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All Textbook Solutions for Physical Chemistry

a Construct Slater determinant wavefunctions for Be and B. Hint: Although you need only include one p orbital for B, you should recognize that up to six possible determinants can be constructed. b How many different Slater determinants can be constructed for C, assuming that the p electrons spread out among the available p orbitals and have the same spin? How many different Slater determinants are there for F?12.25E 12.26E 12.27E Suppose an electron had three possible values of ms, designated , and . What is the Slater determinant for Li? For Be?Using a periodic table or Table 12.1, find the elements whose electron configurations do not follow the aufbau principle strictly. Comment on any relationship between these elements or their place within the periodic table.Write an acceptable electron configuration for these atoms, including ml and ms values. a Li b N c Ne d Sc 12.31E 12.32E 12.33E An anharmonic oscillator has the potential function V=12kx2+cx4 where c can be considered a sort of anharmonicity constant. Determine the energy correction to the ground state of the anharmonic oscillator in terms of c, assuming that H is the ideal harmonic oscillator Hamiltonian operator. Use the integral table in Appendix 1 in this book.12.35E In a particle-in-a-box having length a, the potential energy is given by the function V=kx2 Calculate the average energy of a particle in terms of its mass m, the length of the box a, and the constant k.12.37E 12.38E 12.39E The Stark effect is the change in energy of a system due to the presence of an electric field discovered by German physicist Johannes Stark in 1913. Consider the hydrogen atom. Its normally spherical 1s orbital distorts slightly when exposed to an electric field. If the electric field is considered to be in the z direction, then the field acts to introduce, or mix, some 2pz character in with the 1s orbital. The atom is said to be polarizable, and the extent to which it changes is considered a measure of the atoms polarizability which is designated by the letter , not to be confused with the spin function . The perturbation Hamiltonian is defined as H=eErcos, where e is the charge on the electron, E is the strength of the electric field, and r and represent coordinates of the electron. Evaluate the perturbation energy of the hydrogen atom. You will have to integrate all the three spherical polar coordinates in your evaluation of H. A similar effect, the Zeeman effect, exists for magnetic fields. It too can be treated using perturbation theory.12.41E 12.42E 12.43E Show that a variation theory treatment of H using =ekr as an unnormalized trial function yields the correct minimum-energy solution for the hydrogen atom when the specific expression for k is determined.12.45E Explain why assuming an effective nuclear charge, as used for our treatment of the helium atom in Example 12.11, is unnecessary for a treatment of the hydrogen atom.12.47E Consider a real system. Assume that a real wavefunction is a combination of two orthogonal functions such that H11=15, H22=4, and H12=H21=2.5 arbitrary energy units. Assume S11=S22=1 and S12=S21=0. aEvaluate the approximate energies of the real system and evaluate the coefficients of the expansion a=ca,11+ca,22. b Compare your answer to the answers in exercise 12.12 and comment.12.49E 12.50E 12.51E 12.52E State the Born-Oppenheimer approximation in words and mathematically, and indicate how the mathematical form is implied by the statement.12.54E Spectroscopy deals with differences in energy between levels. Derive an expression for E, the difference in energy, between the two molecular orbitals of H2+.12.56E What is the bond order for the lowest excited state of H2+ ? From this single result, propose a general statement about unstable diatomic molecules and bond orders.The helium atom was defined as two electrons and a single nucleus, and the hydrogen molecule ion was defined as a single electron and two nuclei. It seems that the only difference is an exchange in the identity of the particles in the system; however, their quantum mechanical treatment is completely different, as are the results. Explain why.Explain how we know that the first in equation 12.43 is the wavefunction for the bonding orbital, and that the second