The molecular-orbital diagrams for two- and four-atom linear chains of lithium atoms are shown in Figure 12.21. 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)?

Verified step by step guidance

insert step 1: Understand that a molecular orbital (MO) diagram for a chain of atoms shows the energy levels of the molecular orbitals formed from the atomic orbitals of the atoms in the chain.

insert step 2: Recognize that for a linear chain of lithium atoms, the molecular orbitals are formed by the combination of the 2s atomic orbitals of each lithium atom.

insert step 3: Recall that the number of molecular orbitals formed is equal to the number of atomic orbitals combined. For a chain of six lithium atoms, there will be six molecular orbitals.

insert step 4: Arrange these molecular orbitals in order of increasing energy, starting with the lowest energy (bonding) orbital and ending with the highest energy (antibonding) orbital.

insert step 5: Identify the lowest-energy unoccupied molecular orbital (LUMO) and determine the number of nodes it contains. The number of nodes in a molecular orbital is typically one less than the number of the orbital in the sequence, starting from zero for the lowest energy orbital.

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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 nodes they possess. The arrangement of MOs helps predict the electronic structure and properties of the molecule, including bonding and antibonding interactions.

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; as the energy increases, the number of nodes also increases. For the lowest-energy unoccupied molecular orbital (LUMO), understanding the node count is crucial for predicting its reactivity and stability in chemical processes.

Linear Chains and Orbital Overlap

In linear chains of atoms, such as lithium, the overlap of atomic orbitals leads to the formation of molecular orbitals that can accommodate more electrons as the number of atoms increases. The pattern of bonding and antibonding orbitals changes with the number of atoms, affecting the energy levels and node structure of the MOs. This concept is essential for constructing the molecular orbital diagram for a six-atom chain and analyzing its electronic properties.

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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 12.21. 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 12.21. 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. (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

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.