the heat capacity of a diatomic gas. 15.1 introduction statistical thermodynamics provides deep...
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The Heat Capacity of a Diatomic Gas
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15.1 Introduction• Statistical thermodynamics provides deep
insight into the classical description of a MONATOMIC ideal gas.
• In classical thermodynamics, the principle of equipartition of energy fails to give the observed value of the specific heat capacity for diatomic gases.
• The explanation of the above discrepancy was considered to be the most important challenge in statistical theory.
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15.1 The quantized linear oscillator• A linear oscillator is a particle constrained to move
along a straight line & acted on by a restoring force
F= -Kx = ma =
If displaced from its equilibrium position and released, the particle oscillates with simple harmonic motion of frequency , given by
Note that the frequency depends on K and m, and is
independent of the amplitude X.
m
Kv
2
1
2
2
td
xdm
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• Consider an assembly of N one-dimensional harmonic oscillators, in which the oscillators are loosely coupled so that the energy exchange among them is small.
• In classical mechanics, a particle can oscillate with any amplitude and energy.
• From quantum mechanics, the single particle energy levels are given by
EJ = (J + ½)h , J = 0, 1, 2, ….. • The energies are equally spaced and the ground state
has non-zero energy.
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• The internal degrees of freedom include vibrations, rotations, and electronic excitations.
• For internal degrees of freedom, Boltzmann Statistics applies. The distinguishable property arises from the fact that those diatomic molecules have different translational energy.
• The states are nondegenerate, i.e. gj = 1
• The partition function of an oscillator
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• Introducing the characteristic temperature θ = h/k
• The solution for the above eq. is (in class derivation)
......132
2 TTTT eeee
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• The internal energy is:
U = NkT2
since
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For T → 0
For
thus
1
1
2
1
12
1
TT
T
e
Nk
e
eNkU
0
1
1
Te
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15.3Vibrational Modes of Diatomic Molecules
• The most important application of the above result is to the molecules of a diatomic gas
• From classical thermodynamics
for a reversible process!
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Since
Or
At high temperatures
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At low temperature limit
On has
So
approaching zero faster than the growth of (θ/T)2
as T → 0
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Therefore Cv 0 as T 0
The total energy of a diatomic molecule is made up of four contributions that can be separately treated:
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1. The kinetic energy associated with the translational motion
2. The vibrational motion
3. Rotation motion (To be discussed later)
Example: 15.1 a) Calculate the fractional number of oscillators in the three lowest quantum states (j=0, 1, 2,) for
Sol:
forTandT 4
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J = 0
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• 15.2) a) For a system of localized distinguishable oscillators, Boltzmann statistics applies. Show that the entropy S is given by
• Solution: according to Boltzmann statistics
So
N
NNS J
JJ ln
J
NJ
n
J N
gNW
J
1!
1ln wkS
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!ln!lnln
!ln1ln!lnln
01ln1
lnln!lnln
NNw
NNNw
g
NgNNw
JJ
J
JJJ
JJ
JJ
JJJ
J
JJ
JJ
w
w
w
w
w
lnln
lnlnln
lnlnln
lnlnln
1lnlnln
n
J
jj
jj
S
S
1
ln
ln
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15.4 Rotational modes of diatomic molecules
The moment of inertia,where μ is the reduced mass
r0 is the equilibrium value of the distance between the nuclei.
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From quantum mechanics, the allowed angular momentum states are
where l = 0, 1, 2, 3… …
From classical mechanics, the rotational energy equals with w is angular velocity.
The angular momentum
therefore, the energy
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Define a characteristic temperature for rotation
θrot can be found from infrared spectroscopy experiments, in which the energies required to excite the molecules to higher rotational states are measured.
Different from vibrational motion, the energy levels of the above equation are degenerate.
for level
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Now, one can get the partition function
For , virtually all the molecules are in the few lowest rotational states.
As a result, the series of can be truncated with negligible errors after the first two or three terms!
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For all diatomic gases, except hydrogen, the rotational characteristics temperature is of the order of 10 k (Kelvin degree).
At ordinary temperature,
Therefore, many closely spaced energy states are excited. The sum of may be replaced by an integral.
Define:
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Note that the above result is too large for homonuclear molecules such as H2, O2 and N2 by a factor of 2… why?
