A7 - PHYS 258 - Thermal Physics: Student ID: Assignment 7...

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PHYS 258 - Thermal Physics: Assignment 7 Due Monday March 7 Student ID: Full Name: Score:_______/16___ 1. [5] Temperature and Heat Capacity of an Einstein Solid. In class we found that in the high temperature limit of an Einstein solid, the thermal energy is related to the temperature via U = NkT , (where N is the number of oscillators) . What the corresponding result is in the low temperature limit. In Assignment 5, Problem 1, you found that, when q << N , the multiplicity of a large Einstein solid is given by q q eN q N Ω ) , ( (a) [2] Using this result, find a formula for the temperature of an Einstein solid as a function of its energy. Use U = q ε , where is one quantum of energy for an oscillator. (b) [2] Using your result from part (a), solve for the energy, U , as a function of the temperature. What is the limit of U as T 0? What is the average number of quanta of energy in each oscillator ( q/N ) at temperature T ? Observe that under the assumptions stated, this number is always much less than 1 and so kT << .
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(c) [1] Using your result from part (b), calculate the heat capacity of an Einstein solid in the low temperature limit. Sketch
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A7 - PHYS 258 - Thermal Physics: Student ID: Assignment 7...

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