2004_855_Exams

# 2004_855_Exams - Examination Two MSE 855 Spring 2004 1 For...

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Examination Two, MSE 855, Spring 2004 1. For the initial concentration distribution f(x') given by f(x') = 3A 1 x 3 + 5A 2 x for x' > 0 f(x') = 0 otherwise, (a) Set up the integral that can be used to solve this problem for times t > 0. Use the coordinate transformation Dt 4 x 2 2 = η Write the integrand and the integrator in terms of the transformed coordinate, η . Also, demonstrate the appropriate change of integration limits. (20 pts) (b) State the assumptions involved in the solution to this problem, including the nature of the host. Is D = D(C) an important consideration in this problem? Is your solution in part (a) valid if D = D(C)? Why or why not? (20 points) 2. (a) Sketch (Co/2)[1 + erf(z)] as a function of z. Label the axes. Indicate the values of (Co/2) [1 + erf(z)] at z = 0, z and for -∞ z . (20 points) 3.One can identify three time regimes for solution of diffusion problems, namely short intermediate and long time. (a) Discuss these three time regimes, and for each time regime specify particular criteria for each regime in terms of appropriate parameters. Define each symbol in each relationship you use. Specify physical units for each symbol. (10 points) (b) Green's function type solutions apply best in one of these regimes. Specify which time regime is most appropriate for Green's function solutions and explain why it is appropriate in that regime. Explain why solutions based on Green's functions are not appropriate in the other two regimes? (Specify the particular problems encountered) (10 points) (c) Repeat (b), but for series solutions. (10 points) 4. (a) For the ionic vibrational frequency, ν o , sketch the behavior extending from temperatures near absolute zero to above the Debye temperature. Label the Debye temperature on your plot along with the frequency that corresponds to the Debye frequency. Using additional labeled sketches, explain the connection between the details of your ν o versus T plot and each of the following:

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