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Ch 4 Problems 5Th Edition.Pdf Problems | 111 R e f e r e n c e s 1. M. Gerken, G. J. Schrobilgen, Inorg. Chem. , 2002 , 2. J. A. Boatz, K. O. Christe, D. A. Dixon, B. A. Fir, 41 , 198. M. Gerken, R. Z. Gnann, H. P. A. Mercier, G. J. Schrobilgen, Inorg. Chem ., 2003 , 42 , 5282. General References There are several helpful books on molecular symmetry and Symmetry Through the Eyes of a Chemist , 2nd ed., Plenum Press, its applications. Good examples are D. J. Willock, Molecular New York, 1995. The last two also provide information on space S y m m e t r y , John Wiley & Sons, Chichester, UK, 2009; F. A. groups used in solid state symmetry, and all give relatively gentle Cotton, Chemical Applications of Group Theory, 3 r d e d . , J o h n introductions to the mathematics of the subject. For an explana- Wiley & Sons, New York, 1990; S. F. A. Kettle, Symmetry and tion of situations involving complex numbers in character tables, Structure: Readable Group Theory for Chemists , 2 n d e d . , J o h n see S. F. A. Kettle, J. Chem. Educ. , 2009 , 86 , 6 3 4 . Wiley & Sons, New York, 1995; and I. Hargittai and M. Hargittai, Problems 4.1 Determine the point groups for 4.5 Determine the point groups for Cl a. Ethane (staggered conformation) a. 1,1Ј - Dichloroferrocene b. Ethane (eclipsed conformation) c. Chloroethane (staggered conformation) d. 1,2-Dichloroethane (staggered anti conformation) b. Dibenzenechromium Fe 4.2 Determine the point groups for (eclipsed conformation) a. Ethylene H H b. Chloroethylene CC H H c. The possible isomers of Cl dichloroethylene Cr 4.3 Determine the point groups for a. Acetylene b. H i C ‚ C i F c. H i C ‚ C i CH3 c. d. H i C ‚ C i CH2Cl e. H i C ‚ C i Ph (Ph = phenyl) 4.4 Determine the point groups for Cr a. Naphthalene + d. H3O F e. O F b. 1,8-Dichloronaphthalene Cl Cl 2 2 OO f. Formaldehyde, H2CO F S S g. S (puckered ring) S 8 S S S S S c. 1,5-Dichloronaphthalene Cl h. Borazine (planar) H H B N Cl HHN B d. 1,2-Dichloronaphthalene Cl B N Cl H H 112 Chapter 4 | Symmetry and Group Theory 3- i. [Cr(C2O4)3] O 3- e. A hexagonal pencil with a round eraser O f. The recycle symbol, in three dimensions O O O Cr O O O O O O O j. A tennis ball (ignoring the label, but including the pat- tern on the surface) 4.6 Determine the point groups for g. The meander motif a. Cyclohexane (chair conformation) b. Tetrachloroallene Cl2C “ C “ CCl2 2- c. SO4 d. A snowflake e. Diborane H h. An open, eight-spoked umbrella with a straight handle H H i. A round toothpick BB H H j. A tetrahedron with one green face, the others red H 4.9 Determine the point groups for a. A triangular prism f. The possible isomers of tribromobenzene b. A plus sign g. A tetrahedron inscribed in a cube in which alternate c. A t-shirt with the letter T on the front corners of the cube are also corners of the tetrahedron. d. Set of three wind turbine blades h. B3H8 H H e. A spade design (as on a deck of playing cards) H H BB B f. A sand dollar H H H H i. A mountain swallowtail butterfly. j. The Golden Gate Bridge, in San Francisco, CA 4.7 Determine the point groups for a. A sheet of typing paper b. An Erlenmeyer flask (no label) c. A screw d. The number 96 g. Flying Mercury sculpture, by Giambologna at the e. Five examples of objects from everyday life; select Louvre in Paris, France items from five different point groups h. An octahedron with one blue face, the others yellow f. A pair of eyeglasses, assuming lenses of equal strength i. A hula hoop g. A five-pointed star j. A coiled spring h. A fork with no decoration 4.10 Determine the point groups for the examples of symmetry i. Wilkins Micawber, David Copperfield character who in Figure 4.1. wore a monocle 4.11 Determine the point groups of the molecules in the follow- j. A metal washer ing end-of-chapter problems from Chapter 3: 4.8 Determine the point groups for a. Problem 3.40 a. A flat oval running track b. Problem 3.41 b. A jack (child's toy) 4.12 Determine the point groups of the molecules and ions in a. Figure 3.8 b. Figure 3.15 4.13 Determine the point groups of the following atomic orbit- als, including the signs on the orbital lobes: a. px b. dxy c. dx2 -y2 c. A person's two hands, palm to palm d. dz2 d. A rectangular