Ideal diatomic gas: internal degrees
of freedom
• Polyatomic species can store energy in
a variety of ways:
– translational motion
– rotational motion
– vibrational motion
– electronic excitation
Each of these modes has its own manifold
of energy states, how do we cope?
1
Internal modes: separability of energies
• Assume molecular modes are separable
– treat each mode independent of all others
– i.e. translational independent of vibrational,
rotational, electronic, etc, etc
Entirely true for translational modes
Vibrational modes are independent of:
– rotational modes under the rigid rotor
assumption
– electronic modes under the BornOppenheimer approximation
2
Internal modes: separability of energies
Thus, a molecule that is moving at high speed is
not forced to vibrate rapidly or rotate very fast.
An isolated molecule which has an excess of any
one energy mode cannot divest itself of this
surplus except at collision with another molecule.
The number of collisions needed to equilibrate
modes varies from a few (ten or so) for rotation,
to many (hundreds) for vibration.
3
Internal modes: separability of energies
Thus, the total energy of a molecule j:
  
j
tot
j
trs
j
rot

j
vib

j
el
4
Weak coupling: factorising the energy modes
• Admits there is some energy interchange
– in order to establish and maintain thermal equilibrium
• But allows us to assess each energy mode as if it
were the only form of energy present in the
molecule
• Molecular partition function can be formulated
separately for each energy mode (degree of
freedom)
• Decide later how individual partition functions
should be combined together to form the overall
molecular partition function
5
Weak coupling: factorising the energy modes
• Imagine an assembly of N particles that can store
energy in just two weakly coupled modes a and w
• Each mode has its own manifold of energy states
and associated quantum numbers
• A given particle can have:
- a-mode energy associated with quantum number k
- w-mode energy associated with quantum number r
 tot   ak   wr
6
Weak coupling: factorising the energy modes
The overall partition function, qtot:
qtot 
e
  ( a i   w r )
all states
expanding we would get:
qtot  e   ( a 0   w 0 )  e   ( a 0   w 1 )  e   ( a 0   w 2 )  e   ( a 0   w 3 )
e
  ( a 1   w 0 )
e
  ( a 2   w 0 )
e
  ( a 1   w 1 )
e
  ( a 2   w 1 )
e
  ( a 1   w 2 )
e
  ( a 2   w 2 )
e
  ( a 1   w 3 )
e
  ( a 2   w 3 )
 e   ( a 3   w 0 )  e   ( a 3   w 1 )  e   ( a 3   w 2 )  e   ( a 3   w 3 )
 ...
7
Weak coupling: factorising the energy modes
but e(a+b) = ea.eb, therefore:
qtot  e
e
  a 0
  a 1
.e
.e
  w 0
  w 0
e
e
  a 0
  a 1
.e
.e
  w 1
  w 1
e
e
  a 0
  a 1
.e
.e
  w 2
  w 2
 ...
 ...
 e   a 2 .e   w 0  e   a 2 .e   w 1  e   a 2 .e   w 2  ...
 ...
each term in every row has a common factor of
  a 0
  a 1
in the first row, e
in the second, and
so on. Extracting these factors row by row:
e
8
Weak coupling: factorising the energy modes
qtot  e
  a 0
e
  a 1
e
  a 2
(e
(e
  w 0
  w 0
(e
  w 0
e
  w 1
e
  w 1
e
  w 1
e
  w 2
e
  w 2
e
  w 2
e
  w 3
 ...)
e
  w 3
 ...)
  w 3
 ...)
e
 e   a 3 (e   w 0  e   w 1  e   w 2  e   w 3  ...)
 ...
the terms in parentheses in each row are identical
and form the summation:
all
e

w
  wj
states
9
Weak coupling: factorising the energy modes

qtot  e
  a 0
e
  a 1

all
e
  a 2
e

a
  aj
states
e
  a 3
x
all

  wj 

 ...   e

 allw states 
e

w

  wj
states
If energy modes are separable then we can
factorise the partition function and write:
qtot  qa x qw
10
Factorising translational energy modes
Total translational energy of molecule j:

j
trs, tot

j
trs, x

j
trs, y

j
trs, z
which allows us to write:
qtrs 
e
  trs , tot
all states
qtrs 
e
  trs , x
all x states

