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Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions

One of the low-energy geometries of digermane, Ge2H2, is ethene-like. The Lewis dot structure shown
Use the geometrical construction shown in Example Problem 24.8 to derive the energy levels of the cycloheptatrienyl cation. What is the total π energy of the molecule? How many unpaired electrons will the molecule have? Would you expect this species, the neutral species, or the anion to be
Write down and solve the secular determinant for the π system of ethylene in the Hückel model. Determine the coefficients for the 2pz AOs on each of the carbons and make a sketch of the MOs. Characterize the MOs as bonding and anti-bonding.
The allyl cation CH2 ≡ CH―CH+2has a delocalized π network that can be described by the Hückel method. Derive the MO energy levels of this species and place the electrons in the levels appropriate for the ground state. Using the butadiene MOs as an example, sketch what you would expect
Use the geometrical construction shown in Example Problem 24.8 to derive the π electron MO levels for the cyclopentadienyl radical. What is the total π energy of the molecule? How many unpaired electrons will the molecule have?
Determine the AO coefficients for the lowest energy Huckel π MO for butadiene.
Use the geometrical construction shown in Example Problem 24.8 to derive the π electron MO levels for cyclobutadiene. What is the total π energy of the molecule? How many unpaired electrons will the molecule have?
The occupied MOs of hydrogen cyanide are shown next along with the MO energies. Indicate which AOs are most important in each MO and indicate the relative phases of the AOs. Classify the MOs as localized or delocalized, Ïƒ or Ï€ bonds, and bonding, nonbonding, or antibonding.
The occupied MOs of ethene are shown next along with the MO energies. Indicate which AOs are most important in each MO and indicate the relative phases of the AOs. Classify the MOs as localized or delocalized, Ïƒ or Ï€ bonds, and bonding, nonbonding, or antibonding. +. -305.51
Predict whether the ground state or the first excited state of CH2should have the larger bond angle on the basis of the Walsh correlation diagram shown in Figure 24.11. Explain your answer. 1b, 17u 2a, 1b2 1a, tog 180 160 140 120 100 Bond angle 20/degrees Orbit al energy
The occupied MOs of ammonia are shown next along with the MO energies. Indicate which AOs are most important in each MO and indicate the relative phases of the AOs. Classify the MOs as localized or delocalized, and bonding, nonbonding, or anti-bonding. -31.0 eV -423 eV -17.1 ev -17.1 eV -11.4 eV
Use the VSEPR method to predict the structures of the following:a. PF3b. CO2c. BrF5d. SO2−3
Using your results from Problem P24.10,a. Calculate the s and p character of the water lone pair hybrid orbitalsb. Show that the lone pair orbitals are orthogonal to each other and to the hybrid bonding orbitals.In Problem 24.10Derive two additional mutually orthogonal hybrid orbitals for the lone
In P24.3, the hybrid bonding orbitals for ozone were derived. Use the framework described in Section 24.3 to derive the normalized hybrid lone pair orbital on the central oxygen in O3 that is derived from 2s and 2p atomic orbitals. The bond angle in ozone is 116.8°.
Use the VSEPR method to predict the structures of the following:a. PCl5b. SO2c. XeF2d. XeF6
Use the Boltzmann distribution to answer parts (a) and (b):a. Calculate the ratio of the number of electrons at the bottom of the conduction band to those at the top of the valence band for pure Si at 300. K. The Si band gap is 1.1 eV.b. Calculate the ratio of the number of electrons at the bottom
Derive two additional mutually orthogonal hybrid orbitals for the lone pairs on oxygen in H2O, each of which is orthogonal to Ïˆaand Ïˆb, by following these steps:a. Starting with the following formulas for the lone pair orbitalsuse symmetry conditions to determine d2 and d4,
Predict which of the bent molecules, BH2or NH2, should have the larger bond angle on the basis of the Walsh correlation diagram in Figure 24.11. Explain your answer.Figure 24.11 16, 1Tu + 2a, 1b2 1a1 180 160 140 100 120 Bond angle 20/degrees Orbital energy
Show that the water hybrid bonding orbitals given by ψa = 0.55∅2Pz + 0.71∅2px -0.45 ∅2ps – ψb = 0.55∅2pz - 0.71∅2px - 0.45∅2s are orthogonal.
