10.6: Exercises

Conceptual Problems

1. What is the distinction between an atomic orbital and a molecular orbital? How many electrons can a molecular orbital accommodate?

2. Why is the molecular orbital approach to bonding called a delocalized approach?

3. How is the energy of an electron affected by interacting with more than one positively charged atomic nucleus at a time? Does the energy of the system increase, decrease, or remain unchanged? Why?

4. Constructive and destructive interference of waves can be used to understand how bonding and antibonding molecular orbitals are formed from atomic orbitals. Does constructive interference of waves result in increased or decreased electron probability density between the nuclei? Is the result of constructive interference best described as a bonding molecular orbital or an antibonding molecular orbital?

5. What is a “node” in molecular orbital theory? How is it similar to the nodes found in atomic orbitals?

6. What is the difference between an s orbital and a σ orbital? How are the two similar?

7. Why is a σ_1s molecular orbital lower in energy than the two s atomic orbitals from which it is derived? Why is a σ*​_1s molecular orbital higher in energy than the two s atomic orbitals from which it is derived?

8. What is meant by the term bond order in molecular orbital theory? How is the bond order determined from molecular orbital theory different from the bond order obtained using Lewis electron structures? How is it similar?

9. What is the effect of placing an electron in an antibonding orbital on the bond order, the stability of the molecule, and the reactivity of a molecule?

10. How can the molecular orbital approach to bonding be used to predict a molecule’s stability? What advantages does this method have over the Lewis electron-pair approach to bonding?

11. What is the relationship between bond length and bond order? What effect do antibonding electrons have on bond length? on bond strength?

12. Draw a diagram that illustrates how atomic p orbitals can form both σ and π molecular orbitals. Which type of molecular orbital typically results in a stronger bond?

13. What is the minimum number of nodes in σ, π, σ*, and π*? How are the nodes in bonding orbitals different from the nodes in antibonding orbitals?

14. It is possible to form both σ and π molecular orbitals with the overlap of a d orbital with a p orbital, yet it is possible to form only σ molecular orbitals between s and d orbitals. Illustrate why this is so with a diagram showing the three types of overlap between this set of orbitals. Include a fourth image that shows why s and d orbitals cannot combine to form a π molecular orbital.

15. Is it possible for an np_x orbital on one atom to interact with an np_y orbital on another atom to produce molecular orbitals? Why or why not? Can the same be said of np_y and np_z orbitals on adjacent atoms?

16. What is meant by degenerate orbitals in molecular orbital theory? Is it possible for σ molecular orbitals to form a degenerate pair? Explain your answer.

17. Why are bonding molecular orbitals lower in energy than the parent atomic orbitals? Why are antibonding molecular orbitals higher in energy than the parent atomic orbitals?

18. What is meant by the law of conservation of orbitals?

19. Atomic orbitals on different atoms have different energies. When atomic orbitals from nonidentical atoms are combined to form molecular orbitals, what is the effect of this difference in energy on the resulting molecular orbitals?

20. If two atomic orbitals have different energies, how does this affect the orbital overlap and the molecular orbitals formed by combining the atomic orbitals?

21. Are the Al–Cl bonds in AlCl₃ stronger, the same strength, or weaker than the Al–Br bonds in AlBr₃? Why?

22. Are the Ga–Cl bonds in GaCl₃ stronger, the same strength, or weaker than the Sb–Cl bonds in SbCl₃? Why?

23. What is meant by a nonbonding molecular orbital, and how is it formed? How does the energy of a nonbonding orbital compare with the energy of bonding or antibonding molecular orbitals derived from the same atomic orbitals?

24. Many features of molecular orbital theory have analogs in Lewis electron structures. How do Lewis electron structures represent

    1. nonbonding electrons?
    2. electrons in bonding molecular orbitals?

25. How does electron screening affect the energy difference between the 2s and 2p atomic orbitals of the period 2 elements? How does the energy difference between the 2s and 2p atomic orbitals depend on the effective nuclear charge?

