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Dr. Neelabh Srivastava(Assistant Professor)
Department of PhysicsMahatma Gandhi Central University
Motihari-845401, BiharE-mail: [email protected]
Programme: M.Sc. Physics
Semester: 2nd
Diatomic Molecules: Partition Functions
Lecture NotesPart - 3 (Unit - IV)
PHYS4006: Thermal and Statistical Physics
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Partition Function of a Diatomic
Molecule• Consider a diatomic molecule inside a box at
temperature T,
so the energy of the of ith microstate of this molecule can
be
expressed as –
where εtr is the translational energy of the centre of mass of
the molecule, εrotis the energy associated with the rotation of the
constituent atoms in the
molecule about the centre of mass, εvib is the energy associated
with the
vibrations of the two atoms along the line joining them, εe is
the energy of
atomic electrons and εn is the energy of the atomic nucleus.
2
)...(.......... inevibrottri
Figure: Diatomic molecule having
identical atoms
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So, the partition function of a single diatomic molecule can
be
written as –
from (i),
where Ztr, Zrot, Zvib, Ze and Zn denote the translational,
rotational,
vibrational, electronic and nuclear partition functions
respectively.
Consider a gas consisting of N molecules and each particle
is
free to move throughout the volume. For a perfect gas, as
the
particles are indistinguishable, partition function is-
where Z is single particle partition function. 3
).(..........)(
iieZstates
Di
)........(
)(
iiiZZZZZZ
eeeeeZ
nevibrottrD
states
Dnevibrottr
!N
ZZ
N
D
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Translational Motion: Consider a single diatomic molecule.
It
will have three translational degrees of freedom. Consider a
molecule of mass m enclosed in a rectangular box of sides a,
b and c with volume abc. The translational energy is –
Therefore, one particle translational partition function is
-
After solving, we get the translational partition function
4
)........(8 2
2
2
2
2
22
ivc
n
b
n
a
n
m
h zyxi
)........(8
exp8
exp8
exp1
2
22
12
22
12
22
vkTmc
nh
kTmb
nh
kTma
nhZ
zyx n
z
n
y
n
x
)........(222/3
33
2/3
vih
Vabc
hZ mkT
mkT
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So, partition function for a gas of N diatomic molecules is
–
Total translational energy of N diatomic molecules is
Rotational Motion: The energy level of a diatomic molecule
according to a rigid rotator model is given by,
where I is moment of inertia and J is rotational quantum
number.
The rotational partition function is
5
)........(!
22/3
3vii
Nh
VZ mkT
N
N
N
tr
NkTT
ZkTE trtr
2
3ln2
.......2,1,0,8
)1(2
2
JI
hJJrot
IkT
hJJgZ
J
rotrot 2
2
8
)1(exp
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Rotational levels are degenerate and this degeneracy arises due
to
space quantization of angular momentum. There are (2J+1)
allowed orientations. So, grot = (2J+1)
Where is the rotational temperature.
Case I: High temperature limit: When T >> θrot
After solving, we get
For a gas of N diatomic molecules,
6
Ik
hrot 2
2
8
)........()1(
exp)12(8
)1(exp)12(
2
2
viiiT
JJJ
IkT
hJJJZ rot
JJ
rot
dJT
JJJZ rotrot
)1(
exp)12(0
2
28
h
IkTTZ
rot
rot
2
28
h
IkTN
rotZ
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The mean energy of the gas will be
Contribution to entropy due to rotational motion of N
diatomic
molecules is -
Contribution to specific heat due to rotational motion of N
diatomic molecules is –
The Helmholtz free energy due to rotational motion of N
diatomic molecules is –
7
NkTT
ZkTE rotrot
ln2
Nkh
IkTNkSrot 2
28ln
RNkrotVT
EC
rot
rot
2
28ln
h
IkTNkTTSEF rotrotrot
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Case II: Low temperature limit: When T
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Vibrational Motion: A diatomic molecule has only one degree
of freedom corresponding to the vibrational motion of the
nuclei along the axis joining them.
Vibrational motion of atoms bound in a molecule can be taken
to
be nearly simple harmonic. The energy level of a linear
simple
harmonic oscillator are non-degenerate and vibrational
energy
of a diatomic molecule is given by –
So, the vibrational partition function can be written as –
After further simplification and neglecting higher order terms,
we
get 9
.......2,1,0,2
1
nhnvib
0
0
exp2
exp
2
1exp
n
vib
n
vib
kT
nh
kT
hZ
kT
hnZ
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Vibrational energy: The vibrational energy of a gas of N
diatomic molecules is –
Helmholtz free energy:
Entropy:
10
).......(
exp1
2exp
ix
kT
h
kT
h
Zvib
1exp2
ln2
kT
h
NhNh
T
ZkTE vibvib
kThvibvib eNkTNh
ZNkTF 1ln
2ln
11ln
kTh
kThvibvib
e
kTheNk
T
FS
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Specific heat at constant volume:
For T>>θvib, (CV)vib→Nk
For T
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Electronic and Nuclear Partition
functions
• Molecules can exist with electrons excited to states higher
than
the ground state. The energy spacings of these states vary
in
irregular manner. So, it is not possible to give a general
expression for Ze.
• At ordinary temperatures, most of the molecules are usually
in
their ground state whose energy can be taken as zero. Thus,
where ggr(e) is the degeneracy of the electronic ground
state.
12
)..(..........)(
.........)(exp)( 11
xegZ
egegZ
gre
gre
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• The nuclear energy can be taken to be zero. Except in
atomic
explosions, the nuclei are not excited thermally to states
above
their ground state. Thus,
where, ggr(n, s) is the nuclear spin degeneracy.
• If in a diatomic molecule, nuclei have spins s1 and s2
then,
13
)..(..........),( xisngZ grn
)12)(12(),( 21 sssnggr
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Assignment
• Use the concept of partition function
– to determine the specific heat capacity of Hydrogen.
– to determine the specific heat capacity of Solids.
14
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References: Further Readings
1. Statistical Mechanics by R.K. Pathria
2. Thermal Physics (Kinetic theory, Thermodynamics and
Statistical Mechanics) by S.C. Garg, R.M. Bansal and
C.K. Ghosh
3. Elementary Statistical Mechanics by Gupta & Kumar
4. Statistical Mechanics by K. Huang
5. Statistical Mechanics by B.K. Agrawal and M. Eisner15
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For any questions/doubts/suggestions and submission of
assignment
write at E-mail: [email protected]
mailto:[email protected]