in equation 12.43 is the wavefunction for the antibonding orbital.12.60E 12.61E Use molecular orbital arguments to decide whether or not the difluoride dianion, F22, should exist as a stable ion.12.63E 12.65E 12.67E 12.68E In your own words, explain why an object that has more symmetry elements is said to have higher symmetry than an object with fewer symmetry elements.Identify the symmetry elements present in the following objects. a A ream of blank paper, no holes. b A ream of blank three-holed paper. c A round pencil, unsharpened, with cylindrical eraser. d A round pencil, sharpened, with cylindrical eraser.Identify the symmetry elements present in the following objects. a The Eiffel Tower. You may have to look up a picture of it if you dont remember its shape b Any book ignore the printing. c An octagonal wood block. d A jack from the set of jacks pictured here: Note that some of the points end differently.13.4E 13.5E 13.6E 13.7E 13.8E Any axis of symmetry Cn that rotates an object by =360/n about the z-axis can be generalized by the matrix Cn=[cossin0sincos0001] 13.18 See Table 13.1 Determine the matrix forms of the C2,C3,C6, and C1 rotation operators.13.10E 13.11E 13.12E 13.13E What are the number of classes and the order of the following point groups? a C2v b D2h c D6h d S4 e Cs.13.15E a Show that the C3v point group satisfies the closure property of a mathematical group. b Show that the C3v point group satisfies the associative law by evaluating v(EC3) and (vE)C3.a In the Td point group, an S41 improper rotation is equivalent to what other improper rotation? b In the D6h point group, the symmetry operation labeled C21 is equivalent to what other symmetry operation?Determine which single symmetry operation of the following point groups is equivalent to the given combination of multiple symmetry operations. a In C2v, C2v=? b In C2h, iC2=? c In D6h, C6h=? d In D2d, C2C2=? e In Oh, iS4=?13.19E 13.20E 13.21E Figure 13.27 shows the structure of the molecule porphine. Figure 13.27 The structure of porphine. Determine the symmetry elements present in the molecule, and its point group. Does the point group change if an Fe2+ ion is substituted for the two hydrogen atoms in the center of the porphine ring?13.23E Identify all the symmetry elements present in the tetrahedron, the cube, and the octahedron.Point groups are called such because all of the symmetry elements in the group intersect at one point in space. For point groups that have i as a symmetry operation, why must i be at that point?Determine the point groups of the following molecules. a Hydrogen peroxide, H2O2 it is not planar. b Allene, CH2=C=CH2 c D-alanine Its absolute stereochemistry can be found in any good organic or biochemistry textbook. d L-alanine, and compare with part c above.Determine the point group of the following molecules. a cis1,2 Dichloroethylene b trans1,2 Dichloroethylene c Toluene, C6H5CH3 d 1,3-Cyclohexadiene.Determine the point groups of the following molecules. a Hydrogen selenide, H2Se b Partially deuterated hydrogen sulfide, or HDS c The chair conformer of cyclohexane, C6H12 d The boat conformer of cyclohexane, C6H12 Determine the point groups of the following molecules. a Fe(CO)5, which has a trigonal bipyramidal structure, b CO32, which has three resonance structures that contribute to its overall shape, c The perfectly staggered conformation of ethane, d The perfectly eclipsed conformer of ethane.Structural isomers can have very different point groups. Determine the point groups of 1,4cyclohexadiene and 1,3cyclohexadiene, which both have the molecular formula C6H8.Structural isomers can have very different point groups. There are three isomers of dichlorobenzene. Identify the point groups of the three isomers.13.32E Identify the point group of the wave functions of the following molecules. a Deuterium oxide, D2O where D=2H b Boron trichloride, BCl3 c Phosphorous trichloride, PCl3 Identify the point group of the wave functions of the following molecules. a Hydrogen chloride, HCl b Sulfur dioxide, SO2 c Sulfur trioxide, SO3 13.35E Determine if the following species have permanent dipole moments. a Dichloromethane, CH2Cl2 b Chlorobenzene, C6H5Cl c Ammonia, NH3 d Carbon dioxide, CO2.Determine if the following species have permanent dipole moments. a The carbonate ion, CO32 b The phosphate ion, PO43 c Uranium