Group Theory Problems

Transcription

1 Grup Thery Prblems The fllwing table shws the vibratinal frequencies f CH. Assuming CH belngs t the T d pint grup, fill in the gaps in the table. Use fr and fr t designate type f vibratin. tretchorend ymmetry Iraman 2917cm 1 153cm cm 1 130cm 1 The fllwing table shws the vibratinal frequencies f CD. Assuming CD belngs t the T d pint grup, fill in the gaps in the table. Use fr and fr t designate type f vibratin. tretchorend ymmetry Iraman 2109cm cm cm 1 99cm 1 The fllwing table summarizes the infrared activity CH 2 D 2. Assuming it belngs t the C 2v pint grup, cmplete the table. Use fr and fr t designate the type f vibratin. cm tretchorend ymmetry Iraman The fllwing table summarizes the infrared activity f CH 3 D. Assuming CH 3 D belngs t the C 3v pint grup, fill in the gaps in the table. Use fr and fr t designate type f vibratin. tretchorend ymmetry Iraman 295cm cm cm cm cm cm 1 Frmaldehde has vibratinal frequencies at 283, 277, 1501, 1251, 17, and 117 cm -1. Hw many are I active and hw many are aman active. eparate them int s and es. cm ymmetry tretchorend Iraman 283 tretch I 277 tretch I 1501 end I end I 17 2 tretch I end I

2 PH 3 (C 3v ) has I and aman active vibratins at 221, 2327, 1121, and 991 cm -1. PD 3 has I and aman active vibratins at 198, 19, 80 and 730 cm -1. Make assignments in terms f es and s. The infrared spectrum f Cl 3 shws vibratinal bands at 995, 80, and 2 cm -1, while aman bands appear at 995, 71, and 2 cm -1. Is the gemetry f the mlecule trignal pyramid (C 3v ) r trignal planar (D 3h )? Is yur answer t this questin cnsistent with chemical bnding principles? Assign symmetry labels t the vibratinal bands and identify the es and s. Iraman ymmetry tretchorend 995cm 1 I 80cm 1 I 71cm 1 2cm 1 I CH 3 CN has 12 vibratinal degrees f freedm, but 8 fundamental vibratinal frequencies appear in the infrared at 2999, 292, 229, 10, 137, 112, 918, and 380 cm -1. xplain. The infrared spectrum f methane shws tw absrptins: a at 130 cm -1 and a at 3019 cm -1. Demnstrate that a symmetry analysis assuming tetrahedral symmetry fr methane is cnsistent with this spectrscpic data. Als predict hw many aman active mdes methane shuld have. The infrared spectrum f XeF has absrptins at 11, 291, and 58 cm -1 (tw s, ne ), while the aman spectrum has peaks at 218, 52, and 55 cm -1 (ne, tw es). Is its mlecular structure tetrahedral r square planar? eferences: J. Am. Chem. c. 193, 85, 1927; J. Phys. Chem. 1971, 5, 527. Cubane, C 8 H 8, has 2 vibratinal degrees f freedm, but nly three I active mdes. Cubane belngs t the ctahedral pint grup. hw that grup thery predicts three I active mdes. Determine hw many vibratinal mdes will be aman active. Will there be any cincidences between the I and aman active mdes? The synthesis and characterizatin f cubane was reprted in 19 by Philip atn and Thmas Cle in JAC 19, 8, They reprted three I bands at 3000, 1231, and 851 cm -1. When Paquette's grup synthesized ddecahedrane, C 20 H 20, they measured its infrared and aman spectra (JAC 1983, 105, 5-550). They fund three I active bands at 295, 1298, and 728 cm -1 and eight aman frequencies at 292, 2938, 132, 11, 1092, 80, 7, and 80 cm -1. Use grup thery t shw that these data are cnsistent with the fact that ddecahedrane has icsahedral symmetry.

