MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
Problem Set 10 Solutions
1. We derived Einstein’s famous crystal model in class. Repeat the exercise explaining all
assumptions and steps.
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
2. For each Einstein temperature, θE, of 100K, 200K, 300K, and 500K, use a computer
program other than Excel to calculate and plot the:
a. partition function
θE : 100 K=Blue ;200 K=Red ;300 K=Green ;500 K=Magenta
2.5
2
P
1.5
1
0.5
0
0
100
200
300
T H KL
400
500
b. Cv
hv
. Consider the system as a simple
k
cubic Einstein crystal. Of interest is the range of temperatures from 0-500K. Your plot
should be similar to figure 6.1 in Gaskell. Use a mathematical program that is NOT
Excel. Your plots should be labeled thoroughly.
for the Einstein model as a function of T given θ E =
θE : 100 K=Blue ;200 K= Red ;300 K=Green ;500 K=Magenta
25
Cv H
J
L
mole K
20
15
10
5
0
0
100
200
T HKL
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300
400
500
MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
3. Comment on the similarities and differences between statistical thermodynamics and
classical thermodynamics.
Open ended problem
4. Comment on the partition function.
Open ended problem
5. Derive the equation that we obtained for the Tangent method. Explain any assumptions
and your steps.
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
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MSE 308
Thermodynamics of Materials
Dept. of Materials Science & Engineering
Spring 2005/Bill Knowlton
6. The change in volume of each component for a particular solution two component system
are given by
∆V1 = ao X 22 (1 − 2 X 1 )
∆V2 = 2ao X 12 X 2
Strictly using the following form of the Gibb’s-Duhem equation:
2
∑X
i =1
k
d ∆ Bk = 0
prove the Gibb’s-Duhem equation is zero using the volumes given.
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