partition functions of ideal gases. we showed that if the # of available quantum states is >>...
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Partition functions of ideal gases
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We showed that if the # of available quantum states is >>
N
!
),(),,(
N
TVqTVNQ
N
The condition is valid when
18
2/32
Tmk
h
V
N
B
Examples gases at low densities (independent molecules)2
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Monoatomic ideal gas
eleci
vibi
roti
transi
eleci
transi
general
MIG
)(),(),( TqTVqTVq electrans
3
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Translational partition function MIG
2
2
2
2
2
22
8 c
n
b
n
a
n
m
h zyxnnn zyx
If cubic (a=b=c)
Particle in a parallelepipeda,b,c
222
2
2
8 zyxnnn nnnma
hzyx
nx, ny, nz =1,2,3…
1 1 1
222
2
2
1,,
/
8exp
),( ,,
x y z
zyx
Bznynxn
n n nzyx
nnn
Tk
trans
nnnma
h
eTVq
4
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1 1 12
22
2
22
2
22
8exp
8exp
8exp),(
x y zn n n
zyxtrans ma
nh
ma
nh
ma
nhTVq
...8
exp2
2
2
2
2
2
8
9
8
4
8
12
22
ma
h
ma
h
ma
h
n
eeema
nh
3
12
22
8exp),(
ntrans ma
nhTVq
5
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this sum cannot be expressed as a sum of a series but…
6
3
0
8 2
22
),(
dneTVq ma
nh
trans
2/1
0 4
2
dne n
Vh
TmkTVq B
trans
2/3
2
2),(
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example: average translational energy MIG
7
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Electronic partition function MIG
8
...)( 221 eggegTq l
llelec
the ground state energy is taken as the zero of energy
the other terms are negligible since they are typically in the order of 10-5 and smaller
exceptions: some of the halogens may have contributions from the first terms
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Summary MIG
9
!
)(),(),,(
N
TqTVqTVNQ
Nelectrans
Vh
TmkTVq B
trans
2/3
2
2),(
221)( eggTqelec
elecB
VB
VNB q
eNgTk
T
qTNk
T
QTkE
2222
,
2
2
3lnln
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Diatomic ideal gas
10
eleci
vibi
roti
transi
rot
qqqqTVq vibelectrans),(
!
),,(N
qqqqTVNQ
Nvibrotelectrans
Vh
TkmmTVq B
trans
2/3
221 )(2
),(
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Zeros of energy
• rotational: J=0 (rotational energy =0)• vibrational: a) the bottom of the well or
b) the ground vibrational state; in (a) the ground vibrational state is h/2
• electronic: the energy is zero when the two atoms are completely separated
11
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Dissociation energy (Do)and ground state electronic energy (De)
12
2
hDD eo
Anharmonic oscillator: HCl
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vibrational partition function DIG
13
harmonic oscillator approximation
2
1vhv
0
/)2/1()(
Tkhvib
BeTq Tkh Bex /
T
T
vib vib
vib
e
eTq /
2/
1)(
Bvib k
h
h
h
vib e
exTq
1
)(2/
0
2
1
vibrational temperature:
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average vibrational energy
14
12
ln/
2Tvibvib
Bvib
Bvib vibeNk
dT
qdTNkE
vibrational contribution to Cv
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fraction of molecules in the jth vibrational state
15
vibq
ejyprobabilit
j
)(
hh
h
hhh
eee
eee
112/
2/
see problems 3.35; 336 (we solved them last class)
most molecules are in the ground vibrational state at room T
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rotational partition function DIG
16
0
)2/()1(2
)12()(J
TIkJJrot
BeJTq (sum is over levels)
Brot Ik2
2rotational temperature
0
/)1()12()(J
TJJrot
roteJTq
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17
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rotational partition function DIG
18
0
/)1()12()(J
TJJrot
roteJTq
because the ratio rotational temperature/T is small for most molecules at ordinary Ts
dJeJTq TJJrot
rot
0
/)1()12()(
integrating
TT
dxeTq rotrot
Txrot
rot
for id val)(
0
/
much better at high T; is the high T limit
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average rotational energy
19
TNkdT
qdTNkE B
rotBrot
ln2
total rotational contribution to Cv is R; R/2 per rotational degree of freedom for a diatomic
fraction of molecules in the Jth rotational state
)(
)12( )2/()1(2
Tq
eJf
rot
TIkJJ
J
B
see problem 3.37 that we solved last class
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physical meaning of the rotational temperature
20
It gives us an estimate of the temperature at whichthe thermal energy (kT) equals the separation between rotational levels. At this T, the population of excited rotationalstates is significant.
