CHAPTER 12 2

3. The electromagnetic spectrum:

Region λ Processes gamma <10 pm nuclear state changes x-ray 10 pm – 10 nm inner shell e- transition UV 10 nm – 400 nm electronic trans in valence shell VIS 400 – 750 nm electronic trans in valence shell IR 750 nm – 1 mm Vibrational state changes microwave 1 mm – 10 cm Rotational state changes radio 10 cm – 10 km NMR - spin state changes

4. Classical picture of light absorption – since light is oscillating electric

and magnetic field, interaction with matter depends on interaction with fluctuating dipole moments, either permanent or temporary.

Discuss vibrational, rotational, electronic 5. Quantum picture: Einstein identified three contributions to transitions

between states:

a. stimulated absorption b. stimulated emission c. spontaneous emission

Stimulated absorption transition probability w is proportional to the

energy density ρ of radiation through a proportionality constant B w=B ρ The total rate of absorption W depends on how many molecules N in

the light path, so: W = w N = N B ρ

CHAPTER 12 3

Now what is B?

Consider plane-polarized light (along z-axis), can cause transition if the Q.M. transition dipole moment is non-zero.

€

µij = ψj∫ * ˆ µ zψidτ

Where

€

ˆ µ z = z-component of electric dipole operator of atom or molecule Example: H-atom

€

ˆ µ z = −e r

Coefficient of absorption B (intrinsic ability to absorb light)

€

B =µij

2

6εo2 Einstein B-coefficient of stimulated absorption

if

€

µij ≠0, transition i → j is allowed

€

µij = 0 forbidden 6. Selection rules – QM-derived rules, based on the evaluated transition

dipole moments which say what transitions are allowed or forbidden. Examples: H-atom selection rules. sà p allowed Δml = 0, ±1 Δl = ±1 Δn = anything sà s forbidden

r

e

+

-

1s

2s

CHAPTER 12 4

Molecular vibration – a vibrational mode must result in the oscillation of a dipole moment in order to appear in vibrational spectrum.

Example: HCl has 1 vibr mode. IR active. O2 has 1 vibr mode. IR inactive. A molecule has: 3N–6 vibr modes, if non-linear 3N-5 vibr modes, if linear CO2 3 * 3 – 5 = 4 vibr modes Vibration state changes allowed: Δv = ±1 Since selection rules are usually derived from approximate ψ’s and

without relativity effects, some “forbidden” transitions actually can occur with a small probability.

Example: Vibration Δv = ±2 (1st overtone) Δv = ±3 (2nd overtone) Molecular rotations – molecule must possess a permanent dipole

moment to absorb in a pure rotation spectrum. 7. Stimulated Emission. Stimulated emission transition probability w’ is given by w’=B’ ρ and the total rate of stimulated emission W’ depends on number of

molecules N’ in the excited state as: W’ = w’ N’ = N’ B’ ρ

Einstein was able to show that B=B’ For example, if you had one molecule in the lower state and one

molecule in the upper state, an incident photon would have EQUAL probability of stimulating an emission as stimulating absorption.

Implication: continuous bombardment of a sample with high intensity

resonant light can only induce up to but not beyond a 50/50 population of lower and upper state.

CHAPTER 12 5

Wnet = W - W’ = NBρ - N’B’ρ = NBρ - N’Bρ = (N - N’)Bρ Actually there is one more event happening that negligibly affects the

above equation, and that is spontaneous emission. 8. Spontaneous emission: Upper to lower state transition occurring

even in absence of radiation. So the downward rate would be modified to: W’ = N’(A + Bρ) Where A = coefficient of spontaneous emission:

€

A =8πhν3

c3

$

% &

'

( ) B

9. Two principal factors determine intensity of absorption or emission:

a) inherent probability of transition. B

€

∝ µij

2 as seen before

b) population of atoms or molecules in initial state.