e
  (  trs , x   trs , y   trs , z )
all states
x
e
  trs , y
all y states
e
x
  trs , z
all z states
qtrs  qtrs, x x qtrs, y x qtrs, z
11
Factorising internal energy modes
Total translational energy of molecule j:
qtot  qtrs . qrot .qvib .qel
using identical arguments the canonical partition
function can be expressed:
Qtot  Qtrs . Qrot . Qvib .Qel
but how do we obtain the canonical from the
molecular partition function Qtot from qtot? How
does indistinguishability exert its influence?
12
Factorising internal energy modes
When are particles distinguishable (having
distinct configurations, and when are they
indistinguishable?
• Localised particles (unique addresses) are always
distinguishable
• Particles that are not localised are indistinguishable
– Swapping translational energy states between such
particles does not create distinct new configurations
• However, localisation within a molecule can also
confer distinguishability
13
Factorising internal energy modes
When molecules i and j, each in distinct rotational and
vibrational states, swap these internal states with
each other a new configuration is created and both
configurations have to be counted into the final sum
of states for the whole system. By being identified
specifically with individual molecules, the internal
states are recognised as being intrinsically
distinguishable.
Translational states are intrinsically indistinguishable.
14
Canonical partition function, Q

qtrs 
N
N
N
qrot  qvib  qel 
Qtot 
N
and thus:
N!
1
N
Qtot  qtrs .qrot .qvib .qel 
N!
This conclusion assumes weak coupling. If
particles enjoy strong coupling (e.g. in liquids
and solutions) the argument becomes very
complicated!
15
Ideal diatomic gas: Rotational
partition function
Assume rigid rotor for which we can write
successive rotational energy levels, J, in terms
of the rotational quantum number, J.
EJ 
h2
J ( J  1) joules
8 I
EJ
h
1
J 
 2 J ( J  1) cm
hc 8 Ic
1
 BJ ( J  1) cm
2
where I is the moment of inertia of the
molecule, m is the reduced mass, and B the
rotational constant.
16
Ideal diatomic gas: Rotational
partition function
Another expression results from using the
characteristic rotational temperature, qr,
h2
hcB
qr  2 
 kq r  hcB
8 Ik
k
E J  J ( J  1)kq r ( joules )
• 1st energy increment = 2kqr
• 2nd energy increment = 4kqr
17
Ideal diatomic gas: Rotational
partition function
Rotational energy levels are degenerate and
each level has a degeneracy gJ = (2J+1). So:
qrot   g J e
 J / kT
  (2 J  1)e
 J ( J 1)q r / T
If no atoms in the atom are too light (i.e. if the
moment of inertia is not too small) and if the
temperature is not too low (close to 0 K), allowing
appreciable numbers of rotational states to be
occupied, the rotational energy levels lie
sufficiently close to one another to write:
18
Ideal diatomic gas: Rotational partition
function

qrot   (2 J  1)e
 J ( J 1)q r / T dJ
0
 qrot
8 2 IkT
 
qr
h2
T
• This equation works well for heteronuclear
diatomic molecules.
• For homonuclear diatomics this equation
overcounts the rotational states by a factor of
two.
19
Ideal diatomic gas: Rotational partition
function
• When a symmetrical linear molecule rotates
through 180o it produces a configuration which is
indistinguishable from the one from which it
started.
– all homonuclear diatomics
– symmetrical linear molecules (e.g. CO2, C2H2)
• Include all molecules using a symmetry factor s
qrot 
T
sqr
s = 2 for homonuclear diatomics, s = 1 for heteronuclear diatomics
s = 2 for H2O, s = 3 for NH3, s = 12 for CH4 and C6H6
20
Rotational properties of molecules at 300 K
qr/K
H2
CH4
HCl
HI
N2
CO
CO2
I2
88
15
9.4
7.5
2.9
2.8
0.56
0.054
s
T/qr
qrot
2
12
1
1
2
1
2
2
3.4
20
32
40
100
110
540
5600
1.7
1.7
32
40
50
110
270
2800
21
Rotational canonical partition function
Qrot  q
N
rot
relates the canonical partition function to the
molecular partition function. Consequently, for the
rotational canonical partition function we have:
N
Qrot
 T 
 8 IkT 
 T 


 

2

shcB 
 sh 
sqr 
N
2
N
22
Rotational Energy
 8 2 Ik 

ln Qrot  N ln T  N ln 
2 
 sh 
this can differentiated wrt temperature, since the
second term is a constant with no T dependence
  ln Qrot 

