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
B
T
i
=
0
∞
∑
=
e
−
i
·0.1·1.602·10
−
19
1.38·10
−
23
T
i
=
0
∞
∑
=
e
−
11.6
( )
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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This note was uploaded on 10/26/2009 for the course MSE 330 taught by Professor Kiffer during the Fall '09 term at University of MichiganDearborn.
 Fall '09
 kiffer

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