >c, Ac(t) = Ac(0) exp(-- (-+-)t) V1 2)Assume the molecules are ions. Using Fick's Law, Particle flux j=cv, Terminal speed v=µE, E=-dVldx (V: voltage), and Stokes-Einstein relation to derive the voltage difference across the membrane. KT AV =V, -V, = In(1) C2 Assuming ideal diffusive behavior, for a sufficiently low diffusivity sample and sufficiently large vessels, the concentration profile across the specimen should become practically linear after some initial induction period. At this point, the flux of iodide would be constant across the sample, and the corresponding concentration gradient would also be constant. This behavior is referred to here as the constant gradient approximation (CGA), and has been used elsewhere to analyze diffusion data. V2 V1 Figure 1: Schematic of the constant gradient approximation (CGA) for a sample with thickness I. Vessels 1 and 2 contain an ionic species with concentrations cı and c2, and have volumes vị and v2, respectively. The schematic shown in Fig. 1 depicts a system in the constant gradient state. The thin line depicts the concentration of iodide throughout the system; constant in each vessel and a straight line with constant slope across the sample. The specimen apparent diffusivity is D, the thickness is 1, the area is A, and the volume of each vessel is vi and v2, each with iodide concentration ci and c2, respectively. Under the CGA, the flux is constant and the rate of change in iodide concentration in each vessel is also a constant: ôc DA G -c, (1) Upon making the following substitution for the concentration difference, Ac(t)=c1(t)- c2(t), the time dependent behavior for Ac(t) can be expressed as an exponential: DA,1 1. Ac(t) = Ac(0) exp(-- (÷+-)t) (2)

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