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

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 : 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: Geometric series

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

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.

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:

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

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

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

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: Examples: O2,  = 94 kJ; noble gases,   900 kJ Exception: NO, g0 = 2 e  = 1.5 kJ

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

U Cv S Translation Vibration Rotation Electronic

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

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

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

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 :

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

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

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. 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)

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)

For ethane Vmáx  kBT, and rotation is impeded.