Show that two of the set of four equivalent orbitals appropriate for sp3hybridization,Andare orthogonal. W. =-62, + 2p, + ¢2, + 2p,) ¿(: - O2p. – 2p, + 2p,)
Use the formula cos2θ = −α2 and the method in Section 24.2 to derive the formulas ψa = 1/√2 (−∅2s + ∅2pz) and ψb = 1/√2 (−∅2s −∅2pz for two sp hybrid orbitals directed 180° apart. Show that these hybrid orbitals are orthogonal.
Use the method described in Example Problem 24.3 to show that the sp-hybrid orbitals ψa = 1/√2 (−∅2s +∅2pz) and ψb = 1/√2 (−∅2s −∅2pz) are oriented 180° apart.In Problem 24.3Use the framework described in Section 24.3 to construct normalized hybrid bonding orbitals on the
Are the localized bonding orbitals in Equation (24.13) defined byAndorthogonal? Answer this question by evaluating the integral ˆ« (Ïƒ€²)*Ïƒ€²€²dÏ„. GC4. + C2Be2s lo = 201:4 C3 o = 2g1:B +| CzB02= C3
Use the framework described in Section 24.3 to construct normalized hybrid bonding orbitals on the central oxygen in O3 that are derived from 2s and 2p atomic orbitals. The bond angle in ozone is 116.8°.
Predict whether LiH+2and NHˆ’2should be linear or bent based on the Walsh correlation diagram in Figure 24.11. Explain your answers.Figure 24.11 1b, + 2a, 1b2 1a1 tog 100 120 140 160 180 Bond angle 20/degrees Orbital energy
Show that the determinantal propertyused in the discussion of localized and delocalized orbitals in Section 24.6 is correct. ya + c a ь la b yb + d 9.
A cyclic polyene is known to be non-planar. Are the MO energy levels of this molecule well described by the Huckel model? Justify your answer.
What is the rational for setting Hij = 0 for nonadjacent atoms in the Huckel model?
What is the in-plane amplitude of the wave functions describing the Ï€ network in the conjugated molecules shown in Figures 24.18 and 24.19? a-2B a-B a+B a+B a+2B a-2B a+2B- m=3 m=4 m=5 m=6 Antibonding Bonding || Energy
The angular functions, Î˜ (Î¸)Î¦(ˆ…), for the one-electron Hartree€“Fock orbitals are the same as for the hydrogen atom, and the radial functions and radial probability functions are similar to those for the hydrogen atom. The contour coloring
In explaining molecular structure, the MO model uses the change in MO energy with bond angle. Explain why the decrease in energy of the 1a1 and 2a1 MOs as 2θ decreases more than offsets the increase in energy for the 1b2 MO for water.
In using the sum of the occupied MO energies to predict the bond angle in H2A molecules, the total energy of the molecule is assumed to be proportional to the sum of the occupied MO energies. This assumption can be justified. Do you expect this sum to be greater than or smaller than the total
Why can’t localized orbitals be represented in an MO energy diagram?
The hybridization model assumes that atomic orbitals are recombined to prepare directed orbitals that have the bond angles appropriate for a given molecule. What aspects of the model can be tested by experiments, and what aspects are conjectures that are not amenable to experimental verification?
Why are localized and delocalized models equally valid for describing bonding in closed-shell molecules? Why can’t experiments distinguish between these models?
How is it possible that a semiconductor would become metallic if the nearest-neighbor spacing could be changed sufficiently?
