rotacijska spektroskopija vibracijska...
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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.
Rotacijski energijski nivoi
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.
Rotacijski energijski nivoi
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
Rotacijski energijski nivoi
Primjer 1: rješenje
m1005.232)()(
)()(
mkg10451.244
1235127
35127
245
−
−
⋅=⋅+=⇒
⋅==⇒=
ClmIm
ClmImIr
cBI
cIB
ICl
ππhh
IClr
Rotacijski prijelazi
Rotacijski prijelazi
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
Rotacijski prijelazi
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
Rotacijski energijski nivoi
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
Rotacijski prijelazi
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←
Rotacijski prijelazi
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
Rotacijski prijelazi
Exercise 4
Koliko normalnih modova vibracija imaju sljedece molekule:
H2O, H2O2, C2H4 , HC≡C–C≡CH ?
53ili 63 −=−= NNNN vibvib
Rotacijski prijelazi
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:
Vibracijski energijski nivoi
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:
Vibracijski energijski nivoi
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:
Vibracijski energijski nivoi
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.
Vibracijski energijski nivoi
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:
Vibracijski prijelaziExercise 5: Solution
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
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
=
==∆=∆ ←
Vibracijski prijelaziExercise 6
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
Vibracijski prijelaziExercise 7
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
Vibracijski prijelaziExercise 7: Solution
( ) 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.
Vibracijski prijelazi
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