Probl_4.5_sol - MatSciE 330 Thermodynamics of Materials...

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MatSciE 330 Thermodynamics of Materials 10/24/09 © J. Kieffer, University of Michigan 1 Problem 4.5 Consider a system in which atoms can occupy energy levels that are evenly spaced, and there exist an infinite number of energy levels. The separation between energy levels is 0.1 eV., so that ε 0 = 0 eV, ε 1 = 0.1 eV, ε 2 = 0.2 eV, … ε i = i·0.1 eV … a) Calculate the partition function for such a system at 100K and at 1000K b) Calculate the fraction of atoms that occupy ε 0 and ε 1 at these two temperatures. Solution : a) Energy levels increment by 0.1 eV. To evaluate the partition function we calculate P = e i ·0.1 k B T i = 0 = e i ·0.1·1.602·10 19 1.38·10 23 T i = 0 = e 11.6 T ( ) i i = 0 = x i i = 0 . For T = 100 K x = e 11.6 100 = 9.08·10 –6 , and for T = 1000K x = e 11.6 1000 = 0.313. At 100K the power series then becomes P = 1 + 9.08·10 –6 + 8.256·10
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