EXAM INFORMATION. Harmonic Oscillator. Anharmonic Oscillator 1 ~ 1. Rigid Rotor

Transcription

1 EXAM INFORMATION Harmonc Oscllator Hamltonan: H d dx 1 kx Energy Levels: 1 k mm 1 En n n 0,1,, c m m 1 Anharmonc Oscllator Energy Levels: E n 1 ~ 1 n hc n hcx ~ e n 0,1,,... Rgd Rotor Quantum Numbers: J = 0, 1,, 3,... M=0, ±1, ±,..., ±J h Energy Levels: ~ ~ h E J J J 1 J J 1hcB B 8 I 8 Ic Moment of Inerta: Lnear Polyatomc: m r Datomc: m m I 1 I r m1 m Boltzmann Dstrbuton: N g e E / kt Frst Order Perturbaton Theory: E (0) * H (1) (0) d Quadratc Equaton: b b ax bx c 0 x a 4ac INTEGRALS 1 1 sn xdx x sn x 4 x x sn x cos x xsn xdx x x 1 x cos x x sn xdx snx

2 Constants and Conversons: h = 6.63x10-34 J s ħ = h/ = 1.05x10-34 J s k = 1.38x10-3 J/K c = 3.00x10 8 m/s = 3.00x10 10 cm/s NA = 6.0x10 3 mol -1 1 Å = m 1 amu = 1.66x10-7 kg 1 J = 1 kg m /s 1 N = 1 kg m/s 1 N/m = 1 kg/s Cv Character Table CV E C V(xz) V (yz) A z x, y, z A Rz xy B x, Ry xz B y, Rx yz Transton Moment: x y M 0 M 0 M 0 j M z 0 k M M x 0 y 0 e e x 0 y 0 M z 0 e z 0 Raman Matrx Elements: uv 0 u = x, y, or z and v = x, y, or z

3 CHEM 510 Exam March 7, 015 Name (38) 1. For ths problem, assume that the molecule, dfluorodslyne, SF, s lnear, wth the structure: F-SS-F Note: Assume that the atomc mass of F s 19 amu and the atomc mass of S s 8 amu (8) (a) The second and thrd lnes n the rotatonal Raman spectrum of dfluorodslyne are found at frequences of cm -1 and 0.53 cm -1, respectvely. Calculate the value of the rotatonal constant, B, n cm -1. Note: The selecton rule for the rotatonal Raman spectrum s DJ = + Note: For parts, (b), (c), and (d) you can use the value, B ~ = 0.04 cm -1 f your answer to part (a) s far from ths value. (8) (b) For the Rotatonal Saffy spectrum, the selecton rule s DJ = +3. Calculate (1) the frequency of the fourth (4th.) lne n the Rotatonal Saffy spectrum of dfluorodslyne (n cm -1 ); () the energy of the photon absorbed for ths transton, Eph = DE (n J)

4 Prob. 1 (Cont'd) Note: For parts, (b), (c), and (d) you can use the value, B ~ = 0.04 cm -1 f your answer to part (a) s far from ths value. (10) (c) Calculate the rato of ntenstes of the 150th. lne to that of the 50th. lne n the rotatonal Raman spectrum of dfluoroslyne at 150 o C

5 Prob. 1 (Cont'd) Note: For parts, (b), (c), and (d) you can use the value, B ~ = 0.04 cm -1 f your answer to part (a) s far from ths value. (1) (d) The SS bond length n dfluorodslyne s RSS =.0 Å. Calculate the S-F bond length, RSF, n Å. Hnt: Frst calculate the moment of nerta, I, n kg m, and then convert to amu Å before proceedng. (1 kg m = 6.0x10 46 amu Å ) Note: Wth a bt of thought, you won t need to use the quadratc equaton to solve for RS-H. However, f you do need t, t s on the nfo. sheet Addtonal Space for Prob. 1d on next page f needed

6 Prob. 1d (Cont'd) Addtonal Space f needed

7 (10). The vbratonal force constant of Deuterum Fluorde, D- 19 F, s 840 N/m (Newton/meter) = 840 kg/s. Calculate the Zero-Pont vbratonal energy of D-F, n kj/mol.

8 (10) 3. The Thermodynamc Dssocaton Energy of F s D0 = 151. kj/mol, and the fundamental vbratonal frequency s 890 cm -1. Calculate the Spectroscopc Dssocaton Energy, De, of F.

9 (10) 4. The observed vbratonal frequency of the frst overtone (0) L-H radcal s at 671 cm -1. The frequency of the hot band (1) s at 131 cm -1. Calculate the harmonc frequency ( ~ ) and the anharmoncty constant (xe).

10 x (10) 5. The ground-state wavefuncton for a partcle n a box s: Asn a Consder a partcle n a box wth the perturbng potental: x sn a a V(x) x < 0 V0 V(x) = x a 0 x a V(x) x > a Calculate the frst order perturbaton theory correcton to the energy of a partcle n the ground-state wth the above perturbng potental. Note: Your answer should be of the form of a numercal value tmes V0.

11 () 6. Consder three of the CH bendng vbratons of cs-1,-dchloroethene, whch belongs to the Cv pont group. The vbratons (and symmetres) are: H F C C H F ~ 1 = 870 cm -1 (a) ~ = 1380 cm -1 (b1) ~ 3 = 760 cm -1 (b) (8) (a) Calculate frequency (1) n cm -1 and () n Hz (s -1 ) of the transton: (,0,1) (0,,). (6) (b) Consder the combnaton band 1 + [.e. (0,0,0) (1,,0)]. Determne whether or not ths transton s IR Actve. NOTE: You MUST show your work for credt.

12 (6) (c) Consder the combnaton mode, 1-3 [.e. (0,0,1) (1,0,0)]. Determne whether or not ths combnaton mode s IR actve. NOTE: You MUST show your work for credt. () (d) Whch, f any, of the above ndvdual fundamental vbratons (transton from 0 to 1) are IR Actve (just the answers - no explanaton necessary)