09.30.2011 - Chemistry 260/261 Lecture 11 Today in...

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Unformatted text preview: Chemistry 260/261 Lecture 11 Today in Chemistry 260/261 •  •  •  •  Introduction to spectroscopy Beer ­Lambert Law how matter interacts with electromagnetic radiation describing the interaction quantum ­mechanically: the transition dipole moment •  selection rules Next Week in Chemistry 260/261 • Reading: (Mon) 20.3 (Wed) 20.5 (Fri) 20.5, 20.7 • Rotational, vibrational, electronic spectroscopy • Problem set 4 due Today, by 3:00 pm • Problem set 5 due Monday, October 10 by 3:00 pm September 30, 2011 Quantum systems have energy levels Spectroscopy … Spectroscopy is the use of electromagnetic radiation to study the energy levels of molecular systems. The interaction of light with matter can induce a transition from one stationary state to another. excited state Eexcited hν h! ground state Eground NMR hν http://www.firstrays.com/plants_and_light.htm Quantum systems have energy levels Molecular orbitals Bound systems have discrete energy levels (due to boundary conditions) Hydrogen atom ! RH / 16 ! RH / 9 ! RH / 4 ? ΔE=hν But which are observed? Why? What does this tell us? ! RH l=0 l =1 l=2 l=3 Spectroscopy is the main tool to determine information about these energy levels experimentally IR Spectroscopy probes transitions between quantum levels Rotational Spectroscopy hν = ΔErot Microwave λ  ≅ 10 ­1 – 101 cm ν ≅ 109 – 1012 Hz Erot < Evib < Eel Vibrational Spectroscopy hν = ΔEvib Infrared λ  ≅ 10 ­2 – 10 ­4 cm ν ≅ 1012 – 1014 Hz Electronic Spectroscopy hν = ΔEel Visible/UV/ Soft X ­ray λ  ≅ 100 – 1000 nm ν ≅ 1015 – 1017 Hz Hard X ­ray λ  ≅ 0.1 – 100 nm 1 note: an external magnet is needed to spectroscopically probe spin \lips The wide variety of spectroscopy Spectroscopy Probes Energy Differences Using radiation from different regions of the electromagnetic spectrum gives rise to different types of spectroscopy Want to measure this: Generic absorption spectrum measurement radiowave absorption: nuclear spin \lips hν T (! ) UV/visible absorption: valence electron transitions X ­ray absorption: core electron transitions microwave absorption: molecular rotations, electron spin \lips NMR Light source reference Grating/prism I transmitted = I incident 10 !!cl ε Extinction coef\icient (M ­1 cm ­1) c Concentration (M) l infrared absorption: molecular vibrations Path length (cm) Inidn i c et Spectroscopy Probes Energy Differences ε Ias itd t n te rm Spectroscopy Probes Energy Differences Ninal Want to measure this: Generic absorption spectrum measurement sample Grating/prism initial I t( R ) = I 010 ! A I t( S ) = I 010 ( ( !A ε Concentration (M) l Path length (cm) !  ­log ln I incident (S) +A I t( S ) I 010 = (R) I t( R ) I 010 ! A Extinction coef\icient (M ­1 cm ­1) c λ ε I I (S ) t (R) t I transmitted − log A radio antenna picks up signals because the oscillating electromagnetic \ield causes the electrons in the antenna to oscillate The radio then ampli\ies these oscillations sample Grating/prism ) = 10 ( ) ! A( S ) + A( R ) + A( R ) λ Detector array I t( R ) = I 010 ! A I transmitted = I incident 10 !!cl ) (R) I t( S ) = I 010 (R) ( ! A( S ) + A( R ) ( ) ! A( S ) + A( R ) = 10 ! A (S) = A( S ) I t( S ) (λ ) = A( S ) (λ ) = ε (λ )c l I t( R ) (λ ) How Does Matter Absorb Light? E = h! Light source (R) ! A( S ) + A( R ) initial A (! ) Detector array I transmitted = I incident 10 !!cl Ninal Want to measure this: Generic absorption spectrum measurement T (! ) Light source λ Detector array ε I t( S ) I 010 = (R) I t( R ) I 010 ! A Extinction coef\icient (M ­1 cm ­1) c Concentration (M) l Path length (cm) !  ­log ln I incident ε I I (S ) t (R) t I transmitted − log ) = 10 ( ) ! A( S ) + A( R ) + A( R ) = 10 ! A (S) = A( S ) I t( S ) (λ ) = A( S ) (λ ) = ε (λ )c l I t( R ) (λ ) Resonance Light is an electromagnetic wave All waves have frequencies and amplitudes In order for frequency to mean anything, we need to consider some sort of time, but we have not considered time in quantum mechanics Consider a guitar held near a speaker. From the speaker we hear a sound that ramps through the scale. The guitar string will vibrate when the sound wave from the speaker is in resonance with the characteristic frequency of the string. In resonance the frequencies match. 2 How Does Matter Absorb Light? An example of a resonance in a molecule The electric Nield of light interacts with the charged particles of matter: electrons and protons classical picture h ν Energy transfer occurs as a result of B interaction between oscillating electric (or magnetic) \ields and oscillation of the charge distribution within the molecule (dipole moment). E frequency matching If the frequency of electromagnetic radiation matches that of an oscillating dipole (e.g. a vibration), energy is absorbed. Ⱥ dµ x Ⱥ Ⱥ Ⱥ ≠ 0 Ⱥ dQ3 Ⱥ antisymmetric stretching vibration µ =Q!r Q+ Q ­ µx(t) r t Another example of a resonance. Forget time, let s consider an antenna in a static electric \ield: Another example of a resonance. Forget time, let s consider an antenna in a static electric \ield: ++++ ++++ + - Apply a voltage dipole -- + - Apply a voltage ++ ---- ---- Another example of a resonance. Forget time, let s consider an antenna in a static electric \ield: Another example of a resonance. Now lets put a hydrogen atom in there. What happens? ---- dipole ++ + Switch sign of voltage +e -++++ 3 Another example of a resonance. Now lets put a hydrogen atom in there. What happens? Another example of a resonance. Now lets put a hydrogen atom in there. What happens? +e ---- + - Apply a voltage dipole dipole ++++ ---- Now lets put a hydrogen atom in there. What happens? Now lets put a hydrogen atom in there. What happens? + - Switch sign of voltage dipole dipole ---- ---- Now lets put a hydrogen atom in there. What happens? + Switch sign of voltage Another example of a resonance. Now lets put a hydrogen atom in there. What happens? ---- + - Switch sign of voltage dipole ++++ dipole +e ++++ Another example of a resonance. ---- Switch sign of voltage Another example of a resonance. ++++ +e + ++++ Another example of a resonance. +e +e +e + Switch sign of voltage ++++ 4 Another example of a resonance. Now lets put a hydrogen atom in there. What happens? Another example of a resonance. Now lets put a hydrogen atom in there. What happens? +e ---- + - Switch sign of voltage dipole dipole ++++ ---- Now lets put a hydrogen atom in there. What happens? Now lets put a hydrogen atom in there. What happens? + - Switch sign of voltage dipole dipole ---- ---- Now lets put a hydrogen atom in there. What happens? + Switch sign of voltage Another example of a resonance. Now lets put a hydrogen atom in there. What happens? ---- + - Switch sign of voltage dipole ++++ dipole +e ++++ Another example of a resonance. ---- Switch sign of voltage Another example of a resonance. ++++ +e + ++++ Another example of a resonance. +e +e +e + Switch sign of voltage ++++ 5 Another example of a resonance. Another example of a resonance. Now lets put a hydrogen atom in there. What happens? Now lets put a hydrogen atom in there. What happens? +e ---- + - dipole dipole ++++ Switch sign of voltage +e ---- Another example of a resonance. Now lets put a hydrogen atom in there. What happens? Now lets put a hydrogen atom in there. What happens? ---- + - dipole ++++ dipole Switch sign of voltage ++++ Another example of a resonance. +e + Switch sign of voltage +e ---- + Switch sign of voltage ++++ How Does Matter Absorb Light? The interaction of matter and light Put all the frames together Resonance condition: Hydrogen atom Classical Picture: The frequency of the light wave matches the frequency of the oscillation of the dipole in the atom or molecule. ! RH / 16 ! RH / 9 ! RH / 4 !E = h" E(t) ! RH l=0 t The oscillating electric \ield progressively in\luences the electron only if the wave has the right frequency As the electron oscillates, it looks like a 2p orbital! l =1 l=2 l=3 In general, the energy of the incident radiation (hν) matches the energy difference between a ground state wave function and an excited state wave function. Quantum Picture: The energy of the photon matches the energy difference in the atomic or molecular system. … but what about the dipole? 