Rotational partition function

This includes the derivation of partition function and its relation with different thermodynamic parameters

2. The rotational energy for a single molecule is then
given by
??????
??????=
ℎ
2
8�
2
�
��+1……..1
Where I is moment of inertia of diatomic molecule
J= rotational quantum number (J=0,1,2,3…..)
??????
??????=�??????� ????????????+?????? ………..(2)
�ℎ??????�??????,�=
ℎ
8�
2
��

3. Rotational partition function for single molecule is given as
�
??????= ??????
????????????��
−????????????
????????????
∞
&#3627408445;<0 …….(3)

Since, (2J+1) Eigen states ≅(2&#3627408445;+1)

∴&#3627408478;
??????= (2&#3627408445;+1)??????&#3627408485;&#3627408477;
;&#3627408437;ℎ&#3627408438; &#3627408445;&#3627408445;:1
&#3627408446;??????
∞
&#3627408445;<0
……….4
Ignoring the nuclear spin term
Rotational partition function is expressed as
&#3627408478;
??????= (2&#3627408445;+1)??????&#3627408485;&#3627408477;
;??????&#3627408445;&#3627408445;:1
……..5
∞
&#3627408445;<0

Where, = Bhc/KT

4. Equation (5) can be solved by Euler-Maclaurin formula
&#3627408478;
??????=
1
&#3627409164;
(1+
&#3627409164;
3
+
&#3627409164;
2
15
+
4&#3627409164;
3
315
+⋯……)
 is very small, i.e <0.05 then,
&#3627408478;
??????=
1
&#3627409164;
=
8π
2
IKT
ℎ
2

For the molecule having some symmetry, symmetry number  is
introduced,
&#3627408478;
??????=
8π
2
IKT
 ℎ
2


Hetero nuclear diatomic()= 1
Homo nuclear diatomic()=2
Simple symmetrical (CO
2), S
2()= 2
Asymmetric triatomic HOD()= 1
Benzene ()= 6
Chair form cyclohexane()=12

5. ??????
??????=&#3627408446;&#3627408455;
2
??????????????????&#3627408452;
??????&#3627408455;
??????

Since, &#3627408452;
??????=&#3627408478;
??????
??????

Or, ????????????&#3627408452;
??????=??????????????????
8π
2
IKT
 ℎ
2

Or,
????????????????????????
????????????
??????
=
1
??????
×??????
Substituting this rotational energy is found to be
??????
??????=&#3627408446;&#3627408455;
2
×
1
&#3627408455;
×??????
E
r= NKT
E
r= nRT

6. &#3627408438;
??????=
??????
??????&#3627408455;
&#3627408446;&#3627408455;
2
??????????????????&#3627408452;
??????&#3627408455;
??????

&#3627408438;
??????=
??????
??????&#3627408455;
&#3627408446;&#3627408455;
2
×
1
&#3627408455;
×??????
&#3627408438;
??????=??????&#3627408455;
=??????&#3627408453;
Similarly,
entropy, free energy and enthalpy is related as
∴&#3627408454;
??????=??????&#3627408453;[1+????????????
8π
2
IKT
 ℎ
2
]
∴??????
??????=−??????&#3627408453;&#3627408455;????????????
8π
2
IKT
 ℎ
2

H
r= nRT

7. Q. Calculate the rotational partition function for H
2molecule
at 0°C given that K= 1.38X10
-16
erg deg
-1
, h= 6.624X10
-
27
ergsec, =2 and I= 0.459X10
-40
g cm
2
Solution:
&#3627408478;
??????=
8π
2
IKT
 ℎ
2

=
8π
2
X4.59X10
−48
X1.38X10
−23
??????273
2?????? (6.624??????10
−34
)
2

&#3627408478;
??????=1.554

8. Thank You