Diatomic gases, beyond translational contribution, also possess vibrational, rotational and electronic contributions.
Remember: total = transl + vib + rot + elect
ztotal = ztransl zvib zrot zelect
ztransl: expression identical to that previously obtained
zelect: in many cases the contribution is not significant
For the calculation of the vibrational partition function we will use the linear harmonic oscillator model (OHL). According to quantum mechanics :
hni
2
1
n – vibrational quantum number(n = 0,1,2,.....)
h – Planck’s constant (h = 6.626 10-34 J.s)
- frequency of vibration (IR spectra)
All vibrational levels are non-degenerate (gi = 1). Then:
......1 /2/2/
0
/2
1
TkhTkhTkh
vib
n
Tkhn
vib
BBB
B
eeez
ez
Tk
h
Tk
h
vib
B
B
e
ez
1
2
1Geometric series
For rotational contribution we will use the linear rigid rotor model. The rotational energy levels are given by:
)1(8 2
2
JJI
hJ
J – rotational quantum number (J = 0,1,2,......)
I – Inertial moment of the molecule:2rI
- Reduced mass:
r – Interatomic distance.BA
BA
mm
mm
Each rotational level has degeneration = 2J + 1. Thus, the partition function is:
TJJ
Jrot
reJz /1
0
12
where r is the characteristic temperature of rotation:
Br Ik
h2
2
8
For low r , r / T << 1, and we can write:
dJeJz TJJrot
r /1
0
12
By a change of variable, J(J+1) = x , e (2J+1)dJ = dx
0
0
/ eeT
dxezr
Txrot
r
rrot
Tz
For temperatures of T r , we have:
...4183.1......531 62 eezrot
For T > r , but not >> r , we can use the expression of Mulholland:
....
315
4
15
1
3
11
32
TTT
Tz rrr
rrot
For diatomic homonuclear molecules we have to enter the number of symmetry, : number of indiscernible configurations obtained by rotation of the molecule.
x
z
y
H2: = 2
x
z
y
z
xy xy
Rotation of 180º
Rotation of 180ºHCl : = 1
rrot
Tz
r (K) v (K)
H2 85.4 6100
N2 2.86 3340
O2 2.07 2230
CO 2.77 3070
NO 2.42 2690
HCl 15.2 4140
HBr 12.1 3700
HI 9.00 3200
In most cases, zelect = 1 (gap between electron levels is very high).
Considering the first excited state (for some cases) we obtain:
01101
/10
/ e doing
1
ggg
eggz Tkelect
B
Tkelect
Bgegz /0 1
Examples: O2, = 94 kJ; noble gases, 900 kJ
Exception: NO, g0 = 2 e = 1.5 kJ Tk
electBez /150022
From the expressions for the various contributions of the partition function of the diatomic ideal gas we can get all the thermodynamic quantities. Example :
rotvibtransltotal
Brot
Brot
Tvv
Bvib
Bvib
Btransl
Btransl
UUUU
TNkT
zTNkU
e
T
TTNk
T
zTNkU
TNkT
zTNkU
v
ln
1
/
2
ln
2
3ln
2
/2
2
U Cv STranslation
Vibration
Rotation
Electronic
RT2
3 R2
3
7235.20ln
2
3lnln
2
5MpTR
1
/
2 /Tvv
ve
T
TRT
2/
/2
1
T
Tv
v
v
e
e
TR
T
Tv v
ve
e
TR /
/ 1ln1
/
RT R
1ln
r
TR
RT
RTA
ge
geN/
/
1
2/
/2
1 RT
RT
ge
ge
RTR
general)(in ln
ln
0gRT
UzR elect
For polyatomic gases, the expressions for the partition function must be modified.
2/3
2
2
h
Tkm
VzB
ii
transl
For translation:
For the vibration is necessary to rely on the several normal modes of vibration.
For a molecule with N atoms we have:
3N-6 vibrational coordinates for non-linear molecules
or
3N-5 vibrational coordinates for linear molecules
The molecule has 3N-6 or 3N-5 vibration modes each with a characteristic vibration temperature given by:
B
iiv k
h ,
.....11 /
2/
/
2/
2,
2,
1,
1,
T
T
T
T
vib v
v
v
v
e
e
e
ez
53
63
1/
2/
,
,
1
Nou
N
iT
T
vib iv
iv
e
ez
For the calculation of zrot is necessary to take into account the 3 main moments of inertia, with three characteristic temperatures, r,1, r,2, r,3. For a non-linear polyatomic molecule :
2/1
23
22/1
22
22/1
21
22/1 888
h
TkI
h
TkI
h
TkIz BBB
rot
2/1
3,2,1,
32/1
rrrrot
Tz
The numbers of symmetry can be obtained by analysis of the structure of the molecule.
molecule
Linear asymmetric 1
Linear symmetric 2
H2O 2
NH3 3
CH4 12
C2H4 4
C6H6 12
R2
3
Gas Translational Vibrational Rotational Total
Monatomic 0 0
Diatomic
Polyatomic linearPolyatomic non-linear
R2
3
R2
3
R2
3
R2
3
R R R2
7
RN 53 R RN
2
53
RN 63 R2
3 RN 33
Some molecules have internal rotation (when a part of the molecule rotates in relation to the remaining molecule).
Internal rotation contributes to the thermodynamic properties.
Generally, a molecule with N atoms and r groups that turn freely has (3N-6-r) frequencies of vibration.
2/1
2
2
int,
8
h
TkIz Bred
livrot
Ired –is the moment of inertia reduced along the axis around which the rotation angle is measured.
int –symmetry number of the rotor (methyl group, CH3, int =3)
TNkT
zTNkU B
livrotBlivrot 2
1ln ,2,
In the case of ethane exist repulsion between the C-H bonds of the two rotors (methyl groups). The calorimetric entropy is greater than the calculated based on rigid rotor model but less than the calculated assuming the two methyl groups free rotation.
“staggered“ conformation (more stable)
Eclipsed conformation (more unstable)