Final answer:
Among methane, ethene, benzene, and ethyne, benzene is the molecule with the smallest HOMO-LUMO gap due to its delocalized system of π electrons.
Step-by-step explanation:
The question asks which of the given molecules will have the smallest HOMO-LUMO (Highest Occupied Molecular Orbital - Lowest Unoccupied Molecular Orbital) gap, implying we're considering the difference in energy between the highest occupied and lowest unoccupied molecular orbitals. Among the options given, which are methane, ethene, benzene, and ethyne, the molecule with the smallest HOMO-LUMO gap is typically the one with the greatest degree of conjugation or the most extensive delocalization of π (pi) electrons.
Methane is a simple alkane with no double bonds; thus, it has a relatively large HOMO-LUMO gap. Ethene has a double bond, presenting pi-bonding, but without conjugation. Benzene has a delocalized system of pi-electrons over six carbon atoms, leading to a smaller HOMO-LUMO gap. Finally, ethyne has a triple bond with significant pi character, but lacks conjugation. Therefore, the smallest HOMO-LUMO gap is expected in benzene due to its delocalized electronic structure.
To illustrate, ethene and its structure can be examined in Figure 22.5.1, showing its trigonal planar bond angles, which also aids in understanding the concept of molecular orbitals. However, the presence of the conjugated pi-electron system in benzene creates a lower energy transition between the HOMO and LUMO compared to the non-conjugated alkenes and alkanes.