thermo & stat mech - spring 2006 class 20 1 thermodynamics and statistical mechanics heat...
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Thermo & Stat Mech - Spring 2006 Class 20
1
Thermodynamics and Statistical Mechanics
Heat Capacities
Thermo & Stat Mech - Spring 2006 Class 20
2
Diatomic Gas
A gas of diatomic molecules can have translational, rotational and vibrational energy. Because of the spacing of the energy levels of each type of motion, they are not all equally excited. This shows up in the heat capacity.
VV T
UC
Thermo & Stat Mech - Spring 2006 Class 20
3
Partition Function
For one molecule, = trans + rot + vib
vibrottrans
vibrottrans
jj
jj
ZZZZ
egegegZ
egegZ
vibrottrans
vibrottransj
)(
Thermo & Stat Mech - Spring 2006 Class 20
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Partition Function
vibrottrans
V
vib
V
rot
V
trans
V
vibrottrans
vibrottrans
UUUU
ZZZ
N
U
Z
N
U
ZZZZ
ZZZZ
lnlnln
ln
lnlnlnln
Thermo & Stat Mech - Spring 2006 Class 20
5
Translational Motion
NkC
NkTU
h
mkTVZ
transV
trans
trans
2
32
3
2
,
2/3
2
Thermo & Stat Mech - Spring 2006 Class 20
6
Vibrational Motion
2
2
2
2
,
11
1
1
2
1
1
1
2
1
1
21
kT
h
kT
h
kT
h
kT
h
V
vibvibV
kT
hhvib
h
h
vib
e
ekTh
Nk
e
ekTh
NhT
UC
e
Nhe
NhU
e
eZ
Thermo & Stat Mech - Spring 2006 Class 20
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High and Low Temperature Limits
kT
h
vibV
vibV
kT
h
kT
h
vibV
ekT
hNkChkT
Nk
kTh
kTh
kTh
NkChkT
e
ekTh
NkC
2
,
2
2
,
2
2
,
11
1
1
Thermo & Stat Mech - Spring 2006 Class 20
8
Rotational Motion
NkTUkTZ
N
U
Ie
IdxeZ
dxlxllI
kT
elZllI
rot
V
rot
xI
xI
rot
l
llIkT
rotl
1ln
22
12)1(let ,2
For
)12()1(2
2
0
220
2
2
)1(2
2
22
2
Thermo & Stat Mech - Spring 2006 Class 20
9
Rotational Motion
0
02
For
2For
,
2
,
2
rotV
rot
rotV
rot
C
UI
kT
NkC
NkTUI
kT
Thermo & Stat Mech - Spring 2006 Class 20
10
Diatomic Gas Overall
NkChkT
NkChkTI
NkCI
kT
Ih
V
V
V
2
7
2
5
2
2
3
2
2
2
2
2
Thermo & Stat Mech - Spring 2006 Class 20
11
Graph
Thermo & Stat Mech - Spring 2006 Class 20
12
Einstein Solid
kT
h
VV
kT
h
kT
h
NVV
kT
hh
ekT
hNkChvkTNkChvkT
e
ekTh
NkT
UC
e
Nhe
NhU
2
2
2
,
3:3:
1
3
1
1
2
13
1
1
2
13
Thermo & Stat Mech - Spring 2006 Class 20
13
Einstein Temperature
TEVEVE
EE
T
TE
kT
h
kT
h
NVV
E
E
E
eT
NkCTNkCT
k
hvkhv
e
eT
Nk
e
ekTh
NkT
UC
2
2
2
2
2
,
3:3:
1
3
1
3
Thermo & Stat Mech - Spring 2006 Class 20
14
Einstein Heat Capacity
Heat Capacity
-5
0
5
10
15
20
25
30
0 100 200 300 400
T (K)
Cv
(J/K
)
Thermo & Stat Mech - Spring 2006 Class 20
15
Debye Model of Solid
The solid is treated as a continuum, calculating the number of standing wave states in the frequency range between and + d , and Bose-Einstein statistics is used to determine the number of phonons in each state. Then the energy can be calculated.
Thermo & Stat Mech - Spring 2006 Class 20
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Density of States (Lec 17)
dc
Vdg
cddk
ck
dkkV
dkkg
232
22
2)(
2)(
Thermo & Stat Mech - Spring 2006 Class 20
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Transverse and Longitudinal
2332
232
232
12
2)()()(
alLongitudin2
)(
Transverse2
2)(
ltlt
ll
tt
cc
Vggg
c
Vg
c
Vg
Thermo & Stat Mech - Spring 2006 Class 20
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Density of States
dc
Vdg
ccc
cc
Vg
lt
lt
232
333
2332
2
3)(
312 :Define
12
2)(
Thermo & Stat Mech - Spring 2006 Class 20
19
Phonon Energy
D
D
dgN
dgfU
dc
V
edgfdU
D
0
0
223
)(3
frequency cutoff Debye
)()(
2
3
1
1)()(
Thermo & Stat Mech - Spring 2006 Class 20
20
Debye Cutoff
23
232
3/1
2
3/1
32
3
320
232
0
9
2
3)( :convenient More
66
32
3
2
3)(3
D
D
D
N
c
Vg
V
Nc
V
Nc
c
Vd
c
VdgN
DD
Thermo & Stat Mech - Spring 2006 Class 20
21
Debye Temperature
dk
Nd
Ndg
kk
DD
DDDD
233
32
3
99)(
Thermo & Stat Mech - Spring 2006 Class 20
22
Energy
dkT
e
e
k
N
e
d
Tk
N
T
UC
e
d
k
NdgfU
D
D
DD
kT
kT
kTV
V
kT
20
2
3
33
4
0
3
33
4
0
3
33
4
0
1
9
1
9
1
9)()(
Thermo & Stat Mech - Spring 2006 Class 20
23
Heat Capacity
dxe
exTNk
kTd
e
ekTT
NkC
TkTxd
kTe
e
k
NC
T
x
x
DkT
kT
DV
DDD
kT
kT
DV
DD
D
/
02
43
02
4
3
20
2
3
33
4
19
1
9
1
9
Thermo & Stat Mech - Spring 2006 Class 20
24
Debye Heat Capacity
Debye Heat Capacity
0
5
10
15
20
25
30
0 100 200 300 400
T (K)
Cv
(J/K
)
Thermo & Stat Mech - Spring 2006 Class 20
25
High Temperature
NkC
T
TNkdxx
TNkC
xexT
V
D
D
T
DV
xD
3
3
199
11For 33/
0
2
3
Thermo & Stat Mech - Spring 2006 Class 20
26
Low Temperature
34
43
02
43
5
12
15
49
19
For
DV
Dx
x
DV
DD
TNkC
TNkdx
e
exTNkC
TT