6 The transition dipole moment The transition dipole moment Likelihood of making the connection between Ψi and Ψj is given by the transition dipole moment, µij. Likelihood of making the connection between Ψi and Ψj is given by the transition dipole moment, µij. excited state ψ measure of the change of the dipole moment associated with the shift of electric charge that accompanies a transition. j !!! i ψi j excited state " 1 if i = j $ d" = # $ 0 otherwise % ground state ψ measure of the change of the dipole moment associated with the shift of electric charge that accompanies a transition. j µij = ∫ψ i µψ j dτ ψi ground state If µij = 0, no absorption occurs. What does this mean? Two states with different energy are as different as they can possibly be (orthogonal). Their overlap is zero. What does this mean? If there is no oscillating dipole moment in the classical picture, no transition dipole in the quantum picture, then there is no absorption. Even if the energy is in resonance. The details depend on the speci\ic kind of transitions. The transition dipole moment The interaction of matter and light µij = ∫ψ i µψ j dτ excited state µ = (const )x The transition from 1s to 2s is not observed. Hydrogen atom − /6 R1 H ψ µ1s , 2 s = ∫ψ 1s µψ 2 s dτ = 0 −H 9 R/ −H 4 R/ j × !2 ( x ) = !1 ( x ) = ψ ground state i µ12 = " L 0 # 2" x & 2 sin % ( $L' L # 1" x & 2 sin % ( $L' L !1 ( x ) x !2 ( x ) dx = µ12 = − 8 L2 9π 2 µ13 = 0 2 L " * $ 1 # x '- * $ 2 # x ')/ x ,sin & )/ dx ,sin & 0 + % L (. + % L (. L TDM is non ­zero, so light can make transitions from n=1 to n=2 !E = h" −H R l= 0 l= 1 l= 2 l= 3 In general, the energy of the incident radiation (hν) matches the energy difference between a ground state wave function and an excited state wave function. No oscillating dipole moment to absorb energy from the electromagnetic \ield. µ1s , 2 p ≠ 0 This one does equal zero, so light cannot connect n=1 and n=3 in the same way Selection Rules Classical or Quantum? A selection rule is a statement about when a transition dipole moment is non ­ zero, i.e. under what circumstances the transition occurs. Much talk of quantum nature of light, where is it? There are two parts to a selection rule: 1. A gross selection rule speci\ies the general features that a molecule must have if it is to exhibit a spectrum of a given kind. Always true – e.g. oscillating dipole moment 2. A speci\ic selection rule states which changes in quantum number may occur in a transition. Wave ­particle duality: it depends how you look at it…literally. Much of the time, spectroscopy can be thought of without ever thinking about the wave ­like nature of light. Light is just a beam of photons with energy and angular momentum (value is !) Hydrogen atom ! RH / 16 ! RH / 9 ! RH / 4 ??? ! RH l=0 l =1 l=2 l=3 Speci\ic to the details A transition permitted by a speci\ic and gross selection rule is classi\ied as allowed. Transitions that are disallowed by a speci\ic selection rule or a gross selection rule are called forbidden. E = h! = hc " The selection rules are intuitive if we consider the classical picture of an oscillating dipole and an oscillating electric (or magnetic \ield). Quantized energy levels interacting with individual photons provide the resonance conditions. 7 Think about for next time: N2: 78% O2: 21% Ar: 0.93% Ne,He,Kr: 0.0001% No worries! CO2: 0.0003% Why does CO2 correlate with global warming, but O2 and N2 do not ??? 8 ...
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This note was uploaded on 01/19/2012 for the course CHEM 260 taught by Professor Staff during the Fall '08 term at University of Michigan.

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