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hw7 - Homework 7 PHZ 5156 Due Tuesday November 2 2009 1 In...

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Unformatted text preview: Homework 7 PHZ 5156 Due Tuesday, November 2, 2009 1. In class, we discussed the diffusion equation and also a bit about Boltzmann statistics. In this problem, we will explore diffusion in the presence of an external field. We saw in class that the mass current J z of diffusing particles is given in terms of the diffusion constant D and local density n ( z,t ) as J z =- D ∂n ∂z (in one dimension). Consider now an external field acting so that the diffusing particles feel an external force with z-component F z =- dU dz , where U ( z ) is the potential energy of a particle at z . If the force F z = F (i.e. a constant force independent of z), then the potential energy is (up to a constant) U ( z ) =- F z . With this added force, the mass current is taken to be J z =- D ∂u ∂z + nμF where μ is the mobility . a) Consider the case where J z = 0 (i.e. equilibrium). Recall first that in equilibrium, based on our discussion of Boltzmann statistics, that at temperature T we have n ( z ) = n e- U ( z ) /k B T . Next, show that there is a relationship between μ and D , μ = D k B T b) Use the continuity equation ∂n ∂t + ∂J z ∂z = 0 to obtain the one-dimensional Smolu- chowski equation, ∂n...
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hw7 - Homework 7 PHZ 5156 Due Tuesday November 2 2009 1 In...

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