Question

Construct a Frost circle for a planar eight-membered ring with one \(2 p\) orbital on each atom of the ring and show the relative energies of its eight \(\pi\) molecular orbitals. Which are bonding MOs, which are antibonding, and which are nonbonding?

Short answer

Question: Determine the relative energies of the pi molecular orbitals in a planar eight-membered ring with one 2p orbital on each atom, and identify which are bonding, antibonding, and nonbonding using the Frost circle method.

Step-by-step solution

Draw a Regular Polygon

Draw a regular polygon with eight vertices representing the eight-membered ring. It should look like an octagon.

Label the Vertices

Label the vertices of the octagon with numbers 1 through 8, going clockwise.

Draw a Circle Around the Polygon

Draw a circle around the octagon so that all vertices touch the circle. The circle represents the energy levels of the pi molecular orbitals.

Draw Horizontal Lines

For each vertex of the polygon, draw a horizontal line from the vertex to the center of the circle. These lines represent the pi molecular orbitals.

Determine the Number of Nodes

Where the horizontal lines intersect the circle, count how many times the line passes through the circle going from the bottom to the top. The number of intersections is equal to the number of nodes in the molecular orbital. The lowest energy orbital (LUMO) will have 0 nodes, the next higher energy orbital will have 1 node, and so on, up to the highest energy orbital with 7 nodes.

Identify Bonding, Antibonding, and Nonbonding Orbitals

Molecular orbitals with an even number of nodes (0, 2, 4, 6) are bonding, those with an odd number of nodes (1, 3, 5, 7) are antibonding, and those that match the number of atoms in the molecule (in this case 8) are nonbonding. Based on this, the bonding molecular orbitals are found at energies with 0, 2, and 4 nodes, the antibonding molecular orbitals are found at energies with 1, 3, 5, and 7 nodes, and there are no nonbonding molecular orbitals since no orbital has 8 nodes.