26. For σ versus π, π versus σ*, and σ* versus π*, which of the resulting molecular orbitals is lower in energy?

27. The energy of a σ molecular orbital is usually lower than the energy of a π molecular orbital derived from the same set of atomic orbitals. Under specific conditions, however, the order can be reversed. What causes this reversal? In which portion of the periodic table is this kind of orbital energy reversal most likely to be observed?

28. Is the \( \sigma _{2p_{z}} \) molecular orbital stabilized or destabilized by interaction with the σ_2s molecular orbital in N₂? in O₂? In which molecule is this interaction most important?

29. Explain how the Lewis electron-pair approach and molecular orbital theory differ in their treatment of bonding in O₂.

30. Why is it crucial to our existence that O₂ is paramagnetic?

31. Will NO or CO react more quickly with O₂? Explain your answer.

32. How is the energy-level diagram of a heteronuclear diatomic molecule, such as CO, different from that of a homonuclear diatomic molecule, such as N₂?

33. How does molecular orbital theory describe the existence of polar bonds? How is this apparent in the molecular orbital diagram of HCl?

Answers

1. An atomic orbital is a region of space around an atom that has a non-zero probability for an electron with a particular energy. Analogously, a molecular orbital is a region of space in a molecule that has a non-zero probability for an electron with a particular energy. Both an atomic orbital and a molecular orbital can contain two electrons.

15. No. Because an np_x orbital on one atom is perpendicular to an np_y orbital on an adjacent atom, the net overlap between the two is zero. This is also true for np_y and np_z orbitals on adjacent atoms.

Numerical Problems

1. Use a qualitative molecular orbital energy-level diagram to describe the bonding in S₂²⁻. What is the bond order? How many unpaired electrons does it have?

2. Use a qualitative molecular orbital energy-level diagram to describe the bonding in F₂²⁺. What is the bond order? How many unpaired electrons does it have?

3. If three atomic orbitals combine to form molecular orbitals, how many molecular orbitals are generated? How many molecular orbitals result from the combination of four atomic orbitals? From five?

4. If two atoms interact to form a bond, and each atom has four atomic orbitals, how many molecular orbitals will form?

5. Sketch the possible ways of combining two 1s orbitals on adjacent atoms. How many molecular orbitals can be formed by this combination? Be sure to indicate any nodal planes.

6. Sketch the four possible ways of combining two 2p orbitals on adjacent atoms. How many molecular orbitals can be formed by this combination? Be sure to indicate any nodal planes.

7. If a diatomic molecule has a bond order of 2 and six bonding electrons, how many antibonding electrons must it have? What would be the corresponding Lewis electron structure (disregarding lone pairs)? What would be the effect of a one-electron reduction on the bond distance?

8. What is the bond order of a diatomic molecule with six bonding electrons and no antibonding electrons? If an analogous diatomic molecule has six bonding electrons and four antibonding electrons, which has the stronger bond? the shorter bond distance? If the highest occupied molecular orbital in both molecules is bonding, how will a one-electron oxidation affect the bond length?

9. Qualitatively discuss how the bond distance in a diatomic molecule would be affected by adding an electron to

   1. an antibonding orbital.
   2. a bonding orbital.

10. Explain why the oxidation of O₂ decreases the bond distance, whereas the oxidation of N₂ increases the N–N distance. Could Lewis electron structures be employed to answer this problem?

11. Draw a molecular orbital energy-level diagram for Na₂⁺. What is the bond order in this ion? Is this ion likely to be a stable species? If not, would you recommend an oxidation or a reduction to improve stability? Explain your answer. Based on your answers, will Na₂⁺, Na₂, or Na₂⁻ be the most stable? Why?

12. Draw a molecular orbital energy-level diagram for Xe₂⁺, showing only the valence orbitals and electrons. What is the bond order in this ion? Is this ion likely to be a stable species? If not, would you recommend an oxidation or a reduction to improve stability? Explain your answer. Based on your answers, will Xe₂²⁺, Xe₂⁺, or Xe₂ be most stable? Why?