Population of state i

€

Ni ∝gie−Ei /kBT

Ei = energy of state i gi = degeneracy of state i kB = 1.38 x 10-23 J K-1 Boltzmann constant T = K temperature 10. Emission intensities can be increased by promoting excited state

populations above their room T values by: -flame excitation -electrical discharge

CHAPTER 12 6

11. Emission spectroscopy (spectroscope) – H-atom Experiment. -emitted light generated in e- discharge tube -passed through narrow slit -dispersed: quartz prism; diffraction grating -detected by: photography; photomultiplier tube (pmt) 12. Absorption Spectroscopy (spectrophotometer) – Dye Experiment.

a) need light source in appropriate frequency range. -klystron for microwaves -Nernst filament for IR (a ceramic containing rare earth oxides) -synchrotron – wide range -common incandescent bulb – visible b) grating or prism to select out monochrome light passed through

cell. c) single beam instrument – sample cell and blank cell are placed

alternately in the beam. Spectra are subtracted later. d) double beam – beam is split and passed simultaneously through

sample and blank. e) detectors – photomultiplier tube (PMT); photo plate;

thermocouple (IR); crystal diode (microwaves).

13. Intensity Definition. Itot = total intensity = intensity of light of all λ passing through a unit

area per unit time. I(λ) = intensity per unit wavelength interval I(λ)dλ = intensity of radiation per area per time in wavelength

interval λ to λ+dλ

CHAPTER 12 7

14. The Beer-Lambert Law: dependence of intensity on concentration of substance in sample.

loss of intensity in dx interval = -dI = κ(λ) c I dx where κ(λ) is intrinsic ability of material to absorb light at λ, c is

concentration, and dx is thickness

€

−dII

= κ λ( )cdx

−dII

Io

I b( )

∫ = κ λ( )cdxo

b

∫

−lnI b( )Io

= κ λ( )cb

+ ln Io

I b( )= κ λ( )cb

2.303log Io

I b( )= κ λ( )cb

Define absorbance as

€

Aλ ≡ logIo

I b( ) absorbance at λ (or optical density)

Aλ = ελbc Beer Lambert Law

€

ελ =κ λ( )

2.303 = extinction coeff or molar absorption coeff

CHAPTER 12 8

molar absorptivity units =

€

1mol

L( )cm

b = path length in cm

c = concentration in mol/L (molarity)

T(λ) = transmittance =

€

I λ( )I λ( )o

%T(λ) =% transmittance =

€

I λ( )I λ( )o

x100%

Integrated absorption coefficient

€

A = ε( ˜ ν )d˜ ν band

∫

15. Raman spectroscopy.

Look at scattering of high intensity visible energy photons (from laser) from the sample.

Sample may absorb incident photon and immediately re-emit at slightly different ν. Δε matches vibrational transition in the molecule.

Raman has advantage that some vibrational modes that are IR

inactive may be Raman active. (see lab manual)

CHAPTER 12 9

B. Linewidths - peaks have a finite width in frequency or wavelength.

1. Doppler broadening. Gas phase Doppler effect

Actual freq molecule will absorb if moving veloc v away:

€

" ν =νo

1+ vc

Moving toward:

€

" ν =νo

1− vc

Measure width of line at half-height

€

δν =2νo

c2kBTm

ln2$

% &

'

( ) 1/2

where m is mass and T is temp

€

δν∝Tm

increases with Temp and decreases with mass

2. Lifetime broadening. Let τ = average lifetime of being in excited state.

Uncertainty principle

€

δE ≈τ

shorten the τ, the greater the blurring δE of the energy of the excited

state

€

δ˜ ν ≈5.3cm−1

τ ps( )

Processes responsible for finite lifetimes τ of the excited state

a. collisional deactivation (work at low P to minimize effect) b. spontaneous emission (can’t change this) τ↓ ν↑

CHAPTER 12 10

II. Pure Rotational Spectra. (microwave region) A. Energy levels depend on symmetries, moments of inertia.

€

I = mi

i

atoms

∑ xi2 where

€

xi2 is distance of atom i from rot axis

Molecules have 3 moments of inertia: Ia < Ib < Ic (principal axis)