towel, blue on front, white on back e. fxyz Problems | 113 4.14 a. Show that a cube has the same symmetry elements as c. Honduras an octahedron. b. Suppose a cube has four dots arranged in a square on each face as shown. What is the point group? c. Suppose that this set of dots is rotated as a set 10° clock- wise on each face. Now what is the point group? d. Field of stars in flag of Micronesia 4.15 Suppose an octahedron can have either yellow or blue faces. a. What point groups are possible if exactly two faces are blue? b. What points are possible if exactly three faces are blue? c. Now suppose the faces have four different colors. e. Central design on the Ethiopian flag: What is the point group if pairs of opposite faces have f. Turkey identical colors? g. Japan 4.16 What point groups are represented by the symbols of h. Switzerland chemical elements? i. United Kingdom (be careful!) 4.17 Baseball is a wonderful game, particularly for someone 4.19 Prepare a representation flowchart according to the format interested in symmetry. Where else can one watch a batter of Table 4.8 for SNF3. step from an on-deck circle of a symmetry to a rectangular 4.20 For trans-1,2-dichloroethylene, which has C2h symmetry, batter's box of b symmetry, adjust a cap of c symmetry (it a. List all the symmetry operations for this molecule. has OO on the front, for the Ozone City Oxygens), swing b. Write a set of transformation matrices that describe the a bat of d symmetry (ignoring the label and grain of the effect of each symmetry operation in the C2h group on a wood) across a home plate of e symmetry at a baseball set of coordinates x, y, z for a point (your answer should that has f symmetry (also ignoring the label) that has been consist of four 3 * 3 transformation matrices). thrown by a chiral pitcher having g symmetry, hit a tower- c. Using the terms along the diagonal, obtain as many ing fly ball that bounces off the fence, and race around the irreducible representations as possible from the trans- bases, only to be called out at home plate by an umpire who formation matrices. You should be able to obtain three may have no appreciation for symmetry at all. irreducible representations in this way, but two will be 4.18 Determine the point groups for the following flags or parts duplicates. You may check your results using the C2h of flags. You will need to look up images of flags not character table. shown. d. Using the C2h character table, verify that the irreducible a. Botswana representations are mutually orthogonal. 4.21 Ethylene has D2h symmetry. a. List all the symmetry operations of ethylene. b. Write a transformation matrix for each symmetry opera- tion that describes the effect of that operation on the coordinates of a point x, y, z. c. Using the characters of your transformation matrices, obtain a reducible representation. d. Using the diagonal elements of your matrices, obtain three of the D2h irreducible representations. b. Finland e. Show that your irreducible representations are mutually orthogonal. 4.22 Using the D2d character table, a. Determine the order of the group. b. Verify that the E irreducible representation is orthogo- nal to each of the other irreducible representations. 114 Chapter 4 | Symmetry and Group Theory c. For each of the irreducible representations, verify that 4.30 The structure of 1,1,2,2-tetraiododisilane is shown here. the sum of the squares of the characters equals the order (Reference: T. H. Johansen, K. Hassler, G. Tekautz, K. of the group. Hagen, J. Mol. Struct ., 2001 , 598 , 171.) d. Reduce the following representations to their compo- nent irreducible representations: H I I Ј D2d E 2S4 C2 2C2 2sd Si Si I ⌫ 6 0 2 2 2 1 I H ⌫2 6 4 6 2 0 a. What is the point group of this molecule? 4.23 Reduce the following representations to irreducible repre- b. P r e d i c t t h e n u m b e r o f I R - a c t i v e Si i I s t r e t c h i n g sentations: vibrations. c. Predict the number of Raman-active Si i I stretching vibrations. C E 2C 3s 3v 3 v - 4.31 Both cis and trans isomers of IO2F4 have been observed. ⌫1 6 3 2 Can IR spectra distinguish between these? Explain, sup- ⌫2 5 -1 -1 porting your answer on the basis of group theory.
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