Rotacijska spektroskopija Vibracijska spektroskopijafizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/afs/8_Rot... · Rotacijski energijski nivoi The quantum-mechanical

Rotacijska spektroskopijaVibracijska spektroskopija

Dr. Marco Tessari/D. Kirin

Presentation can be found at the website: http://www.nmr.kun.nl/Education

Atomska fizika i spektroskopija Predavanje 8

Pregled

Uvod: Rotacijska spectroskopija

Momenti inercije

Rotacijski energetski nivoi

Rotacijski prijelazi

Introduction: Vibrational Spectroscopy

Normal Vibration Modes

The vibrational energy levels

Vibrational transitions

Material from Atkins Physical Chemistry 7th Edition, ch. 16.

Uvod : Rotational Spectroscopy

UvodRotacijska spectroskopija koristi EM zračenje u mikrovalnom području:

100GHz-1GHz

mm3cm30

=−=

νλ

Koji uredajirade u tom području?

UvodOsobine:

- primjenjuje se na molekule s dipolnim momentom

- uzorak je u plinovitom stanju

Prednosti:

- visoka rezolucija

- visoka osjetljivost

- nedestruktivna metoda

UvodPrimjene:

- proučavanje geometrijskih parametara molekule,

duljine veza, kuteva

- proučavanje planarnosti/ne-planarnosti molekula

- odredivanje elektricnih svojstava molekula(dipolni i kvadrupolni moment)

- kvantitativna i kvalitativna analiza plinovitih mješavina (posebno analiza kemijskog sastava u meduzvjezdanom prostoru)

Moment inercije

Moment inercije sistemačestica(molekule) je dan izrazom:

2i

iirmI ∑=

I je mjerainercije sistema s obzirom rotacijsko gibanje. Ovisi o rasporedu mase u sistemu.

Rotacijsko gibanje molekule može se točno opisati kada znamomoment inercije.

Moment inercije

Ovisno o simetriji molekule, jedan ili više momenatainercije moraju biti definirani da bi se opisala rotacijska svojstva.

Za molekularne strukture s niskom simetrijom trebamoTRI momenta inercije s obzirom na tri okomite osi.

Momenti inercije

Kruti rotori se klasificiraju kao:

cba III :rotori Sferni ==

cba III :rotori Simetricni ≠=

0I,II:rotori Linearni cba ==

cba III :rotori iAsimetricn ≠≠

2COCO,

64 SF,CH

3NH

OH2

Rotacijski nivoi

Rotacijski energijski nivoi

The quantum-mechanical expression for the rotational energy of a molecule depends on the symmetry of the molecular structure. For linear rotors only one moment of inertia needs to be defined, and the corresponding rotational energy is:

)cm(in Constant Rotational:4

,...3,2,1,0)1(2

)1(

1-

2

cIB

JhcBJJI

JJEJ

πh

h

=

=+=+=

The same equation applies for spherical rotors. The expression of the rotational energy for symmetric and asymmetric rotors is (slightly) more complex and will not be treated.


JM

JhcBJJI

JJE

J

J

±±±=

=+=+=

,...,2,1,0

,...2,1,0)1(2

)1(2

h

Energetski nivo rotacije odreden je s dva kvantna broja (J, MJ).

Rotacijska energija ovisi SAMO oJ; to znaci, za svaki energijski nivo EJ postoji2J+1 stanja, tj. svaki nivo EJ je 2J+1-struko degeneriran.


Primjer 1

Rotacijska konstanta127I35Cl je 0.1142 cm-1. Izračunajte duljinu veze molekule.

kg1066054.1u1

u9688.34)(

u9045.126)(

)()(

)()(I

)cm(in Constant Rotational:4

27

35

127

235127

35127

1-

−⋅===

⋅+⋅=

=

Cm

Im

rClmIm

ClmIm

cIB

ICl

πh IClr

1-8

-134

sm109979.2

2

sJ1062608.6

⋅=

=

⋅= −

c

h

h

πh


Primjer 1: rješenje

m1005.232)()(

)()(

mkg10451.244

1235127

35127

245

−

−

⋅=⋅+=⇒

⋅==⇒=

ClmIm

ClmImIr

cBI

cIB

ICl

ππhh

IClr



EM zračenje može apsorbirati ili emitirati rotacijske prijelaze samoako molekula ima permanentni dipolni moment.