2 
U rot  kT 
ln T 
  NkT 
 T

 T V
 U rot  NkT (for diatomic molecules)
2
23
Rotational heat capacity
U rot  NkT
(for diatomic molecules)
this equation applies equally to all linear molecules
which have only two degrees of freedom in rotation.
Recast for one mole of substance and taking the T
derivative yields the molar rotational heat capacity,
Crot, m. Thus, when N = NA, the molar rotational
energy is Urot,m
U rot, m  RT
Crot,m  R (linear molecules)
24
Rotational entropy
S rot
U rot
  ln Q 
 kT 
 k ln Q
  k ln Q 
T
 T V
 8 IkT 
NkT


 k ln 
2
T
 sh 
2
 S rot
N

 8 2 k 
IT
 Nk 1  ln
 ln  2 
s
 h 

Srot is dependent on (reduced) mass (I = mr2), and
there is also a constant in the final term, leading to:



Srot / R  ln I / kg m T / K s  106.53
2
1
25
Rotational entropy
Typically, qrot at room T is of the order of hundreds
for diatomics such as CO and Cl2. Compare this with
the almost immeasurably larger value that the
translational partition function reaches.
qrot  10
2
but qtrs  10
28
26
Extension to polyatomic molecules
• In the most general case, that of a non-linear
polyatomic molecule, there are three independent
moments of inertia.
• Qrot must take account of these three moments
– Achieved by recognising three independent characteristic
rotational temperatures qr, x, qr, y, qr, z corresponding to the
three principal moments of inertia Ix, Iy, Iz
• With resulting partition function:
qrot


s
 T  T  T







q
q
 r , x  r , y  q r , z




1
2
27
Conclusions
• Rotational energy levels, although more widely
spaced than translational energy levels, are still
close enough at most temperatures to allow us to
use the continuum approximation and to replace
the summation of qrot with an integration.
• Providing proper regard is then paid to rotational
indistinguishability, by considering symmetry,
rotational thermodynamic functions can be
calculated.
28
Ideal diatomic gas: Vibrational partition
function
Vibrational modes have energy level spacings that
are larger by at least an order of magnitude than
those in rotational modes, which in turn, are 25—
30 orders of magnitude larger than translational
modes.
– cannot be simplified using the continuum approximation
– do not undergo appreciable excitation at room Temp.
– at 300 K Qvib ≈ 1 for light molecules
29
The diatomic SHO model
We start by modelling a diatomic molecule on a
simple ball and spring basis with two atoms, mass
m1 and m2, joined by a spring which has a force
constant k.
The classical vibrational frequency, wosc, is given
by:
wosc
1

2
k
m
Hz
There is a quantum restriction on the available energies:
 vib
1

  v  hwosc
2

( v  0, 1, 2, ...)
30
The diatomic SHO model
1
The value hwosc is know as the zero point energy
2
• Vibrational energy levels in diatomic molecules
are always non-degenerate.
• Degeneracy has to be considered for polyatomic
species
– Linear: 3N-5 normal modes of vibration
– Non-linear: 3N-6 normal modes of vibration
31
Vibrational partition function, qvib
• Set 0 = 0, the ground vibrational state as the
reference zero for vibrational energy.
• Measure all other energies relative to reference
ignoring the zero-point energy.
– in calculating values of some vibrational thermodynamic
functions (e.g. the vibrational contribution to the
internal energy, U) the sum of the individual zero-point
energies of all normal modes present must be added
32
Vibrational partition function, qvib
The assumption (0 = 0) allows us to write:
1  hw ,  2  2hw ,  3  3hw ,  4  4hw , ...
Under this assumption, qvib may be written as:
qvib   e
  vib
 1 e
 hw
e
 2 hw
e
3 hw
 ...
a simple geometric series which yields qvib in closed
form:
qvib
1
1


 hw
1 e
1  e q vib / T
where qvib = hw/k = characteristic vibrational
33
temperature
Vibrational partition function, qvib
• Unlike the situation for rotation, qvib, can be
identified with an actual separation between
quantised energy levels.
• To a very good approximation, since the
anharmonicity correction can be neglected for low
quantum numbers, the characteristic temperature
is characteristic of the gap between the lowest and
first excited vibrational states, and with exactly
twice the zero-point energy, 1 hwosc .
2
34
Ideal diatomic gas: Vibrational
partition function
Vibrational energy level
spacings are much larger
Species
than those for rotation, so H
2
typical vibrational
HD
temperatures in diatomic
D2
molecules are of the order
N2
of hundreds to thousands
of kelvins rather than the CO
Cl2
tens of hundreds
characteristic of rotation. I2
qvib/K
qvib
(@ 300 K)
5987
1.000
5226
1.000
4307
1.000
3352
1.000
3084
1.000
798
1.075
307
1.556
35
Vibrational partition function, qvib
• Light diatomic molecules have:
– high force constants
– low reduced masses
wosc
1