hexafluoride, UF6 d Bromine, Br2.Which of the following species will not have permanent dipole moments? a Hydrogen cyanide, HCN b Carbonyl sulfide, OCS c Phosphorous pentachloride, PCl5 d Trimethylamine, N(CH3)3 e Boron trifluoride, BF3 f Diborane, which has the following structure: Where the bridging hydrogens are perpendicular to the four terminal hydrogens, which are all coplanar 13.39E Explain why a molecule with a center of inversion cannot have a dipole moment.a Unlike methane, bromochlorofluoromethane (CHBrClF) is chiral. Determine all symmetry elements that are present in CHBrClF and identify its point group. b If the fluorine in this molecule were substituted with a hydrogen atom, what is the point group for the new molecule? Is it chiral?13.42E 13.43E 13.44E Show that the irreducible representations of the D2d point group satisfy the closure requirement. You will have to use the great orthogonality theorem to reduce one combination.Show that any two of the irreducible representations of the following point groups are orthogonal to each other. a C2 b C2v c D2h d Oh e Td Show that any irreducible representation of these point groups is normalized. a C2 b D2d c Oh d C4h e C6v Explain why this proposed irreducible representation for C2v is impossible. EC2A?1100 13.49E 13.50E Why is it unnecessary to consider whether an irreducible representation from C4h is orthogonal to an irreducible representation of D6h?13.52E 13.53E 13.54E 13.55E 13.56E 13.57E 13.58E Reduce the following reducible representations using the great orthogonality theorem. a In the C2 point group: EC251 b In the C3v point group: E2C23v600 c In the D4 point group: E2C4C22C22C262224 d In the Td point group: E8C33C26S46d72311 Determine the resulting representations for the following products of irreducible representations. a In C3v, A1A2 b In C6v, E1E2 c In D3h, A2A1E d In D6h, B2gB2u 13.61E Without using the great orthogonality theorem, reduce the given irreducible representation in C2v symmetry. Does your answer make sense? EC25555 Assume that you are evaluating the integral of products of functions having symmetry labels in exercise 13.60, parts a-d. Which integrals, if any, are exactly zero due to symmetry considerations?13.64E Assume that x- polarized light can be assigned an E symmetry species in a system that has C4v symmetry. Can a transition from E state to a B2 state occur? Why or why not?13.66E 13.67E 13.68E 13.69E 13.70E Construct the symmetry-adapted linear combination molecular orbitals for hydrogen sulfide, H2S.13.72E 13.73E 13.74E 13.75E 13.76E 13.77E Suppose you use p0,p1 and p+1 along with s orbitals to construct hybrid orbitals. Will they be the same hybrid orbitals defined by the px,py and pz orbitals? Justify your answer.Show that the individual sp orbitals, as written in equation 13.16, are orthogonal.13.80E What is the rough hybridization of the carbon orbitals in the methyl carbonium ion, CH3+, which is almost perfectly planar triangular in shape?Determine the symmetry species of the D3h point group for the sp2 hybrid orbitals, assuming that the C3 axis is coincident with the z-axis and that one of the orbitals lies along the positive x-axis. See Example 13.16.Determine the D3h symmetry species of the sp3d hybrid orbitals, assuming that the C3 axis is coincident with the z-axis and that one of the orbitals lies along the positive x-axis. See Example 13.16.13.84E In propene CH3CH=CH2, the first carbon has sp3 hybrid orbitals and the second carbon has sp2 hybrid orbitals. These orbitals interact to make a bond. Why are these hybrid orbitals not orthogonal?13.87E 13.88E 13.89E 14.1E Determine if the following integrals can be nonzero if the molecular or atomic system has the given local symmetry. Use the great orthogonality theorem if necessary. a AuOB2uAud in D2h symmetry b A1OA1A2d in C3v symmetry c g+Oggd in Dh symmetry d EOA2T1d in Td symmetry What is the frequency of light having the following wavelengths? a 1.00m b 4.77105m c 7894A d 1.903103m What is the wavelength of light having the given frequencies? a 1.441013s1 b 7.30107Hz c 106.5MHz d 6.63103s1 What is the energy of light having each characteristic? a v=6.031014s1 b =9.27nm c v=4320cm1 d =5.69 The Cu(H2O)62+ complex has octahedral symmetry. Is a transition from a T2g state to an Eg state allowed if the transition moment operator has a T1u symmetry