Samo polarne molekule daju čisti rotacijski spektar. As a consequence, homonuclear diatomic molecules and molecule with spherical symmetry are not directly observable with rotational spectroscopy.

The intensity of the signals in a rotational spectrum increase with the molecular dipole moment.

U jeziku IZBORNIH PRAVILA, dozvoljeni prijelazi su definirani kao:

;1,0;1 ±=∆±=∆ JMJ


The appearance of a rotational absorption spectrum can be explained using the expression for the rotational energy and theselection rules:

1,01

,...3,2,1,0)1(

±=∆±=∆=+=

J

J

MJ

JhcBJJE

The spectrum consists of a set of lines at position with a fixed separation 2B. The intensity of the lines can be explained considering the relative population of the rotational states and their degeneracy.

,...3,2,1,0,)1(2~ =+= JBJν


Frequency

Absorb

ed Inte

nsity

Frequency

Absorb

ed Inte

nsity

kTE >>∆ kTE <<∆

EEEE1111

EEEE2222

EEEE1111

EEEE2222

Rotacijski energijski nivoiPrimjer 2

Rotational absorption lines from 1H35Cl gas were found at the following wavenumbers: 83.32 cm-1, 104.13 cm -1, 124.73 cm -1, 145.37 cm -1, 165.9 cm -1, 186.23 cm -1, 206.60 cm -1, 226.86 cm -1. Calculate the moment of inertia and the bond length of the molecule.

u96885.34)(

u007825.1)(

)()(

)()(

.4

22~

35

1

2351

351

==

+⋅=

==∆

Clm

Hm

rClmHm

ClmHmI

cIB

HCl

πν h

18

134

27

sm109979.2

2

sJ1062608.6

kg101.66054u1

−

−−

−

⋅=

=

⋅=⋅=

c

h

h

πh


Exercise 2:Solution

m105.129)()(

)()(

mkg10731.2~4

2cm50.20~

~4

2

4

22~

12351

351

2471

−

−−

⋅=⋅+=

⋅=∆

=⇒=∆

∆=⇒==∆

ClmHm

ClmHmIr

cI

cI

cIB

HCl

νπν

νππν

h

hh

Rotacijski prijelaziExercise 3

Is the bond length in 1HCl the same as in 2HCl? The wavenumbersof the J = rotational transition for 1HCl and 2HCl are 20.8784 cm-1 and 10.7840 cm -1, respectively.

u96885.34)(

u007825.1)(

4

22~

35

1

2

21

21

01

==

+⋅=

==←

Clm

Hm

rmm

mmI

cIB

πν h

18

134

27

2

sm109979.2

2

sJ1062608.6

kg101.66054u1

u 2.0140 )(

−

−−

−

⋅=

=

⋅=⋅=

=

c

h

h

Hm

πh

01←


Exercise 3: Solution

HClHClHClHCl

HCl

HCl

HCl

HCl

HCl

HCl

HCl

HCl

HCl

HCl

HCl

HClHCl

HClHCl

rrrr

r

r

r

r

r

ClmHm

ClmHmClmHm

ClmHm

I

I

I

I

cIcI

112

1

2

1

2

1

2

1

2

1

2

2

1

2

2

1

1

99795.094400.1

93605.193605.194400.1

94400.1

)()()()()()()()(

93605.1~

~4

2~;4

2~

2

2

2

2

2

2

351

351

352

352

⋅==⇒=

=

+⋅+⋅

=

==

==

νν

πν

πν hh

Vibracijska Spektroskopija

Kod vibracijske spektroskopije EM zračenje koje seupotrebljava je u infracrvenom (IR) području spektra:

Hz10-Hz10

m1.0mm3.01411=

−=ν

µλ

UvodOsobine:

- dipolni moment se mora mijenjati pri vibraciji

- primjena u krutim, tekućim i plinovitim uzorcima

Primjena:

- široka primjena

- visoka osjetljivost

- ne-destruktivna tehnika

Uvod

Primjene:

- odredivanje duljine veza

- odredivanje snage veza

- odredivanje energije disocijacije

- kvalitativna i kvantitativna kemijska analiza

Normalni modovi

Normalni (vibracijski) modovi

In general, the description of a molecule consisting of N atoms requires 3N coordinates.

