Thermodynamics of Materials
© John Kieffer, University of Michigan (2009)
1
HW
3
Problem 6.12
a)
Calculate the heat capacity based on the Debye model for aluminum at all tempera
tures between 0 K and 600 K spaced by 10 degrees intervals.
Use a numerical inte
gration procedure to calculate these data.
Present the results in graphical and in
tabulated form.
b)
Calculate the enthalpy change in Al when heated between 100 K and 300 K.
c)
For bonus points: determine the Einstein temperature for which the Einstein model
provides the best match to the data calculated in (a) using the Debye model.
Please provide printouts of any code that you wrote for solving these questions and ex
plain the numerical procedures that you used.
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Thermodynamics of Materials
© John Kieffer, University of Michigan (2009)
2
Answer
a) We numerically integrate the expression
c
V
=
9
nk
B
T
θ
D
⎛
⎝
⎜
⎞
⎠
⎟
3
x
4
e
x
e
x
−
1
(
)
2
dx
0
x
D
∫
, where
x
=
θ
D
T
, e.g., using the trapezoid rule. The code is posted on the Ctools site.
The
results are
T (K)
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 Fall '09
 kiffer
 Thermodynamics, Absolute Zero, θe, John Kieffer

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