19.2) consider a kilomole of 3 He gas atoms under STP conditions
No. of 3 He atoms at STP = N = 1 kilomole = 6.023 10 26
Volume at STP =V = 22.4 dm 3
Mass of 3 He atom = m =
3
27
4
.
98
10
6.023 10 26
a) What is the Fermi temperature of the gas?
Fermi Temperature = TF ?
We know that
h2 N
TF
2mk 1.504v
2
3
6.626 10
6.023 10 26
2 4.98 10 27 1.38 10 23 1.504 22.4
0.069 K 0.07 K
Fermi Temperature TF 0.07 K
34 2
b) Calculate
kT
and exp
kT
2
3
?
kT
we knowthat
2mkT 3 2 V
ln 2
2
kT
N
h
3
27
23
2
2
4
.
98
10
1
.
38
10
273
22.4
ln 2
26
34 2
6
.
023
10
6.626 10
kT
12.7
exp e
kT
e
kT
kT
?
3.28 10 5
c) Find the average occupancy f of a single particle state that has energy of 3 kT
2
Energy : 3 kT
2
Chemical Potential : 12.7kT
Average Occupancy : f ?
We knowthat
f
e
1
KT
1
1
3
12.7 kT
2
kT
e
1
1
1.469 10 6
f 6.8 10 7
Average
Occupancy f 6.8 10 7
19.3) For a system of noninteracting electrons, show that the probability f of
finding an electron in a state with energy ∆ above the chemical potential μ is the same as
the probability of finding an electron absent from the with energy ∆ below μ at any given
temperature T
N
1
g e kT 1
where
1
f
e kT 1
for the probability of absent at
1
1
1 f
1
e kT 1
e kT 1
f
1 f
e
kT
1
1
1 1 1
1 e kT
e kT
therefore they are equal to each other
e
kT
19.4a) Verify that the average energy per fermion is
a direct calculation of U 0
Use equation 19.18
35
F
at absolute zero by making
N
3
U N F for T 0
5
U
3
T o F
N
5
b) Similarly, prove that the average speed of a fermion gas particle at T=0 is
where the Fermi velocity
VF
v f v g v dv
0
N
v
is defined by
2
F 3 5 mvF
3 4
VF
,
f v
N v
g v
1
1 2
mv
2
kT
e
1
g d g v dv
8 2V 3 2 1 2 2 1
g v dv
m mv mv dv
h3
2
2
1
1
1 2 2 1
mv mv dv
8 2V 3 2 2
2
v
m
1
3
mv
0
h
2
kT
e
1
2
8 2V
v
h3
3
1 2 3
m
0
2
v3
1 2
mv
2
kT
dv
e
1
need to use int egration in part !
19.6) At very low temperature, the electronic specific heat capacity of a metal is
approximately ce AT where A can be determined by experiment. For gold,
the constant A is found to be A 0.73J kilomole K . Compare this to the
value obtained from Equation (19.20). (The atomic weight of gold is 197 and its
3
3
density is18.9 10 kgm )
1
1
From eqn (19.20), one identifies
2 Nk
2
TF
1
1
6.63 1034 3 18.9 103
with TF F
6.02 1026
5
1
31
k
8.62 10 eVk
2 9.11 10 8 197
TF 62028k
2
A
2 6.02 1026 1.381 1023 Jk 1
2
62028k
1
0.66 J kilomole K 2
19.7 a) Calculate
F
for aluminum assuming three electrons per aluminum atom.
3 Kg
N 2.69 10
m 3 6.02 10 26 atoms 5.99 10 28
V
kilo mole
27 Kg
kilomole
N
# Density for electrons 3
V
1.8 10 29
F
6.63 10
3 1.8 10 29
2 9.1110 31
8
34 2
b) Show that the aluminum at T=100 K, μ differs from
aluminum is
2.69 10 kgm
3
3
2
3
5.6 2.1 11.8eV
F by less than 0.01%. (The density of
and its atomic weight is 27.)
