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?

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insert step 1> Start by understanding that each lithium atom contributes one atomic orbital to the molecular orbital (MO) diagram. For a chain of six lithium atoms, you will have six atomic orbitals combining to form molecular orbitals.

insert step 2> Recognize that the number of molecular orbitals formed is equal to the number of atomic orbitals combined. Therefore, for six lithium atoms, you will have six molecular orbitals.

insert step 3> Arrange these molecular orbitals in order of increasing energy. In a linear chain of lithium atoms, the molecular orbitals will be formed by the combination of the 2s atomic orbitals.

insert step 4> The molecular orbitals will include bonding, non-bonding, and antibonding orbitals. The lowest energy orbitals will be bonding, the highest energy orbitals will be antibonding, and any orbitals in between may be non-bonding.

insert step 5> Consider the symmetry and energy levels of the molecular orbitals. The bonding orbitals will have lower energy and be more stable, while the antibonding orbitals will have higher energy and be less stable.

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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 based on their energy levels and the number of atomic orbitals involved. For a chain of lithium atoms, the number of MOs corresponds to the total number of atomic orbitals contributed by the atoms in the chain.

Atomic Orbitals in Lithium

Lithium (Li) has an electron configuration of 1s² 2s¹, meaning it has two electrons in the 1s orbital and one in the 2s orbital. In a linear chain of lithium atoms, each atom contributes its 1s and 2s atomic orbitals to the molecular orbital formation. Understanding how these atomic orbitals combine is crucial for determining the total number of molecular orbitals in the chain.

Counting Molecular Orbitals

The total number of molecular orbitals in a system is equal to the number of atomic orbitals that combine to form them. For a chain of six lithium atoms, each contributing two atomic orbitals (1s and 2s), the total number of molecular orbitals can be calculated as 6 atoms × 2 orbitals = 12 molecular orbitals. This principle is fundamental in constructing and analyzing molecular orbital diagrams.

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An increase in temperature causes most metals to undergo thermal expansion, which means the volume of the metal increases upon heating. How does thermal expansion affect the unit cell length? What is the effect of an increase in temperature on the density of a metal?

Textbook Question

State whether each sentence is true or false: (a) Metals have high electrical conductivities because the electrons in the metal are delocalized. (c) Metals have large thermal conductivities because they expand when heated. (d) Metals have small thermal conductivities because the delocalized electrons cannot easily transfer the kinetic energy imparted to the metal from heat.

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: (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. (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)?