• 3 koor. → translacija

• 3 koor. (2 za lin. mol.) → rotacija

• 3N – 6 (3N – 5 za lin. mol.) → relativno gibanje atoma, tzv. vibracijski modovi

Normalni vibracijski modoviNormalni modovi 3N – 6 (or 3N – 5)

♀ pojednostavljen opis mol. vibracija

♀ svaka jezgra vrsi harmonijsko titranje oko ravnoteznog polozaja

♀ sve se jezgre gibaju istom frekvencijom, u fazi su, a centar mase je nepromjenjen

Primjer: CO2 modovi vibracija


Exercise 4

Koliko normalnih modova vibracija imaju sljedece molekule:

H2O, H2O2, C2H4 , HC≡C–C≡CH ?

53ili 63 −=−= NNNN vibvib


Exercise 4:Solution

13)(18)(

12)(6)(

6)(4)(

3)(3)(

53or63

4242

2222

22

=≡−≡⇒=≡−≡=⇒==⇒=

=⇒=

−=−=

CHCCHCNCHCCHCN

HCNHCN

OHNOHN

OHNOHN

NNNN

vib

vib

vib

vib

vibvib

Normalni (vibracijski) modovi

Svaki normalni mod vibracije je u stvari harmonijski oscilator kojiodgovara jednojnormalnojkoordinati. Normalna koor. se konstruira tako da odrazava sve individualne pomake jezgri.

Linearna dvoatomna molekula→

• JEDAN normalni mod

• jedina koordinata – promjenaudaljenosti Normalna koordinata

Vibracijski energijski nivoi

Jednostavni model – HARMONIJSKI OSCILATOR opisuje dvoatomnumolekulu:


ωπ

ν

ννω

c

hcGhcEv

2

1~

,...2,1,0v

)~(~)2

1v()

2

1v(

=

=

=+=+= h

eRRxkxV −== )2

1( 2

U ovom slucaju, dozvoljeni vibracijski energijski nivoi su dani s:

Kutna frekvencija titranjaω ovisi o atomskim masama i konstantama silaoscilacija:


21

21,mm

mmm

m

keff

eff +==ω

1m 2m

Dozvoljeni energetski nivoi► ekvidistantni s

ωh

For larger deviations (anharmonicity) the internuclear potential energy is better described by a Morse potential:


The harmonic oscillator is an acceptable approximation only for small deviation from the equilibrium internuclear distance

(vibrations of small amplitude).

{ }

e

eff

ee

hcD

ma

RRahcDV

2

)(exp(1

2

2

ω=

−⋅−−=

Note that the energy levels are no longer equally spaced. In addition, the number of vibrational levels is finite: for energies larger than Evmax

dissociation of the chemical bond takes place.


constantsityanharmonic:4

~

2

v,...,2,1,0v

)~(~)2

1v()

2

1v(

2

max

2v

ee

e

D

ax

hcGxhcE

νµω

νν

==

=

=

+−+=

h

When the Morse anharmonic potential is used, the permitted

vibration energy levels are defined by the expression:

De

Do

Vibracijski prijelaziExercise 5

The 1H35Cl molecule is quite well described by the Morse potential. Assuming that the potential De does not change on deuteration, predict the dissociation energies D0 of 1H35Cl and 2H35Cl.

J10602177.1eV1

sm109979.2

sJ1062608.6

19

18

134

−

−

−−

⋅=

⋅=

⋅=

c

h

1

1

cm05.52~cm7.2989~

eV33.5

−

−

=

=

=

νν

e

e

x

D

De

Do

00

2v

~)2

1v()

2

1v(

EDD

xhcE

e

e

−=

+−+= ν

Vibracijski prijelaziExercise 5: Solution

eV15.5J102452.8

J105396.8eV33.5

J109436.2~4

1

2

1

01741.0~05.52

1900

19

200

=⋅=−=

⋅==

⋅=

−=

==

−

−

−

EDD

D

xhcE

x

e

e

e

e

ν

ν

For 1HCl:


eV20.5J103284.8

J101111.2~4

1

2

1

1900

200 2

=⋅=−=

⋅=

−=

−

−

EDD

xhcE

e

HCle ν

( )1

2

1

cm3.2144~2

1~

~2

2

1cm7.2989~

2

1

122

1

1

11

−

−

===

⋅⋅=

==

HCleff

HCleff

HClHCleff

HCl

HCleffHCl

HCleff

HCl

m

m

m

k

c

mck

m

k

c

νπ

ν

νπ

πν

Vibracijski prijelazi


♂ EM radiation can induce vibrational transitions only for vibration modes that induce changes in the molecular dipole moment.