13. Draw a molecular orbital energy-level diagram for O₂²⁻ and predict its valence electron configuration, bond order, and stability.

14. Draw a molecular orbital energy-level diagram for C₂^2– and predict its valence electron configuration, bond order, and stability.

15. If all the p orbitals in the valence shells of two atoms interact, how many molecular orbitals are formed? Why is it not possible to form three π orbitals (and the corresponding antibonding orbitals) from the set of six p orbitals?

16. Draw a complete energy-level diagram for B₂. Determine the bond order and whether the molecule is paramagnetic or diamagnetic. Explain your rationale for the order of the molecular orbitals.

17. Sketch a molecular orbital energy-level diagram for each ion. Based on your diagram, what is the bond order of each species?

    1. NO⁺
    2. NO⁻

18. The diatomic molecule BN has never been detected. Assume that its molecular orbital diagram would be similar to that shown for CN⁻ in Section 5.3 "Delocalized Bonding and Molecular Orbitals" but that the <( \sigma _{2p_{z}} \) molecular orbital is higher in energy than the \( \pi _{2p_{x,y}} \)​ molecular orbitals.

    1. Sketch a molecular orbital diagram for BN.
    2. Based on your diagram, what would be the bond order of this molecule?
    3. Would you expect BN to be stable? Why or why not?

19. Of the species BN, CO, C₂, and N₂, which are isoelectronic?

20. Of the species CN⁻, NO⁺, B₂²⁻, and O₂⁺, which are isoelectronic?

Answers

1. 

   The bond order is 1, and the ion has no unpaired electrons.

3. The number of molecular orbitals is always equal to the number of atomic orbitals you start with. Thus, combining three atomic orbitals gives three molecular orbitals, and combining four or five atomic orbitals will give four or five molecular orbitals, respectively.

5. 

   Combining two atomic s orbitals gives two molecular orbitals, a σ (bonding) orbital with no nodal planes, and a σ* (antibonding) orbital with a nodal plane perpendicular to the internuclear axis.

9. 1. Adding an electron to an antibonding molecular orbital will decrease the bond order, thereby increasing the bond distance.
   2. Adding an electron to a bonding molecular orbital will increase the bond order, thereby decreasing the bond distance.

11. Sodium contains only a single valence electron in its 3s atomic orbital. Combining two 3s atomic orbitals gives two molecular orbitals; as shown in the diagram, these are a σ (bonding) orbital and a σ* (antibonding) orbital.

    

    Although each sodium atom contributes one valence electron, the +1 charge indicates that one electron has been removed. Placing the single electron in the lowest energy molecular orbital gives a \( \sigma ^{1}_{3s} \) electronic configuration and a bond order of 0.5. Consequently, Na₂⁺ should be a stable species. Oxidizing Na₂⁺ by one electron to give Na₂²⁺ would remove the electron in the σ_3s molecular orbital, giving a bond order of 0. Conversely, reducing Na₂⁺ by one electron to give Na₂ would put an additional electron into the σ_3s molecular orbital, giving a bond order of 1. Thus, reduction to Na₂ would produce a more stable species than oxidation to Na₂²⁺. The Na₂⁻ ion would have two electrons in the bonding σ_3s molecular orbital and one electron in the σ_3s* antibonding molecular orbital, giving a bond order of 0.5. Thus, Na₂ is the most stable of the three species.

17. 

    1. The NO⁺ ion has 10 valence electrons, which fill all the molecular orbitals up to and including the σ_2p. With eight electrons in bonding molecular orbitals and two electrons in antibonding orbitals, the bond order in NO⁺ is (8 − 2)/2 = 3.
    2. The NO⁻ ion contains two more electrons, which fill the \( \sigma ^{\star }_{2p} \)​ molecular orbital. The bond order in NO⁻ is (8 − 4)/2 = 2.

19. BN and C₂ are isoelectronic, with 12 valence electrons, while N₂ and CO are isoelectronic, with 14 valence electrons.

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