The energy-level diagram in Figure 9.36 shows that the sideways overlap of a pair of \(p\) orbitals produces two molecular orbitals, one bonding and one antibonding. In ethylene there is a pair of electrons in the bonding \(\pi\) orbital between the two carbons. Absorption of a photon of the appropriate wavelength can result in promotion of one of the bonding electrons from the \(\pi_{2 p}\) to the \(\pi_{2 p}^{*}\) molecular orbital. (a) Assuming this electronic transition corresponds to the HOMO-LUMO transition, what is the HOMO in ethylene? (b) Assuming this electronic transition corresponds to the HOMO-LUMO transition, what is the LUMO in ethylene? (c) Is the \(\mathrm{C}-\mathrm{C}\) bond in ethylene stronger or weaker in the excited state than in the ground state? Why? (d) Is the \(\mathrm{C}-\mathrm{C}\) bond in ethylene easier to twist in the ground state or in the excited state?

Step by step solution

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1. Identifying the HOMO in Ethylene

In ethylene, there is a pair of electrons in the bonding π orbital between the two carbons, which means that the carbon-carbon bond is formed. The HOMO corresponds to the highest energy occupied molecular orbital. Since the pair of electrons in the bonding π orbital are involved in forming the C-C bond, the HOMO in ethylene is this bonding π orbital. So, the HOMO corresponds to the \(\pi_{2 p}\) molecular orbital.

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2. Identifying the LUMO in Ethylene

The LUMO is the lowest unoccupied molecular orbital. When a photon is absorbed, one of the electrons in the \(\pi_{2 p}\) bonding molecular orbital (HOMO) can be promoted to the higher energy \(\pi_{2 p}^{*}\) molecular orbital (LUMO). Hence, the LUMO corresponds to the \(\pi_{2 p}^{*}\) molecular orbital.

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3. Comparing C-C bond strength in the ground state and the excited state

In the ground state, both the bonding electrons are in the \(\pi_{2 p}\) molecular orbital. Upon absorbing a photon and exciting an electron to the \(\pi_{2 p}^{*}\) molecular orbital, there is now only one electron left in the \(\pi_{2 p}\) bonding orbital, which weakens the C-C bond. Therefore, the C-C bond in ethylene is weaker in the excited state compared to the ground state.

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4. Comparing C-C bond twisting in the ground state and the excited state

In the ground state, with both electrons in the \(\pi_{2 p}\) bonding orbital, the overlap and bonding between the p orbitals are strong, which makes it difficult to twist the C-C bond. However, in the excited state, with one electron promoted to the \(\pi_{2 p}^{*}\) molecular orbital, the overlap and bonding are weakened, making it easier to twist the C-C bond. Thus, the C-C bond in ethylene is easier to twist in the excited state than in the ground state.

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