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 :

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

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

T

T

vib v

v

e

ez

1

2

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

1/lim ,

RC vibvT

U Cv

Translation

Vibration

Rotation

Total (t+v+r)

RT2

3R

2

3

RT R

RT R

RT2

7R

2

7

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 ,

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

.....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)

3cos12

1máx VV

For ethane Vmáx kBT, and rotation is impeded.