3 cm ymmetry I tretchorend T 1u I 1298 T 1u I 728 T 1u I ben ulfur tetrafluride represents a difficult case which can't be reslved t cmplete satisfactin n the basis f I and aman data alne. The experimental spectra shw eight (five certain and three likely) I bands and eight (five certain and three likely) aman bands, and eight (five certain and three likely) cincidences between the tw. This infrmatin plus nmr spectra lead t the cnclusin that the symmetry is C 2v r the see-saw structure predicted by VP. The March 28, 2003 issue f cience reprted the synthesis and characterizatin f Al 2 H, the aluminum analg f dibrane (therefre, dialane). The researchers reprted the fllwing experimental I frequencies in cm -1 : 1u (1932, 83); 2u (128, 32); 3u (1915, 108,702). D a symmetry analysis f Al 2 H which belngs t the D 2h pint grup. Hw many vibratinal mdes are there? Which nes are I active and what are their symmetry designatins. The researchers reprted that ne f the expected lw frequency vibratins (~200 cm -1 ) was t bserved. What is the symmetry f this vibratinal mde? uckminsterfullerene (C 0 ) has fur I active vibratinal mdes (528, 577, 1180, 130 cm -1 ) and ten aman acitve mdes (273, 3, 9, 710, 773, 110, 1250, 135, 170, 1570 cm -1 ). Demnstrate that the assumptin f icsahedral symmetry fr C 0 is cnsistent with this data. D a symmetry analysis f the 0 rbitals f C 0 and shw that the results are in agreement with a Huckel calculatin. The fllwing aman and I frequencies have been bserved fr the tetrahedral P mlecule. Is the assignment f tetrahedral gemetry t this mlecule in agreement with the spectrscpic data? xplain. 1 cm 372 cm I cm Istpic subsitutin fr ne f the 31 P atms with a 32 P atm reduces the symmetry t C 3v. ed the symmetry analysis and predict the number f I and aman active vibratinal mdes. Tetrahedrane, C H, belngs t the T d pint grup. Use grup thery t predict the number f I and aman acitive vibratinal mdes it has. Predict als the number f es and s will appear in each type f spectrscpy. T date tetrahedrane has t been synthesized. D symmetry analyses n cis-diflurethene (C 2v ) and trans-diflurethene (C 2h ). Can spectrscpic methds (I and aman) be used t distinquish between these ismers. xplain. What abut 1,1 diflurethene? What pint grup des it belng t? Can spectrscpic methds distinquish it frm the cis and trans ismers examined abve? Cis-MA 2 2 has C 2v symmetry and trans-ma 2 2 has D 2h symmetry. Determine the I and aman active mdes fr each mlecule and discuss hw such spectrscpic evidence can be used t distinguish the tw ismers. CH 3 Cl has I and aman active mdes at 3017, 2937, 152, 1355, 1017, and 732 cm -1. Is this data cnsistent with C 3v symmetry assignment fr chlrmethane?

4 Dibrane, D 2h, has 18 vibratinal degrees f freedm. Nine mdes are aman active and nine are I active. There are cincidences. D a symmetry analysis f dibrane t cnfirm the assignments made in the table belw. Identify es and s. The xy plane is the plane f the paper. The fur terminal H atms f dibrane lie in the xz plane and the tw bridging atms lie in the xy plane. D 2h aman cm tretchorend D 2h I cm tretchorend 252 A u ia 210 1u u u 38 1g 178 2u g u 973 2g u g 920 3u 10 3g u 1177 The fllwing I and aman spectrscpic data is available fr cyclprpane, C 3 H. Demnstrate that this data is cnsistent with a D 3h symmetry assignment fr cyclprpane. In additin cmplete the table belw. cm 1 ymmetry tretchorend 3038 I 179 I 1188 I 3025 I 138 I 1029 I 8 I 3103 I 85 I cm 1 ymmetry 3038 A1' 179 A1' 1188 A1' 3025 I ' 138 I ' 1029 I ' 8 I ' 3103 I A2'' 85 I A2'' 3082 '' 1188 '' 73 ''

5 PX 5 has trignal bipyramidal gemetry and therefre belngs t the D 3h pint grup. Use the mdel prvided t d a symmetry analysis f PX 5 by determining uma and bnds. The fllwing spectrscpic infrmatin is available. ymmetry tretchorend 81cm 1 8cm 1 97cm 1 I 525cm 1 I 102cm 1 I 533cm 1 I 17cm 1 I 520cm 1 Is the symmetry analysis cnsistent with the spectrscpic data? xplain in detail. ymmetry tretchorend 81cm 1 8cm 1 97cm 1 I A 2 525cm 1 I A 2 102cm 1 I 533cm 1 I 17cm 1 I 520cm 1 The fllwing I and aman vibratinal data is available fr tetrahedral methane. Cmplete the table. Als shw that vibratinal data is t cnsistent with a square planar (D h ) r square pyramid (C v ) gemetry. ymmetry tretchorend 3019cm 1 I 2717cm 1 153cm 1 130cm 1 I The fllwing I and aman spectrscpic data is available fr ethene, C 2 H. Demnstrate that this data is cnsistent with a D 2h symmetry assignment fr ethene. In additin cmplete the table belw. cm 1 tretchorend ymmetry 3108 Ag 310 I 2u 3019 Ag 2990 I 3u 123 1g 1 I 1u 132 Ag 123 1g 99 2g 93 I 2u 810 I 3u I and aman data fr XeOF which has C v symmetry. stablish that the symmetry assignment is crrect. cm ymmetry tretchorend 92 I 57 I 28 I I 3 I 11 I

