Repeat Exercise 12.51 for a linear chain of eight lithium atoms. (b) How many nodes are in the lowest-energy molecular orbital? (c) How many nodes are in the highestenergy molecular orbital? (d) How many nodes are in the highest-energy occupied molecular orbital (HOMO)? (e) How many nodes are in the lowest-energy unoccupied molecular orbital (LUMO)?

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To solve this problem, we need to understand the concept of molecular orbitals in a linear chain of atoms. For a linear chain of N atoms, the number of molecular orbitals is equal to N.

The number of nodes in a molecular orbital is determined by its energy level. The lowest-energy molecular orbital (bonding) has zero nodes, and the number of nodes increases with energy.

For a linear chain of eight lithium atoms, there are 8 molecular orbitals. The lowest-energy molecular orbital (part b) will have 0 nodes.

The highest-energy molecular orbital (part c) will have the maximum number of nodes, which is N-1. For 8 lithium atoms, this is 7 nodes.

The highest-energy occupied molecular orbital (HOMO) and the lowest-energy unoccupied molecular orbital (LUMO) are determined by the electron configuration. For lithium, each atom contributes one electron, so the HOMO will be the 4th orbital (3 nodes), and the LUMO will be the 5th orbital (4 nodes).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Molecular Orbitals

Molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO) when atoms bond together. They can be classified as bonding, antibonding, or non-bonding, and their energy levels determine the stability and reactivity of the molecule. Understanding MOs is crucial for predicting the electronic structure and properties of molecular systems.

Nodes in Molecular Orbitals

Nodes are regions in a molecular orbital where the probability of finding an electron is zero. The number of nodes in an orbital is related to its energy level; higher energy orbitals have more nodes. For linear chains of atoms, the number of nodes can be calculated based on the number of atoms and the specific orbital being considered.

HOMO and LUMO

The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are critical in determining a molecule's electronic properties. The HOMO is the highest energy orbital that contains electrons, while the LUMO is the lowest energy orbital that is empty. The energy gap between these two orbitals is significant for understanding chemical reactivity and electronic transitions.

Related Practice

Textbook Question

The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.22. Construct a molecular-orbital diagram for a chain containing six lithium atoms and use it to answer the following questions: (a) How many molecular orbitals are there in the diagram?

Textbook Question

The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.22. Construct a molecular-orbital diagram for a chain containing six lithium atoms and use it to answer the following questions: (c) How many nodes are in the highest-energy molecular orbital?

Textbook Question

The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.22. Construct a molecular-orbital diagram for a chain containing six lithium atoms and use it to answer the following questions: (e) How many nodes are in the lowest-energy unoccupied molecular orbital (LUMO)?

Textbook Question

Repeat Exercise 12.51 for a linear chain of eight lithium atoms. (f) How does the HOMO–LUMO energy gap for this case compare to that of the four-atom case?

Textbook Question

Which would you expect to be the more ductile element, (a) Ag or Mo? (b) Zn or Si? In each case explain your reasoning.

Textbook Question

Which of the following statements does not follow from the fact that the alkali metals have relatively weak metal–metal bonding? (a) The alkali metals are less dense than other metals. (b) The alkali metals are soft enough to be cut with a knife. (c) The alkali metals are more reactive than other metals. (d) The alkali metals have higher melting points than other metals. (e) The alkali metals have low ionization energies.