1. Diatomic and Polyatomic Gases
Valentim M. B. Nunes
ESTT-IPT - May 2015

2. 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

3. For the calculation of the vibrational partition function we will use the linear harmonic oscillator model (OHL). According to quantum mechanics :
n – vibrational quantum number(n = 0,1,2,.....)
h – Planck’s constant (h =  J.s)
 - frequency of vibration (IR spectra)

4. All vibrational levels are non-degenerate (gi = 1). Then:
Geometric series

5. Defining v = h/kB, as the characteristic temperature of vibration, we obtain:

6. For rotational contribution we will use the linear rigid rotor model
For rotational contribution we will use the linear rigid rotor model. The rotational energy levels are given by:
J – rotational quantum number (J = 0,1,2,......)
I – Inertial moment of the molecule:
- Reduced mass:
r – Interatomic distance.

7. Each rotational level has degeneration = 2J + 1
Each rotational level has degeneration = 2J + 1. Thus, the partition function is:
where r is the characteristic temperature of rotation:

8. For low r , r / T << 1, and we can write:
By a change of variable, J(J+1) = x , e (2J+1)dJ = dx

9. For temperatures of T  r , we have:
For T > r , but not >> r , we can use the expression of Mulholland:

10. 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 x z y
Rotation of 180º
HCl :  = 1
H2:  = 2

11. 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

12. In most cases, zelect = 1 (gap between electron levels is very high).
Considering the first excited state (for some cases) we obtain:
Examples: O2,  = 94 kJ; noble gases,   900 kJ
Exception: NO, g0 = 2 e  = 1.5 kJ

13. From the expressions for the various contributions of the partition function of the diatomic ideal gas we can get all the thermodynamic quantities. Example :

14. U Cv S
Translation
Vibration
Rotation
Electronic

16. U Cv
Translation
Vibration
Rotation
Total (t+v+r)

17. For polyatomic gases, the expressions for the partition function must be modified.
For translation:

18. 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:

19. 1= 1351 cm-1; 2 = 3 = 672.2 cm-1 e 4 = 2396 cm-1.

21. 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 :

22. The numbers of symmetry can be obtained by analysis of the structure of the molecule.
Linear asymmetric 1
Linear symmetric 2
H2O
NH3 3
CH4 12
C2H4 4
C6H6

23. Polyatomic non-linear
Gas
Translational
Vibrational
Rotational
Total
Monatomic
Diatomic
Polyatomic linear
Polyatomic non-linear

24. 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.

25. Generally, a molecule with N atoms and r groups that turn freely has (3N-6-r) frequencies of vibration.
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)

26. 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.
Eclipsed conformation (more unstable)
“staggered“ conformation (more stable)

27. For ethane Vmáx  kBT, and rotation is impeded.