What evidence can you find in Table 24.1 that C€•C sp bonds are stronger than sp3bonds? TABLE 24.1 C–C Bond Types Carbon-Carbon Single Bond Types o Bond Carbon- Carbon Single Bond Length (pm) Angle between Equivalent o Bonds () s-to-p Hybridization Ratio 109.4 1:3 154 sp 1:2 120 146 c-c
On the basis of what you know about the indistinguishability of electrons and the difference between the wave functions for bonding electrons and lone pairs, discuss the validity and usefulness of the Lewis structure for the fluorine molecule (#-E).
Explain why all possible wave functions between the fully bonding and the fully anti-bonding are possible for the bands shown in Figure 24.22.Figure 24.22 N atomic orbitals 48 3.22B 28 2 atoms, N atoms 4 atoms ... Energy
What experimental evidence can you cite in support of the hypothesis that the electronegativity of a hybridized atom increases with increasing s-character?
How do the values of the AO coefficients in a MO differ for a delocalized and a localized bond?
Why can it be unclear whether a material is a semiconductor or an insulator?
Calculate the bond order in each of the following species. Which of the species in parts (a)–(d) do you expect to have the shorter bond length?a. Li2 or Li+2b. C2 or C+2c. O2 or O+2d. F2 or F−2
Predict the bond order in the following species:a. N+2b. Li+2c. O−2d. H−2e. C+2
Why are the magnitudes of the coefficients ca and cb in the H2+ wave functions ψg and ψu equal?
Identify the molecular orbitals for F2in the images shown here in terms of the two designations discussed in Section 23.7. The molecular axis is the z axis, and the y axis is tilted slightly out of the plane of the image. +
Show that the Slater determinant formalism automatically incorporates the Pauli exclusion principle by evaluating the He ground-state wave function of Equation (21.9), giving both electrons the same quantum numbers.
A surface displaying a contour of the total charge density in LiH is shown here. The molecular orientation is the same as in Problem P23.13. What is the relationship between this surface and the MOs displayed in Problem P23.13? Why does this surface closely resemble one of the MOs?In problem
Sketch the molecular orbital energy diagram for the radical OH based on what you know about the corresponding diagram for HF. How will the diagrams differ? Characterize the HOMO and LUMO as anti-bonding, bonding, or non-bonding.
The bond dissociation energies of the species NO, CF–, and CF+ follow the trend CF+ > NO > CF–. Explain this trend using MO theory.
Evaluate the energy for the two MOs generated by combining a H1s and a F2p AO. Use Equation (23.12) and carry out the calculation for SHF = 0.075, 0.18, and 0.40 to mimic the effect of increasing the atomic separation in the molecule. Use the parameters H11 = -13.6 eV, H22 = -18.6 eV, and H12 =
The expressions (c11)2 + c11c21S12 and (c12)2 + c11c21S12 for the probability of finding an electron on the H and F atoms in HF, respectively, were derived in Section 23.8. Use your results from Problem P23.21 and these expressions to calculate the probability of finding an electron in the bonding
Follow the procedure outlined in Section 23.2 to determine cu in Equation (23.17).
Evaluate the energy for the two MOs generated by combining two H1s AOs. Carry out the calculation for S12 = 0.15, 0.30, and 0.45 to mimic the effect of decreasing the atomic separation in the molecule. Use the parameters H11 = H22 = –13.6 eV and H12 = −1.75S12 √H11H22. Explain the trend that
Calculate the dipole moment of HF for the bonding MO in Equation (23.33). Use the method outlined in Section 23.8 to calculate the charge on each atom. The bond length in HF is 91.7 pm. The experimentally determined dipole moment of ground-state HF is 1.91 debye, where 1 debye = 3.33 × 10–30 C
What is the electron configuration corresponding to O2, O−2, and O+2?What do you expect the relative order of bond strength to be for these species? Which, if any, have unpaired electrons?