label?What are the wavelength, speed, and energy of a photon that has a frequency of 8.0411012s1?14.8E 14.9E 14.10E 14.11E 14.12E 14.13E 14.14E Diatomic sulfur, S2, was detected in the tail of Halleys comet when it last approached Earth in 198586. It has a bond length of 1.8801010m. Calculate the value of B, in units of cm1 and J, for S2.14.16E 14.17E 14.18E 14.19E 14.20E 14.21E 14.22E Which of the following molecules should have pure rotational spectra? a Deuterium, D2 D is 2H b Carbon monoxide, CO c cis1,2Dichloroethylene d trans1,2Dichloroethylene e Chloroform, CHCl3 f Buckminsterfullerene, C60 Which of the following molecules should have pure rotational spectra? a Dimethyltriacetylene, H3CCCCCCCCH3 b Cyanotetraacetylene, HCCCCCCCCCN Such molecules have been detected in intersteller space. c Nitric oxide, NO d Nitrogen dioxide, NO2 e Sulfur tetrafluoride, SF4 f Sulfur hexafluoride, SF6 The following are sets of rotational quantum numbers (J,MJ,K). Label each indicated transition as either allowed or forbidden. Hint: Remember the rules for allowed values of the various quatum numbers. a (0,0,0)(1,1,0)b (0,0,0)(1,0,0) c (3,2,1)(3,1,1)d (4,4,1)(2,4,1)The following are sets of rotational quantum numbers (J,MJ,K). Label each indicated transition as either allowed or forbidden. Hint: Remember the rules for allowed values of the various quantum numbers. a (5,4,0)(3,6,0)b (8,2,2)(9,2,2) c (7,4,2)(7,4,2)d (4,2,5)(3,2,5)Derive equation 14.21 from the E expression immediately preceding it.14.28E 14.29E Lithium hydride, 7Li1H, is a potential fuel for fusion reactors because it is one of the few compounds of very small elements that exists as a solid, and is therefore more dense than gaseous fuels even under extreme conditions. LiH in the gas phase has a pure rotational spectrum consisting of lines spaced by 15.026cm1. Calculate the bond distance in LiH.14.31E 14.32E 14.33E 14.34E 14.35E 14.36E From the data in Table 14.2, predict B for DCl D is 2H.A colleague states that the pure rotational spectrum of CH4 is a set of absorptions spaced by 10.48cm1. How would you refute this?14.39E 14.40E 14.41E 14.42E 14.43E Determine E for J=20J=21 for HBr assuming it acts ideally and assuming it experiences centrifugal distortion. See Table 14.2 for data.Determine the number of total degrees of freedom and the number of vibrational degrees of freedom for the following molecules. a Hydrogen fluoride, HF b Hydrogen telluride, H2Te c Buckminsterfullerene, C60 d Phenylalanine, C6H5CH2CHNH2COOH e Naphthalene, C10H8 f The linear isomer of the C4 radical g The bent isomer of C4 radical.Determine the number of total degrees of freedom and the number of vibrational degrees of freedom for the following species. a Hydrogen sulfide, H2S b Carbonyl sulfide, OCS c The sulfate ion, SO42 d Phosgene, COCl2 e Elemental chlorine, Cl2 f A linear molecule having 20 atoms g A nonlinear molecule having 20 atoms 14.47E 14.48E 14.49E 14.50E 14.51E 14.52E 14.53E 14.54E 14.55E 14.56E 14.57E 14.58E 14.59E 14.60E 14.61E 14.62E 14.63E 14.64E 14.65E 14.66E 14.68E 14.69E 14.70E 14.71E 14.72E 14.73E 14.74E 14.75E 14.76E 14.77E 14.78E 14.79E 14.80E 14.81E 14.82E 14.83E 14.84E 14.85E Dioctyl sulfide, (C8H17)2S, and hexadecane, C16H34, have very similar vibrational spectra. Use a correlation table why.Where would you expect vibrations for ethyl alcohol, CH3CH2OH, to appear in a vibrational spectrum?14.88E 14.89E 14.90E 14.91E 14.92E 14.93E 14.94E The mutual exclusion rule states that for certain molecules, vibrations that are IR-active are not Raman-active, and vice versa. Molecules must have a certain symmetry element in order for the mutual exclusion rule to apply. Examine the character tables in Appendix 3 and determine what that symmetry element is.14.96E 14.97E 14.98E 14.99E Construct and compare the energy level diagrams for the rotations of a diatomic molecule assuming it acts as a rigid rotor equation 14.21 and a rigid rotor with centrifugal distortion corrections equation 14.26. Use HBr as a model system, where B=8.473cm1 and DJ=3.72104cm1. Compare rotational levels up to J=20.14.101E Linearly polarized light can be assigned a specific irreducible representation of a symmetry point group. If the electronic ground state of benzene has A1g symmetry and y- polarized light has the label E1u, what are the symmetry labels of allowed excited electronic