Classical rotational energy

€

E =Ja

2

2Ia

+Jb

2

2Ib

+Jc

2

2Ic

Jα = ang mom about α axis Convert to QM each

€

J2 → J J+1( )2 B. Four types of Rigid Rotors. 1. Spherical rotors, 3 equal moments of inertia (CH4) 2. Symmetric rotors, 2 equal moments of inertia, one different (NH3) 3. Linear rotors, one moment of inertia is zero (CO2, HCl) 4. Asymmetric rotors, 3 different moments of inertia (H2O) Table 12.1 Moments of inertia HCl HCN CO2

CHAPTER 12 11

CH3Cl NH3 SF4Cl2 CH4 SF6

CHAPTER 12 12

C. Linear rotor. Rotation about internuclear axis has zero moment of inertia, no energy Other two rotations have equal moment of inertia = I

€

EJ = J J+1( ) 2

2I

ΔJ = + 1 Selection rule (also molecule must be polar to have pure rot spectrum) ΔmJ = 0, + 1 degeneracy of rot level J = 2J + 1 = gj For rotational absorption transitions: J-1 à J Derive ΔEJ defined as EJ - EJ-1 Set hν = ΔEJ Solve for

€

˜ ν wavenumber Result:

€

˜ ν (J← J−1) = 2BJ where

€

B ≡

4πcI the rotation constant

CHAPTER 12 13

D. Pure Rotational Spectrum. “Pure” meaning only rotation state transitions, no vibration state

changes. Pure rotational transitions are in microwave region of spectrum.

At right is pure rot spectrum of a linear molecule. Wavenumbers of transitions are equally spaced at 2B, 4B, 6B, 8B, etc. spacing between them = 2B Typical B falls in range: 0.1 to 10 cm-1, so in microwave Peak heights vary according to equilibrium population of the energy levels and their degeneracy

Population of level J

€

NJ ∝gJe−EJ /kBT

gJ = 2J+1

E. Centrifugal distortion effect. We have treated rotors as rigid. In reality, they can flex, and the

magnitude of their flex increases as rotational energy increases.

€

˜ ν (J← J−1) = 2BJ−4DJJ3

CHAPTER 12 14

III. Vibrational Spectra of Diatomics.

A. Molecular vibrations.

1. Here we are concerned with motions of nuclei in a potential well which, to a first approximation and for small amplitude vibrations, is parabolic.

€

V(x) =12

kx2 where x ≡R −Re

R=instantaneous internuclear separation Re=equilibrium internuclear separation (bond length) k = force constant of the bond (curvature of parabola) 2. We have previously solved the Schrod

Eq for this system Energy levels are evenly spaced, given by:

Ev= (υ + 1

2)ν υ = 0,1,2,3...

where :

ν =12π

kµ

$

%&&'

())

12

µ=reduced mass

3. Vibration terms G(v) of a molecule: These are energy levels but expressed in wavenumbers (cm-1)

G(υ) = (υ + 12)ν

where

ν = 12πc

kµ

$

%&&'

())

12

G(υ) =Ev

hc

CHAPTER 12 15

4. Selection Rules for vibration of molecules.

a. The electric dipole moment of the molecule must change when the atoms vibrate in order to absorb in the infrared (be infrared active).

For diatomics this means they must possess a dipole moment.

HCl vs N2 For polyatomics, there will be several modes of vibration, some

of which are IR active and some which are not, depending on whether the motion produces a change in the dipole.

A polyatomic need not have a permanent dipole moment to have

some modes which are IR active. b. For purely harmonic vibrations, the vibration state can only

change by one.

€

Δυ = ±1 This means that there should be only one absorption line in the

pure IR spectrum, where

Transition wavenumber = G(υ +1) −G(υ)= ν

5. At room temperature the equilibrium distribution of energy in

molecules has almost all molecules in the ground vibrational state, so the fundamental transition.

v=0 to v=1

is the only transition contributing to the spectrum. ν = 1← 0( ) = G 1( ) −G 0( )

6. Real molecules show anharmonic vibrations, especially in higher vibr levels.

The true potential energy curve is not parabolic, and the levels are thus not evenly spaced but becomes shortened at higher energy:

Do De

CHAPTER 12 16

Potential energy curve can be approx. by so-called Morse potential:

€

V(R) = hcDe 1−e−a(R−Re ){ }2

An anharmonicity correction term can be derived for the vibrational energy level equation since the Schrodinger Eq can be solved in the Morse potential. (See text)

Anharmonicity also loosens up the selection rule such that weak

overtones appear in the spectrum, in which level changes of + 2 are possible.