2
3
2
0 1
12
T
TF
2
2
1000k
0 1
11.8eV
12
8.62 10 5 eVk 1
0 1 4.38 10 5
2
less than 0.01%
c) Calculate the electronic contribution to the specific heat capacity of aluminum at room
temperature and compare it to 3R
T 2
kT
Nk
Nk
2
TF 2
F
8.617 10 5 eVk 1 2.43 10 3
3 n R
2
1
297 J kilomole k
Ce
19.8) In sodium there are approximately 2.6 10 conduction electrons per cubic meter,
which behave as a free electron gas. Give an approximate value for the electronic specific
heat of sodium at room temperature. (The atomic weight of sodium is 23.) In the series
expansion for c e , show that at this temperature the cubic term is negligible compared
with the linear term.
26
N
2.6 1028
V
6.63 10
28
3 2.6 10
F
2 9.11 1031
8
5.6eV 0.579
3.243eV
34 2
2
3
k 298
Ce Nk
R 0.008
2 3.243 2
325J kilomole1k 1
2
19.10) calculate the isothermal compressibility of the fermion gas consisting of the free electrons
in silver. Compare your answer with the experimental value for silver of
0.99 1011 Pa 1
1 2v
V 2 p T
from eq 19.24
P
2 NkTF
5 V
2
h2 N 3 1
TF
2mk 1.504 V
2
2
3
eq 19.8
2
h2 N 3 1
P Nk
5
2mk 1.504 V
2
Nh 2 N 3 1
P
5
5m 1.504 V 3
5
3
2
V
5
3
Nh 2 N 3 1
5m 1.504
p
3
Nh 2 N 3 5 1
V
35
5
m
1
.
504
p
2
3
5
dv Nh 2 N 3
3 58
P
dp 5m 1.504
5
2
3
5
1 Nh 2 N 3
3 58
P
v 5m 1.504
5
3
this equation leads to P with P 2.1 1010 Pa from textbook
5
3
3
P 2.1 1010 Pa 1.26 1011 Pa 1
5
5
2
19.12) For the white dwarf star Sirius B, do the computation leading to Rmin 7 10 6 m
Use Equation (19.28) and (19.30) with the appropriate parameter value.
Rmin 7 10 6 m
We knowthat
2a
19.32
b
calculatin g b
Rmin
3
b GM 2 19.30
5
G Gravitational energy 6.673 10 11
M Mass of the sun 2.09 10 30 kg
3
b 6.673 10 11 2.09 10 30
5
b 1.75 10 50
2
calculatin g a
2
3h 2 9 3 5 3
a
N
10me 32 2
calculatin g N
5 1.82 10 22 7.23 10 20
6.10 10 56
2
5.33 10 14
so value of a is
N
2
2
3 6.626 10 34 9 3
a
6.10 10 56
31
2
10 9.1 10
32
a 1.45 10 37 0.0933 4.388 10 94
a 5.92 10 56
put values of a and b in 19.32
Rmin
2 5.92 10 56
1.75 10 50
6.77 10 6 7 10 6
Rmin
5
3
19.13) consider the collapse of the sun into a white dwarf. For the sun, M= 2 10
R= 7 10
8
30
kg
m , V=1.4 1027 m3
a) Calculate the Fermi energy of the sun’s electrons
No. of electrons
2 10 30
1.205 10 57
1.66 10 27
1
of nucleous
2
N 1.205 2 10 57
V
1.4 10 27
# of electrons
h2
F
2me
3N
8V
2
3
1.205 7.23 10 27
F 0.33me v
27
1
.
26
1
.
4
10
5
F 0.33me v 6.248 10
2
3
F 20.6eV
b) What is the Fermi Temperature?
TF
F
k
20.6eV
2.39 105 k
5
1
8.62 10 eVk
c) What is the average speed of the electrons in the fermion gas (see problem 19-4). Compare
your answer with the speed of light.
N
1.205 2 1057
31
Density 9.11 10 9.11 10
V
1.4 1027
0.39 kg 3
m
31
0