states? Use the D6h character table in Appendix 3.Linearly polarized light can be assigned a specific irreducible representation of a symmetry point group. If the electronic ground state of methane has A1g symmetry and x- polarized light has the label T2, what are the symmetry labels of allowed excited electronic states? Use the Td character table in Appendix 3.15.3E Determine the value of R for deuterium (2H) and tritium (3H) using equation 15.3. You will have to look up the masses of the D and T nuclei. By what percentage do these R values differ from that for hydrogen?15.5E Draw a Grotrian-type diagram for the first four lines of the Lyman series in the hydrogen atom spectrum. Be sure to include all allowed transitions.What is the value of the change in energy, E, for the (n,l) transition of the hydrogen atom labeled (3,2)(3,1)? Justify your answer.15.8E List the possible values of L, ML, S, MS, J, and MJ for the following: a two coupled p electrons in different shells, b two coupled f electrons in different shells, c two coupled electrons, one a p electron and one a d electron. Remember that the z-component quantum numbers depend on the values of the total angular momentum quantum numbers.15.10E 15.11E 15.12E 15.13E What is the term symbol for the ground state of a Li, b Al,c Sc?15.15E 15.16E 15.17E 15.18E 15.19E 15.20E 15.21E 15.22E Use Hunds rules to predict the term symbol of the ground electronic state of an atom having a d6 electron configuration for its valence subshell.15.24E 15.25E An Mg atom is in the excited electron configuration [Ne]4s2. What are the possible term symbols for this atom?15.27E 15.28E 15.29E 15.30E Use group-theoretical arguments to determine the term symbols for B2, which has two unpaired electrons in doubly degenerate u molecular orbitals.The GeF molecule, which exists in the gas phase at high temperature, has a 2 electronic ground state. What are the term symbols of the electronic states that can be accessed by allowed electronic transitions?The SrH molecule in a 2+ electronic ground state. What are the term symbols of the electronic states that can be accessed by allowed electronic transitions?15.34E What are the values of and for NF in its 3 ground electronic state?15.36E Determine the ground-state term symbol of the acetylide ion, C22. Use the molecular orbital diagram for oxygen for assistance.15.38E 15.39E 15.40E 15.41E 15.42E The 1, 1'-diethyl-2, 2'-dicarbocyanine cation has a conjugated carbon-carbon double bond system. If there are 10 electrons in the conjugated system, and the molecule has an initial strong absorption at 713nm, what is the approximate length of the conjugated system?15.44E Naphthacene, C18H12, consists of four benzene rings in a row, each connected by sharing a CC bond. Naphthacene absorbs light at 450.2nm. a If this corresponds to an electron going from n=9 to n=10 in a particle-in-a-box, what is the length of the naphthacene molecule? b If each benzene unit is 0.280nm wide, how good is this approximation?15.46E 15.47E Construct the Hckel determinants for cyclobutadiene and cyclopentadiene. In what ways are they alike? In what ways are they different?15.49E Using the Internet, find a Hckel determinant solver and determine the relative energies of the molecular orbitals for a the butadiene molecule, which has four electrons and b the cyclobutadiene molecule, which also has four electrons. Can you explain why cyclobutadiene is labeled with the word antiaromatic?15.51E Explain why cyclopentadiene easily accepts an electron to become cyclopentadienyl anion C5H5, which is abbreviated Cp in organic or organometallic chemistry.15.53E 15.54E 15.55E 15.56E 15.57E 15.58E Would the light from fireflies be considered an example of a fluorescence or a phosphorescence process?15.60E 15.61E 15.62E How many 632.8-nm photons must a He-Ne laser emit per second to achieve a power of 1J/s?15.64E Green He-Ne lasers are also available; they emit light at 543.5nm. What is the A/B value at this wavelength?Blu-ray disks and high-density DVDs use an indium gallium nitride diode laser that emits light having a wavelength of 405nm. a What is the energy of 1.00mol of photons emitted by the laser? b What is the A/B value at this wavelength?15.67E Explain why X-ray lasers would be extremely difficult to build.15.69E

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