B. Vibration-rotation spectra. When heteronuclear diatomics (e.g., HCl) are studied in the gas phase,

their IR spectra will show rotational state changes accompanying the vibrational transition v=0 à 1.

This is called rotational fine structure. The rotation of the molecule will change by ΔJ = + 1 while vibration is changing v=0 to 1.

CHAPTER 12 17

Equation generating this spectrum:

€

˜ ν R−branch = ˜ ν + 2B J+1( ) J = 0,1,2...

˜ ν P−branch = ˜ ν −2BJ J =1,2,3...

Derive by using a combined vibration-rotation term S(v,J), which is the

energy when molecule is in vibrational state v and rotational state J. Let’s do this without anharmonicity effects in the vibration, and without

centrifugal distortion of the rotor.

€

S(υ,J) = G(υ) + F(J)orS(υ,J) = (υ +1/2) ˜ ν + BJ(J+1)

R-branch ΔJ = +1

€

˜ ν R−branch = S(υ +1,J+1) −S(υ,J)or˜ ν R−branch = ˜ ν + 2B(J+1)

where

€

˜ ν is the wavenumber of pure vibrational trans P- branch ΔJ = -1

€

˜ ν P−branch = S(υ +1,J−1) −S(υ,J)or˜ ν P−branch = ˜ ν −2BJ

Corrections for real diatomics: vibrational anharmonicity xe centrifugal stretching De (very small) *vibr-rot coupling αe

CHAPTER 12 18

IV. Vibrational Spectra of Polyatomics. One bond vibr is coupled to another – leads to complicated motion. Simplified by resolving nuclear vibrations into “normal modes” of vibration. Only normal modes of motion resulting in change in dipole moment can

absorb from electromagnetic field (infrared active). IF N atoms in molecule: 3N-6 normal modes of vibration (non-linear molecules) 3N-5 (linear) H2O CO2 IR spectra useful in compound identification.

CHAPTER 12 19

Greenhouse Gases

CHAPTER 12 20

V. Raman Scattering Spectra.

A. Overview.

1. Means of studying spectroscopically inactive vibr and rotational transitions.

2. Based on inelastic scattering of photons and molecules. 3. In elastic collision – no energy transfer. 4. In inelastic collision – molecule changes quantum states.

€

νscattered ≠ νincident 5. Mechanism of light scattering: -oscill field induces an oscillating dipole moment in molecule (even if

it has no permanent dipole). (induced dipole strength ~ α, polarizability)

-oscill dipole emits electromagnetic radiation, predominantly of same

frequency. (Rayleigh scattering) -molecule may also emit light of slightly lower frequency (Stokes) or

higher frequency (anti-Stokes).

€

hν + # # E = h # ν + # E

where

€

hν is incident light,

€

" " E is molec energy in initial state,

€

h " ν is scattered light, and

€

" E is final state.

€

" E − " " E = h ν − " ν ( ) = hΔν

where

€

" E − " " E is determined by quantum levels and

€

hΔν is called “Raman shift” (Δν independent of ν)

B. Selection rules for Raman:

1. rotational Raman ΔJ = ±2 2. vibrational mode is Raman active if polarizability of molecule changes

during vibration. Δν = ±1 (Stokes and anti-Stokes) 3. If molecule possess a center of inversion, then an IR active vibration

will be Raman inactive, and vice versa.

CHAPTER 12 21

C. Examples.

1. CO2 linear molecule with center of inversion.

IR inactive (no dipole fluct) but Raman active (polarizability changes). IR active, Raman inactive IR active, Raman inactive IR active, Raman inactive

2. Tetrahedral molecule like Methane or carbon tet (no center of

inversion) 3N-6 = 3x5-6 = 9 modes all modes Raman active

1 of these. 3 of these. IR inact. IR active. 2 of these. 3 of these. IR inactive. IR active.

CHAPTER 12 22

NOTES:

hc hν hcν E λ - TTU CAE Network

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