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Solution3 - Materials Science and Engineering 206/196...

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Materials Science and Engineering 206 /196 Prof. W.D. Nix Imperfections in Crystalline Solids Spring 200 8 Homework Assignment #3 SOLUTIONS Problem 1 In this problem, you are asked to find the formation energy for Sn solute atoms in Al, given the concentration of Sn in Al at a given temperature. You are then to use this information to calculate the equilibrium concentration at room temperature. We begin by writing down the equation for the defect concentration at equilibrium: x exp g f kT , (1.1) where g f is the formation free energy of the defect. It can be expanded as g f = ∆ e f + P ext v f T s f . (1.2) We are told that the formation energy term, e f , dominates this expression, so Equation 1.1 can be written as x exp e f kT . (1.3) We can invert this equation to solve for e f : e f = − kT ln x . (1.4) So, for x = 0.001 at T = 600 °C = 873 K, we have e f = − (1.381 × 10 23 J /K)(873 K)ln(0.001) = 8.33 × 10 20 J = 0.52 eV (1.5) This energy does not vary with temperature, so at room temperature (25°C = 298 K) we can calculate the new equilibrium concentration of Sn as x = exp 8.33 × 10 20 J (1.381 × 10 23 J /K)(298 K) = 1.63 × 10 9 (1.6) 1
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Problem 2 In the first part of this problem, you determine the equilibrium concentration of Si in Al at room temperature, given the data on an Al-Si phase diagram. Since the phase diagram does not go all the way down to room temperature, and since the resolution of the horizontal axis is not very good, we must determine the concentration at room temperature in a manner similar to that employed in Problem 1. We use the phase
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