Images of molecular orbitals for LiH calculated using the minimal basis set are shown here. In these images, the smaller atom is H. The H1s AO has a lower energy than the Li2s AO. The energy of the MOs is (left to right) €“63.9 eV,€“7.92 eV, and +2.14 eV. Make a molecular
Arrange the following in terms of decreasing bond energy and bond length: O+2, O2, O-2, and O2-2.
Using the method of Mulliken, calculate the probabilities of finding an electron involved in the chemical bond on the H and F atoms for the bonding and anti-bonding MOs for Problem P23.9.
Calculate the value for the coefficients of the AOs in Example Problem 23.4 for S12 = 0.45. How are they different from the values calculated in that problem for S12 = 0.3? Can you offer an explanation for the changes?
A Hartree€“Fock calculation using the minimal basis set of the 1s, 2s, 2px, 2py, and 2pzAOs on each of N and O generated the energy eigenvalues and AO coefficients listed in the following table.a. Designate the MOs in the table as Ïƒ or Ï€ symmetry and as bonding or
The ionization energy of CO is greater than that of NO. Explain this difference based on the electron configuration of these two molecules.
Make a sketch of the highest occupied molecular orbital (HOMO) for the following species:a. N+2b. Li+2c. O−2d. H−2e. C+2
Calculate the bond order in each of the following species. Predict which of the two species in the following pairs has the higher vibrational frequency:a. Li2 or Li+2b. C2 or C+2c. O2 or O+2d. F2 or F−2
Explain the difference in the appearance of the MOs in Problem P23.13 with those for HF. Based on the MO energies, do you expect LiH+to be stable? Do you expect LiH€“to be stable? За* Li2s H1s 20 Lils lo
Sketch out a molecular orbital energy diagram for CO and place the electrons in the levels appropriate for the ground state. The AO ionization energies are O2s: 32.3 eV; O2p: 15.8 eV; C2s: 19.4 eV; and C2p: 10.9 eV. The MO energies follow the sequence (from lowest to highest) 1σ, 2σ, 3σ, 4σ,
The overlap integral for Ïˆgand Ïˆuas defined in Section 23.3 is given byPlot Sab as a function of R/a0 for Î¶ = 0.8, 1.0, and 1.2. Estimate the value of R/a0 for which Sab = 0.4 for each of these values of Î¶. Sab = e 5R/a|1+ 5 R 3 ao 2.
Using Î¶ as a variational parameter in the normalized functionallows one to vary the size of the orbital. Show this by calculating the probability of finding the electron inside a sphere of radius a0 for different values of Î¶ using the standard integrala. Obtain an expression
In discussing Figure 23.2, the following statement is made: Interchanging red and blue does not generate a different MO. Justify this statement.Figure 23.2 H1s His H2 Energy
Identify the molecular orbitals for F2in the images shown here in terms of the two designations discussed in Section 23.7. The molecular axis is the z axis, and the y axis is tilted slightly out of the plane of the image. + +
Explain why the nodal structures of the 1σg MOs in H2 and F2 differ.
Why do we neglect the bond length in He2when discussing the trends shown in Figure 23.20?Figure 23.20 O, На Не, Liz Be, B2 C2 N, F2 Nez N, На Не* 300 200 100 10 15 20 Number of electrons Force constant (N/m) Bond length (pm) Bond energy (eV) Bond order 2. 2. пи
By considering each term inAndExplain why the values of J and K are positive for H+2. e? |Ятe, dr - j анль Алеоь (Фнь, к Hs, dt J = [@n| dt 4TEQ7B.
Identify the molecular orbitals for F2in the images shown here in terms of the two designations discussed in Section 23.7. The molecular axis is the z axis, and the y axis is tilted slightly out of the plane of the image. +
If there is a node in ψu, is the electron in this wave function really delocalized? How does it get from one side of the node to the other?
For the case of two H1s AOs, the value of the overlap integral Sab is never exactly zero even at very large separation of the H atoms. Explain this statement.