♂ Such vibrations are called infrared active.

♂ The stretching vibration in heteronuclear diatomic molecule is always infrared active.

♂ Conversely, such a vibration is infrared inactive for homonucleardiatomic molecules – homonuclear diatomic molecules are transparent to infrared radiation.

♂ For complex molecules, an analysis of the symmetry of the normalmodes is necessary to define the active normal modes.

♂ The intensity of the signals in a vibrational spectrum increase with the molecular dipole moment.

then dominated by the fundamental transitionfrom the ground state to the first vibrational excited state .

Vibracijski prijelaziIzborna pravila dozvoljeni prijelazi za harmonijski oscilator:

1v ±=∆

)01( ←

From the Boltzmann distribution it follows that for almost all molecules only the vibrational ground state (v=0) is populated at room temperature (kT=207 cm-1). The IR absorption spectrum is

wavenumber (cm-1)A

bsorb

ed Inte

nsity

ν~effm

k

c

GG

π

ν

2

1

~2/101

=

==∆=∆ ←


Calculate the relative number of Cl2 molecules in the ground and first vibrational state at 298 K and 500K

123

18

134

KJ1038065.1

sm109979.2

sJ1062608.6

−−

−

−−

⋅=⋅=

⋅=

Bk

c

hν

ν~

cm7.559~

2/101

1

hcGhcE =∆=∆=

←

−

ee

e kT

E

kT

E

kT

E

E

E

j

i

j

i

n

n ∆−

−

−

==

Vibracijski prijelaziPrimjer 6: Solution

200.0500

expN

N:K 005T

067.0298

expN

N:K 298T

J101118.1~cm7.559~

01

0v

1v

01

0v

1v

202/101

1

=

⋅∆−==

=

⋅∆−==

⋅==∆=∆=

←

=

=

←

=

=

−←

−

B

B

k

E

k

E

hcGhcE νν


The wavenumber of the vibrational fundamental transition of 1H35Cl is 2990.95 cm-1. Calculate the force constant of the bond.

u96885.34)(

u007825.1)(35

1

=

=

Clm

Hm18

134

27

sm109979.2

2

sJ1062608.6

kg101.66054u1

−

−−

−

⋅=

=

⋅=⋅=

c

h

h

πh)()(

)()(

2

1~

351

351

2/101

ClmHm

ClmHmm

m

k

cGG

eff

eff

+⋅=

==∆=∆ ← πν


( ) 12

27351

351

2/101

mN3.516~2

kg1062665.1)()(

)()(

2

1~

−

−

←

=⋅⋅=

⋅=+⋅=

==∆=∆

eff

eff

eff

mck

ClmHm

ClmHmm

m

k

cGG

νπ

πν

Vibracijski prijelazi☺ If the molecules are brought into vibrational excited states, emission signals involving other energy levels than 0 and 1 can be observed.

☺ In the harmonic approximation these emission signals should appear at the same frequency.

☺ In reality, they are observed at slightly different positions as a consequence of the breakdown of the harmonic approximation:

νν ~)1v(2~2/1vv1v

ex

GG

+−==∆=∆ +→+

wavenumberE

mitte

d Inte

nsity

ν~2 ex

presence of weak bands (overtones) in the absorption spectrum, corresponding to the transitions , , etc.


IZBORNA PRAVILA ∆v = ±1 is derived assuming a parabolic curve for the potential energy curve (harmonic oscillator approximation). When anharmonicity is taken into account, the selection rule ∆v = ±1 does not hold rigorously, and in fact all values of ∆v are allowed. Experimentally this is confirmed by the

wavenumber (cm-1)A

bsorb

ed Inte

nsity

ν~

02 ← 03←

Therefore, when several vibrational transitions are detectable, the dissociation energy of the bond can be estimated (with the so-called Birge-Sponer plot, for example).

Vibracijski prijelaziThe sum of the all the vibrational transition energies of a diatomic oscillator corresponds to the bond dissociation energy D0:

∑ +∆==+∆+∆=

v1/2v

2/32/1 ...

G

GGDo

Documents

Rotacijska spektroskopija Vibracijska spektroskopijafizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/afs/8_Rot... · Rotacijski energijski nivoi The quantum-mechanical