88 K for H2, 15.2 K for HCl and 0.561 K for CO2
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21
most molecules are in the excited rotational levels at ordinary Ts
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Symmetry effects
22
TT
Tq rotrot
rot
for id val)( is for heteronuclear DIG
For homonuclear DIG TT
Tq rotrot
rot
for id val2
)(
the factor of 2 comes from the symmetry of the homonuclear molecule; 2 indistinguishable orientations
TT
Tq rotrot
rot
for id val)(
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23
molecular partition function DIG
elecvibtrans qqqqTVqrot
),(
TkDT
T
rot
B Be
vib
vib
ege
eTV
h
TMkTVq /
1/
2/2/3
2 1
2),(
restrictions:Trot
only the ground state electronic state is populatedzero (electronic) taken at the separated atomszero (vibrational) taken at the bottom of the potential well
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average energy DIG
24
V
B T
qTNkE
ln2
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average Cv DIG
25
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Vibrational partition function of a polyatomic molecule
26
...2,1,0 )2
1(
1
jj
n
jjvib
vib
h
nvib is the number of vibrational degrees of freedom3n-5 for a linear molecule3n-6 for a nonlinear moleculenormal modes are independent
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since the normal modes of a polyatomic molecule are
independent
27
vib
jvib
jvibn
jT
T
vibe
eq
1/
2/
,
,
1
vib
jvib
jvibn
jT
Tjvibjvib
Bvibe
eNkE
1/
/,,
,
,
12
vib
jvib
jvibn
jT
Tjvib
BvibV
e
e
TNkC
12/
/2
,,
,
,
1
B
jjvib k
h ,
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28
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29
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Rotational partition function of a linear polyatomic molecule
30
linear )1(8 2
2
JJIJ
J = 0, 1, 2, …
12 Jg Jdegeneracy
n
jjjdmI
1
2
mj is the distance from nucleus j to the center of mass of the molecule
TT
Tq rotrot
rot
for id val)(
is 1 for nonsymmetrical molecules (N2O, COS) and 2 for symmetrical such as CO2
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Importance of rotational motion
31
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32
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symmetry number
33
is 1 for nonsymmetrical molecules (N2O, COS) is 2 for symmetrical such as CO2
how about NH3?
symmetry number is the number of different waysin which a molecule can be rotated into a configuration indistinguishable from the original
For water, =2, successive 180o rotations about an axis through the O atom bisectingthe two H atoms result in two identical configurations
for CH4, for any axis through one of the four CH bonds there are 3 successive 120o rotationsthat result in identical configurations, therefore = 4x3 =12
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Linear polyatomic moment of inertia
34
Example HCN
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Non linear rigid polyatomic
35
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rigid non-linear polyatomic
36
3 moments of inertia
AcI
hA
28
BcI
hB
28
CcI
hC
28
if the three are equal, spherical top
only two equal, symmetric top
three different, asymmetric top
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37
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38
examples of rotational symmetry
http://www.learner.org/courses/learningmath/geometry/session7/part_b/
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examples
39
spherical top symmetrical top
Bjjrot kI2
2
,
J=A, B, C
3 rotational temperatures
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Spherical top molecules
40
0
/)1(2)12()(J
TJJrot
roteJTq
dJeJTq TJJrot
rot
0
/)1(2)12(1
)(
)1(2
2
JJIJ
Allowed energies:
J = 0, 1, 2, …2)12( Jg J
Degeneracy:
Trot for
large JTrot
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Spherical top molecules
41
dJeJTq TJJrot
rot
0
/)1(2)12(1
)(
large JTrot dJeJTq TJrot
rot
0
/2 2
41
)(
Can be solved analytically
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Rotational partition functions
42
2/32/1
)(
rot
rot
TTq
Spherical top
2/1
,,
2/1
)(
CrotArotrot
TTTq
Asymmetric top2/1
,,,
32/1
)(
CrotBrotArotrot
TTq
Symmetric top
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Average rotational energy (nonlinear polyatomic)
43
dT
TqdTNkE rot
Brot
)(ln2
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Partition function ideal gas of linear
polyatomic molecule
44
elecvibtrans qqqqTVqrot
),(
TkDN
jT
T
rot
B Be
jvib
jvib
ege
eTV
h
TMkTVq /
1
53
1/
2/2/3
2 ,
,
1
2),(
Tk
D
e
T
TTNk
U
B
eN
jT
jvibjvib
Bjvib
53
1/
,,
1
/
22
2
2
3,
53
12/
/2
,
,
,
122
2
2
3 N
jT
Tjvib
B
v
jvib
jvib
e
e
TNk
C
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45
Partition function ideal gas of nonlinear polyatomic molecule
TkDN
jT
T
CrotBrotArot
B
Be
jvib
jvib
ege
ex
TV
h
TMkTVq
/1
63
1/
2/
2/1
,,,
32/12/3
2
,
,
1
.2
),(
Obtain U and Cv
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Comparison to experiments
46
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47
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Summary• Considering the molecules that constitute a
macroscopic material, we construct q, and from q we construct Q, and from Q any thermodynamic property.
• For example, U and Cv are not just numbers in tables. We have some new insights about why different materials have different thermodynamic properties
• Next, we will discuss the laws that govern the macroscopic thermodynamic properties.
48