Midterm1reviewlec11

Midterm1reviewlec11 - C566 Midterm 1 review Weeks 1 - 6...

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1 C566 Midterm 1 review Weeks 1 - 6 Speed of light = 3 x 10 8 m s 1 =  Visible light is on the order of 500 nm = 5 x 10 -7 m Molecular sizes are on the order of 10 10 m (1 Å) 1 cm 1 = 1.986 x 10 -23 J = 0.01196 kJ mol -1 = 1.12398 x 10 -4 eV k = 1.38 x 10 -23 J K -1 = 0.69 cm -1 K -1 Marination of the material to date Some interesting chemical phenomena have been explored quite beautifully with vibrational spectroscopy. Here are particularly elegant vibrational spectroscopy explorations that you might find inspiring: Several areas in which vibrational spectroscopy is important/relevant Global climate modeling (greenhouse gas contributions) Astrophysical Chemistry (what elements make up cosmic bodies of mass, what kind of chemical bonding/chemistry is occurring in different interstellar media?) Chemical dynamics (how does vibrational energy/ specific mode activation impact on reaction rates?) Structural determination of just about everything in the gas or condensed phases. Spectroscopy- Interaction between light and matter 1. Bound molecules are described by wavefunctions with both a time-independent and a time-dependent portion, o o o o o o o o E x x t f dt d t f i gives t x t x dt d i ) ( ) ( 1 ) ( ) ( ) , ( ) , ( H H n o ( x , t) = n o ( x ) exp(-iE n t/ )
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2 2. An electromagnetic field perturbs the stationary states of the molecule: i t ( x , t)= ( H o + H ) ( x , t) H = V = E where x = q i x i , with y and z analogs E x = 2 E x o cos t = E x o [e i t + e -i t ] = E x o [e ih t/ + e -ih t/ ] With y and z analogs 3. Where the new wavefunction is approximated as a superposition of eigenfunctions, ( x ,t) = a 1 1 o ( x , t) + a 2 2 o ( x , t) + a 3 3 o ( x , t) + … For simplicity, we truncated at j = 2, but other terms only important when h v is resonant with E j1 . 4. We assumed that before the perturbation, a 1 = 1, a 2 = 0, at t >0, a 2 increases because of coupling to 2 o via interaction between dipole moment of molecule and the electric field. 5. The dipole moment integral, M x21 (aka X 21 , R x21 , plus y and z analogs) has veto power, and figures prominently in dP 21 /dt. Prob = |a 2 (t)| 2 = |  2 o | x | 1 o | 2 (E x o ) 2 2 21 ) ( ) ( ) ( 2 21 21 h E e e t h E i t h E i Prob = -2 |  2 o | x | 1 o | 2 (E x o ) 2 t The intensity of light, I E o2 , so P 21 I For an isotropic radiation field, E x o = E y o = E z o Energy density, ( ) = (2 ) -1 (E x o2 + E y o2 + E z o2 ) = (3/2 ) (E x o ) 2
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3 P 21 (per unit time) = ) ( ] [ 8 21 21 21 2 2 z y x M M M h = B 21 ( ) B 21 is the well-known Einstein coefficient. Rotational spectra of molecules- rot is approximated as a rigid rotor. In center-of- mass coordinates, X = M -1  m i x i , etc. for Y,Z, M = m 1 + m 2 ; R is fixed, reduced mass is = m 1 m 2 /M; m 1 r 1 = m 2 r 2 gives center of mass is along the internuclear bond 1. OVERALL SOLUTION to Schr. Eqn for rigid rotor, spherical harmonics: Y J,M ( , ) = J,M ( ) M ( ) = iM M J J J e P M J M J J ) (cos )! ( )! ( 4 1 2 | |
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Midterm1reviewlec11 - C566 Midterm 1 review Weeks 1 - 6...

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