2
k
m
• Thus:
– vibrational frequencies (wosc) and characteristic
vibrational temperatures (qvib) are high
– just one vibrational state (the ground state) accessible at
room T
• the vibrational partition function qvib ≈ 1
36
Vibrational partition function, qvib
• Heavy diatomic molecules have:
– rather loose vibrations
– Lower characteristic temperature
• Thus:
– appreciable vibrational excitation resulting in:
• population of the first (and to a slight extent higher)
excited vibrational energy state
• qvib > 1
37
Vibrational partition function, qvib
• Situation in polyatomic
species is similar
complicated only by the
existence of 3N-5 or 3N-6
normal modes of vibration.
Species
CO2
1.091
954(2)
NH3
4880(2)
1.001
4780
2330(2)
1360
CHCl3
tot
( n)
(1)
( 2)
( 3)
qvib
  qvib
 qvib
x qvib
x qvib
x ...
(1), (2), (3), … denoting individual normal
modes 1, 2, 3, …etc.
3360
1890
• Some of these normal modes
are degenerate
 
qvib/K
∏(qvib)
(@ 300 K)
4330
2.650
1745(2)
1090(2)
938
523
374(2)
38
Vibrational partition function, qvib
As with diatomics, only the heavier species show
values of qvib appreciably different from unity.
Typically, qvib is of the order of ~3000 K in many
molecules. Consequently, at 300 K we have:
qvib
1

1
10
1 e
in contrast with qrot (≈ 10) and qtrs (≈ 1030)
For most molecules only the ground state is
accessible for vibration
39
High T limiting behaviour of qvib
1

1  e q vib / T
At high temperature the equation qvib
gives a linear dependence of qvib with temperature.
If we expand 1  e
qvib 
q vib / T
, we get:
1
1  1  (q vib / T )  ...

T
q vib
High T limit
40
T dependence of vibrational partition
function
2.0
1.8
As T increases, the
linear dependence of
qvib upon T becomes
increasingly obvious
qvib
1.6
1.4
1.2
1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Reduced Temperature T/q
1.4
41
The canonical partition function, Qvib
Qvib  q
N
vib
1



q vib / T 
1 e

N
Using U  kT 2   ln Q  we can find the first
 T V
differential of lnQ with respect to temperature
to give:
U vib
Nkq vib
  ln Q 
 kT 
  qvib / T
e
1
 T V
2


42
The vibrational energy, Uvib
U vib, m
Rq vib
 q vib / T
(e
 1)
This is not nearly as simple as:
3
U trs, m  kT
2
U rot, m  RT linear molecules
43
The vibrational energy, Uvib
U vib, m
Rq vib
 q vib / T
(e
 1)
This does reduce to the simple form at
equipartition (at very high temperatures) to:
U vib, m  RT
Normally, at
room T:
U vib, m
(equipartition)
3000 R 1
 10
 R
(e  1) 7
44
The zero-point energy
• So far we have chosen the zero-point energy
(1/2hw) as the zero reference of our energy scale
• Thus we must add 1/2hw to each term in the
energy ladder
• For each particle we must add this same amount
– Thus, for N particles we must add U(0)vib, m = 1/2Nhw
U vib, m
Rq vib
 q vib / T
 U (0) vib, m
(e
 1)
Rq vib
1
 q vib / T
 N A hw
(e
 1) 2
45
Vibrational heat capacity, Cvib
The vibrational heat capacity can be found using:
q vib / T
 U vib, m 
e
 q vib 
  R
 
 q vib / T
2
 1)
 T  (e
 T V
2
Cvib, m
The Einstein Equation
This equation can be written in a more compact
form as:
Cvib, m
 q vib 
 RF E 