The slight modification has no effect on the thermodynamics properties of the system such as the internal energy and the heat capacity!
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Using
(Note: )
again
At low temperature
Keeping the first two terms
VT
ZNkTU
ln2
NkT
UC
vrotV
,
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Using the relationship
(for )
And
(for )
2
222 23
ln
TeNkT
T
ZNkTU rotT
V
rot
2
2
, 26T
eNkT
UC rotT
rotV
rotV
rot
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Characteristic TemperaturesCharacteristic temperature of rotation of diatomic molecules
Substance θrot(K)
H2 85.4
O2 2.1
N2 2.9
HCl 15.2
CO 2.8
NO 2.4
Cl2 0.36
Characteristic temperature of vibration of diatomic molecules
Substance θvib(K)
H2 6140
O2 2239
N2 3352
HCl 4150
CO 3080
NO 2690
Cl2 810
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15.5 Electronic Excitation
The electronic partition function is
where g0 and g1, are, respectively, the degeneracies of the ground state and the first excited state.
E1 is the energy separation of the two lowest states.
Introducing
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For most gases, the higher electronic states are not excited (θe ~ 120, 000k for hydrogen).
therefore,
At practical temperature, electronic excitation makes no contribution to the external energy or heat capacity!
0ln2
VT
ZNkTU
0
V
V T
UC
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15.6 The total heat capacity
For a diatomic molecule system
Since
Discussing the relationship of T and Cv (p. 288-289)
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Heat capacity for diatomic molecules
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• Example I (problem 15.7) Consider a diatomic gas near room temperature. Show that the entropy is
• Solution: For diatomic molecules
,2
2ln
2
7 25
23
2
rot
T
h
mkNkS
)!(0
1ln1
121
0
contributenotdoes
e
eNk
Te
Nk
S
S
etemperaturroomAt
SSSSS
T
TT
vib
excit
excitrotvibtrans
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• For translational motion, the molecules are treated as non-distinguishable assemblies
23
2
23
2
2ln
2
5
1lnln
2
freedomofdegreesthree2
3
h
mkT
N
VNkNkS
NZNkT
US
h
mkTVZ
NkTU
tr
tr
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• For rotational motion (they are distinguishable in terms of kinetic energy)
rotsystem
rotsystem
rot
rot
T
h
mk
N
VNkNkS
TNkNk
h
mkT
N
VNkNkS
TNkNk
ZNkT
US
TZNkTU
2
2ln
2
7
2ln
2ln
2
5
2ln
assemblyhabledistinguisforln
moleculerhomonucleatodue2
1
2
252/3
2
23
2
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• Example II (problem 15-8) For a kilomole of nitrogen (N2) at standard temperature and pressure, compute (a) the internal energy U; (b) the Helmholtz function F; and (c) the entropy S.
• Solution:
calculate the characteristic temperature first!
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29.23352 Nforkandk
rvib
J
kkmolJkmol
eNk
eNk
eNkNkTU
Tbecause
NkTe
NkNkTU
UUUU
vib
r
Tvib
rotvibtrans
7
113
25.11
25.11
2983352
1089.1
227210314.80.1
1
33522272
1
33523352
2
12985.2
1
1
2
1
2
5
1
1
2
1
2
3
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4.16
11015.5ln11002.6
10195.3ln
10195.3
103.274.22
1086.2734.22
10626.6
29810381.11065.424.22
2
1lnln
&ln
1lnln
626
33
33
23
2023
23
19
23
682
123263
23
2
NkT
NkTNkTF
Jkm
h
mkTVZ
NZNkTF
rotationvibassuchparticleshableDistinguisZNkTF
particleshabledistinguisnonNZNkTF
t
t
tt
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J
NkT
NkTNkTNkT
FFFF
NkTF
ZZ
TforT
ZZNkTF
eZ
e
e
e
eZ
ZNkTF
tvibrot
rot
rotrot
rotrot
rotrotrot
vib
vib
vibvib
72326
24.11
24.11
62.5
2983352
29823352
10817.32981038.11002.641.15
41.15
4.1662.563.4
63.4
63.4ln72.1029.2
298
2ln
62.51
1ln62.5ln
11
ln