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)?

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Identify the molecular orbitals for a chain of six lithium atoms from the provided diagram.

Determine the number of molecular orbitals by counting the distinct energy levels shown in the diagram.

Locate the lowest-energy unoccupied molecular orbital (LUMO) by finding the first unoccupied orbital above the highest occupied molecular orbital (HOMO).

Count the number of nodes in the LUMO. Nodes are regions where the probability of finding an electron is zero.

Verify the number of nodes by comparing with the pattern observed in the molecular orbitals of the lithium chain.

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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. In a molecular orbital diagram, these orbitals are depicted with their energy levels, showing how electrons are distributed among them. The arrangement of MOs helps predict the bonding and antibonding characteristics of molecules, influencing their stability and reactivity.

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 the lowest-energy unoccupied molecular orbital (LUMO), the number of nodes can be determined by analyzing the molecular orbital diagram, which indicates how many times the wave function changes sign.

Linear Chains of Atoms

In a linear chain of atoms, such as lithium atoms, the molecular orbitals are influenced by the number of atoms in the chain. As more atoms are added, the number of molecular orbitals increases, and their energy levels become more closely spaced. This affects the distribution of electrons and the characteristics of the LUMO, which can be analyzed to determine properties like conductivity and reactivity in materials.

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Related Practice

Textbook Question

State whether each sentence is true or false: (b) Metals have high electrical conductivities because they are denser than other solids.

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

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)?

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.