 T 
46
Vibrational heat capacity, Cvib
FE with the argument qvib/T is the Einstein function
2 u
ue
FE  u
2
(e  1)
q vib 

u 

T 

The Einstein function
47
The Einstein heat capacity
1.0
FE
0.5
low T
High T
0.0
0.1
1
10
Reduced temperature T/q
48
The Einstein function
• The Einstein function has applications beyond
normal modes of vibration in gas molecules.
• It has an important place in the understanding
of lattice vibrations on the thermal behaviour
of solids
• It is central to one of the earliest models for
the heat capacity of solids
49
The vibrational entropy, Svib
S vib
U vib  U vib (0) Avib  Avib (0)


T
T
U vib  U vib (0)

 k ln Qvib
T
N and N = N for one mole,
• We know Qvib  qvib
A
thus:
ln Qvib  N A k ln qvib  R ln qvib

S vib, m
R

q vib / T
q vib / T
(e
 1)

 ln 1  e
q vib / T

50
Variation of vibrational entropy with
reduced temperature
3.0
2.5
Svib/R
2.0
1.5
1.0
0.5
T
0 T
0.0
0.1
1
10
Reduced temperature T/q
51
Electronic partition function
• Characteristic electronic temperatures, qel,
are of the order of several tens of thousands
of kelvins.
• Excited electronic states remain unpopulated
unless the temperature reaches several
thousands of kelvins.
• Only the first (ground state) term of the
electronic partition function need ever be
considered at temperatures in the range
from ambient to moderately high.
52
Electronic partition function
It is tempting to decide that qel will not be a
significant factor. Once we assign 0 = 0, we
might conclude that:
qel   e
  el , i / kT
0
 e  0 (higher terms)  1
i
To do so would be unwise!
One must consider degeneracy of the ground
electronic state.
53
Electronic partition function
The correct expression to use in place of the
previous expression is of course:
qel   g i e
  el , i / kT
 g 0 e 0  0 (higher terms)  g 0
i
Most molecules and stable ions have nondegenerate ground states.
A notable exception is molecular oxygen, O2,
which has a ground state degeneracy of 3.
54
Electronic partition function
Atoms frequently have ground states that are
degenerate.
Degeneracy of electronic states determined
by the value of the total angular momentum
quantum number, J.
Taking the symbol G as the general term in the
Russell—Saunders spin-orbit coupling
approximation, we denote the spectroscopic
state of the ground state of an atom as:
spectroscopic atom ground state =
(2S+1)G
J
55
Electronic partition function
spectroscopic atom ground state = (2S+1)GJ
where S is the total spin angular momentum
quantum number which gives rise to the term
multiplicity (2S+1). The degeneracy, g0, of the
electronic ground states in atoms is related to
J through:
g0 = 2J+1 (atoms)
56
Electronic partition function
For diatomic molecules the term symbols are
made up in much the same way as for atoms.
• Total orbital angular momentum about the
inter-nuclear axis.
Determines the term symbol used for the
molecule (S, P, D, etc. corresponding to S, P, D,
etc. in atoms).
As with atoms, the term multiplicity (2S+1) is
added as a superscript to denote the
multiplicity of the molecular term.
57
Electronic partition function
In the case of molecules it is this term
multiplicity that represents the degeneracy of
the electronic state.
For diatomic molecules we have:
spectroscopic molecular ground state = (2S+1)G
for which the ground-state degeneracy is:
g0 = 2S + 1 (molecules)
58
Electronic partition function
Species
Li
C
N
Term
Symbol
2S
1/2
3P
0
4S
3/2
gn
g0 = 2
g0 = 1
g0 = 4
O
3P
2
g0 = 5
F
2P
3/2
g0 = 4
2P
1/2
g1 = 2
2P
1/2
g0 = 2
3/2
g1 = 2
NO
2P
O2
qel/K
3Sg
g0 = 3
1D
g
g1 = 1
590
178
11650
59
Electronic partition function
Where the energy gap between the ground and
the first excited electronic state is large the
electronic partition function simply takes the
value g0.
When the ground-state to first excited state
gap is not negligible compared with kT (qel/T is
not very much less than unity) it is necessary
to consider the first excited state.
The electronic partition function becomes:
qel  g 0  g1e
q el / T
60
Electronic partition function
For F atom at 1000 K we have:
qel  g 0  g1e
q el / T
 4  2e
590 / 1000
 5.109
For NO molecule at 1000 K we have:
qel  g 0  g1e
q el / T
 2  2e 178/1000  3.674
61