Consider the molecular electrostatic potential map for the H2O molecule shown here. Is the hydrogen atom (shown as a white sphere) an electron acceptor or an electron donor in this molecule?
Consider the molecular electrostatic potential map for the LiH molecule shown here. Is the hydrogen atom (shown as a white sphere) an electron acceptor or an electron donor in this molecule?
The total energy of a molecule is lowered if the orbital energy of the anti-bonding MO is negative, and raised if the orbital energy of the anti-bonding MO is positive. The zero of energy is the electron and the molecule with n – 1 electrons at infinite separation.
Why can you conclude that the energy of the anti-bonding MO in H+2 is raised more than the energy of the bonding MO is lowered?
Justify the Born–Oppenheimer approximation based on vibrational frequencies and the timescale for electron motion.
Explain why s–p mixing is more important in Li2 than in F2.
What is the justification for saying that, in expanding MOs in terms of AOs, the equality ψj (1) = Σcij∅i(1) can in principle be satisfied?
Identify the molecular orbitals for F2in the images shown here in terms of the two designations discussed in Section 23.7. The molecular axis is the z axis, and the y axis is tilted slightly out of the plane of the image. +
Why are MOs on hetero-nuclear diatomics not labeled with g and u subscripts?
Consider the molecular electrostatic potential map for the BeH2molecule shown here. Is the hydrogen atom (shown as a white sphere) an electron acceptor or an electron donor in this molecule?
Using Figures 23.9 and 23.10, explain why Î” Ïˆ2g< 0 and Î”Ïˆ2u> 0 outside of the bonding region of H+2.Figure 23.9Figure 23.10 Bonding Antibonding H.
Using Figures 23.9 and 23.10, explain why Î” Ïˆ2g< 0 and Î”Ïˆ2u> 0 outside of the bonding region of H+2.Figure 23.9Figure 23.10 Bonding На На Л- Antibonding Нь Н. Н Н, Bonding Antibonding Но На Н, Но
Consider the molecular electrostatic potential map for the BH3molecule shown here. Is the hydrogen atom (shown as a white sphere) an electron acceptor or an electron donor in this molecule?
Distinguish between the following concepts used to describe chemical bond formation: basis set, minimal basis set, atomic orbital, molecular orbital, and molecular wave function.
For H+2, explain why Haa is the total energy of an undisturbed hydrogen atom separated from a bare proton by the distance R.
The molecular electrostatic potential maps for LiH and HF are shown here. Does the apparent size of the hydrogen atom (shown as a white sphere) tell you whether it is an electron acceptor or an electron donor in these molecules?
Why is it reasonable to approximate H11 and H22 by the appropriate ionization energy of the corresponding neutral atom?
Give examples of AOs for which the overlap reaches its maximum value only as the internuclear separation approaches zero in a diatomic molecule. Also give examples of AOs for which the overlap goes through a maximum value and then decreases as the internuclear separation approaches zero.
The first ionization potential of ground-state He is 24.6 eV. The wavelength of light associated with the 1s2 p 1P term is 58.44 nm. What is the ionization energy of the He atom in this excited state?
Two angular momenta with quantum numbers j1 = 3/2 and j2 = 5/2 are added. What are the possible values of J for the resultant angular momentum states?
Derive the ground-state term symbols for the following configurations:a. d2b. f9c. f12
What atomic terms are possible for the following electron configurations? Which of the possible terms has the lowest energy?a. ns1np1b. ns1nd1c. ns2np1d. ns1np2
Using Table 22.3, which lists the possible terms that arise from a given configuration, and Hund€™s rules, write the term symbols for the ground state of the atoms K through Cu, excluding Cr, in the form(2s+1)LJ. Possible Terms for Indicated Configurations TABLE 22.3 Electron
The spectrum of the hydrogen atom reflects the splitting of the 1s2S and 2p2P terms into levels. The energy difference between the levels in each term is much smaller than the difference in energy between the terms. Given this information, how many spectral lines are observed in the

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