Ponderable: Activity - Heat Capacity
WID 1172920
Aluminum Calc.xls
They will do (Ch 11 HW 3) on WebAssign a calculation of the following kind for q = 20,
21, 22, but without the calculation of the heat capacity per atom. Review with Aluminum.
Have students calculate the energy corresponding to one quantum of energy in aluminum
(atomic mass = 27, with the effective spring stiffness 4*16 = 64 N/m. Explain why we go
with the factor of 4 from the Einstein solid—in each direction there are two springs for
twice the strength and each is half as long as the interatomic separation for an additional
factor of two increase in strength.)
E1  N
ks
m
 
 1 1.0546  10
34
Js

64 mN



1 mole
kg 
0.027 mole
 6.02  1023 atoms 
 3.97  1021 J
Show them 100 atoms of Aluminum calculations in the spreadsheet:
k = 1.38E-23
Aluminum
one quantum = 3.98446E-21
q
omega
E
S
20
4.9100E+26
7.9689E-20
8.4856E-22
21
4.4400E+27
8.3674E-20
8.7896E-22
22
3.8500E+28
8.7658E-20
9.0878E-22
heat cap/atom
1.5650E-23
from
midpoint
to
mid-point
ks = 64
hbar = 1.05E-34
Mmole = 0.027


T
3.9845E-21
3.0403E-23
131.06
3.9845E-21
2.9823E-23
133.60
Show that adding one quantum of energy raises temp. That tells us per atom heat
capacity. C = E/T, divide by 100. So for 20-21-22 (middle of energy jump to middle
of next energy jump is just one quantum of energy) get
E E2122  E2021
3.9845  1021
J/K
C


 100 atoms  1.565  1023 atom
T T2122  T2021 133.60  131.06
Have them build their own spreadsheet and analysis for 100 atoms of different metals
(gold, tungsten, and copper: a,b,c groups).
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They will need the q and omega values, ks, and Mmole.
Gold
ks =
20
Mmole =
0.197
E
S
one quantum = 1.6492E-21
q
omega
E
S
20
4.9100E+26
3.2984E-20
8.4856E-22
21
4.4400E+27
3.4633E-20
8.7896E-22
22
3.8500E+28
3.6282E-20
9.0878E-22
ks =
90
Mmole =
0.185
E
S
T
1.6492E-21 3.0403E-23 54.246
1.6492E-21 2.9823E-23 55.299
1.5650E-23
Tungsten
one quantum = 3.61017E-21
q
omega
E
S
20
4.9100E+26
7.2203E-20
8.4856E-22
21
4.4400E+27
7.5813E-20
8.7896E-22
22
3.8500E+28
7.9424E-20
9.0878E-22
ks =
28
Mmole =
0.064
E
S
T
3.6102E-21 3.0403E-23 118.75
3.6102E-21 2.9823E-23 121.05
1.5650E-23
Copper
one quantum = 3.42358E-21
q
omega
E
S
20
4.9100E+26
6.8472E-20
8.4856E-22
21
4.4400E+27
7.1895E-20
8.7896E-22
T
3.4236E-21 3.0403E-23 112.61
3.4236E-21 2.9823E-23 114.80
22
3.8500E+28
7.5319E-20
9.0878E-22
1.5650E-23
 values are given, E = q*1 quantum, S = kln(), differences are by subtraction,
T = 1/(dS/dE).
Comment that the heat capacity rises with temperature. Make the comment that at high
temperatures (which may be room temperature!) the heat capacity reaches the classical
value of 3k per atom (4.2e-23 J/K/atom).
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C
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Would have won Nobel prize 100 years ago.
T
12
Reaches a plateau of 3k =04.2 e-23 per atom at high temp (like room).
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