6 I and aman data fr [PtCl ] 2- which has D h symmetry. stablish that the symmetry assignment is crrect. ymmetry tretchorend 332cm 1 g 320cm 1 I u 31cm 1 1g 183cm 1 I u 170cm 1 2g 93cm 1 I A 2u The vibratinal mdes f the ins [rf 2 ] - and [rf 2 ] + are given belw. Identify which ne is A and which ne is. A 59( I) 2( ) 198( I) 715( I ) 70( I ) 3( I ) The fllwing spectrscpic data fr [ClO ] - is available. hw that it is cnsistent with Td symmetry. ymmetry tretchorend 1102cm 1 I 935cm 1 28cm 1 I 2cm 1 MF 5 has trignal bipyramidal gemetry and therefre belngs t the D 3h pint grup. Use the mdel prvided t d a symmetry analysis f MF 5 by determining uma and bnds. The fllwing spectrscpic infrmatin is available. ymmetry tretchorend 77cm 1 732cm 1 I 703cm 1 85cm 1 I 500cm 1 I 0cm 1 239cm 1 I 203cm 1 I The aman and I spectra f a sample f N 2 F 2 are measured and the results are shwn belw. Is the sample cis-diflurdiazine (C 2v ) r trans-diflurdiazine (C 2h )? ymmetry tretchorend 13cm cm 1 989cm 1 I 592cm 1 12cm 1 I 30cm 1 I enzene has I active mdes at 75, 1035, 179, and 303 cm -1. Demnstrate that this is cnsistent with D h symmetry. eparate the vibratins int es and s. D a symmetry analysis f the rbitals f benzene and shw that it is cnsistent with the results f a Huckel calculatin n benzene.

7 Allene, C 3 H, belngs t the D 2d pint grup. cm 1 ymmetry endortretch The fllwing data is available fr ethene and deuterated ethene which have D 2h symmetry. T each f the vibratinal mdes determine whether it is a r a and its symmetry. C2H C2D I I I IA I I tretch end ymmetry H 3 has D 3h symmetry. Determine the reducible representatin fr the hydrgen 1s rbitals (cllectively). What linear cmbinatin f brn valence rbitals has the same symmetry. CH has T d symmetry. Determine the reducible representatin fr the hydrgen 1s rbitals (cllectively). What linear cmbinatin f carbn valence rbitals has the same symmetry. The fllwing vibratinal frequencies are available fr cis-dichlrethene. Cmplete the table assuming that the mlecule has C 2v symmetry. cm ymmetry I aman A 2 0 A

8 The fllwing vibratinal frequencies are available fr trans-dichlrethene. Cmplete the table assuming that the mlecule has C 2h symmetry. cm ymmetry I aman g 900 A u 227 A u 3090 u 1200 u 828 u 250 u The fllwing vibratinal frequencies are available fr 1,1-dichlrethene. Cmplete the table assuming that the mlecule has C 2v symmetry. cm ymmetry I aman A Pyrazine has D 2h symmetry. Cnfirm the entries in the fllwing table. ymmetry cm I ymmetry cm I 305 tretch A u na end I 1578 tretch A u 33 end I 1230 tretch 1u 30 tretch I 1015 end 1u 18 tretch I 59 end 1u 1135 end I 1g 757 end 1u 1021 end I 2g 919 end 2u 30 tretch I 2g 703 end 2u 118 tretch I 3g 301 tretch 2u 13 tretch I 3g 152 tretch 2u 103 end I 3g 1118 end 3u 80 end I 3g 1 end 3u 1 end I

9 Is the fllwing spectrscpic data cnsistent with the assignment f D h symmetry t tetrachlrplatinate, PtCl 2-? ymmetry tretchorend 332cm 1 31cm 1 170cm 1 320cm 1 I 183cm 1 I 93cm 1 I ymmetry tretchorend 332cm 1 g 31cm 1 1g 170cm 1 2g 320cm 1 I u 183cm 1 I A 2u 93cm 1 I 2u Given the fllwing spectrscpic data determine whether InCl 5 2- has C v r D 3h symmetry. cm ymmetry tretchorend 29 I I 27 I I 10 I 108 I ymmetry analysis supprts C v symmetry as indicated belw. xp C v D 3h I Cincidences 3 D symmetry analyses n the three C 10 H 8 ismers: azulene (C 2v ), fulvalene (D 2h ) and napthalene (D 2h ). Determine whether I and aman spectrscpy can be used t distinguish between the ismers. Cmpleting the table belw shuld facilitate answering the questin. Azulene Fulvalene Napthalene Mlecule Azulene Fulvalene Napthalene I tretches aman tretches Cincidences I ends aman ends Cincidences InActive Mdes 0

10 Prpellane has, as shwn belw, has D 3h symmetry. T date it hasn't been synthesized, but thereticians debate whether r t it has a bridging carbn-carbn bnd as shwn in the figure. D a symmetry analysis with and withut the bridging bnd t determine whether vibratinal spectrscpy culd decide the issue if the mlecule ever became available. Prpellane nd Nnd I tretches aman tretches 9 8 I ends aman ends 7 InActive Mdes 3 3 aman spectrscpy wuld be required t answer this questin, because it indicates ne less and ne mre in the vibratinal spectrscpy. This makes sense since the number f vibratinal degrees f freedm must be cnserved. If there is ne less bnd, there must be ne less and therefre ne mre. In this case, the change is bservable (hypthetically) in aman spectrscpy.