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CHAPTER 12 Molecular Spectroscopy 1: Rotational and Vibrational Spectra

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CHAPTER 12 Molecular Spectroscopy 1: Rotational and Vibrational Spectra CHAPTER 12 Molecular Spectroscopy 1: Rotational and Vibrational Spectra I. General Features of Spectroscopy. A. Overview. 1. Bohr frequency rule: hc Ephoton = hν = = hcν˜ = Eupper −Elower λ 1 Also remember ν˜ = (usually in cm-1) λ € 2. Types of spectroscopy (three broad types): -emission € -absorption -scattering Rayleigh Stokes Anti-Stokes CHAPTER 12 1 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 2 Now what is B? Consider plane-polarized light (along z-axis), can cause transition if the Q.M. transition dipole moment is non-zero. µ = ψ *µˆ ψdτ ij ∫ j z i Where µˆ z = z-component of electric dipole operator of atom or molecule Example: H-atom € € - e r + µˆ z = −er Coefficient of absorption B (intrinsic ability to absorb light) 2 µij B = 6ε 2€ Einstein B-coefficient of stimulated absorption o if 0, transition i j is allowed µ ij ≠ → µ = 0 forbidden € ij € 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 2s 1s sà s forbidden CHAPTER 12 3 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 4 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: $ 8πhν3 ' A = & )B c3 % ( 9. Two principal factors determine intensity of absorption or emission: € a) inherent probability of transition. 2 B as seen before ∝ µij b) population of atoms or molecules in initial state. € −Ei /kBT Population of state i N ∝ge i i Ei = energy of state i gi = degeneracy of state i -23 -1 kB = 1.38 x 10 J K 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 5 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 6 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 −dI = κ(λ)cdx I I(b) b dI − = κ(λ)cdx ∫ I ∫ Io o I(b) −ln = κ(λ)cb Io I +ln o = κ(λ)cb I(b) I 2.303log o = κ(λ)cb I b ( ) Define absorbance as I A ≡ log o absorbance at λ (or optical density) € λ I b ( ) Aλ = ελbc Beer Lambert Law € κ(λ) ε = = extinction coeff or molar absorption coeff λ 2.303 CHAPTER 12 7 € 1 molar absorptivity units = mol cm ( L) b = path length in cm c = concentration in€ mol/L (molarity) I(λ) T(λ) = transmittance = I λ ( )o I(λ) %T(λ) =% transmittance = x100% I λ € ( )o Integrated absorption coefficient € A = ∫ ε(ν˜ )dν˜ 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 8 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 ν " = o 1 v + c Moving toward: ν € ν " = o 1− v c Measure width of line at half-height 1/2 2ν $ 2k T ' € δν = o & B ln2) where m is mass and T is temp c % m ( T δν ∝ increases with Temp and decreases with mass € m 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 € 5.3cm−1 δν˜ ≈ 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 9 II. Pure Rotational Spectra. (microwave region) A. Energy levels depend on symmetries, moments of inertia. atoms 2 2 I = mixi where x is distance of atom i from rot axis ∑ i i Molecules have 3 moments of inertia: € € Ia < Ib < Ic (principal axis) 2 2 2 Ja Jb Jc Classical rotational energy E = + + 2I 2I 2I a b c 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 10 CH3Cl NH3 SF4Cl2 CH4 SF6 CHAPTER 12 11 C. Linear rotor. Rotation about internuclear axis has zero moment of inertia, no energy Other two rotations have equal moment of inertia = I 2 EJ = J(J+1) 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: ˜ J J 1 2BJ ν ( ← − ) = where B ≡ the rotation constant € 4πcI CHAPTER 12 12 € 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 −EJ /kBT NJ ∝gJe g = 2J+1 J € 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.
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