Final answer:
To represent the conjugated pi system of beta-carotene, which consists of 11 conjugated double bonds, 22 molecular orbitals are required - 11 bonding and 11 antibonding.
Step-by-step explanation:
The question asks about the number of molecular orbitals needed to represent the conjugated pi system of beta-carotene. Beta-carotene is a molecule with a chain of 11 conjugated double bonds. According to molecular orbital theory, each pi bond is made up of a pair of pi molecular orbitals - one bonding and one antibonding. Hence, for 11 double bonds, there will be 22 molecular orbitals - 11 bonding (pi) and 11 antibonding (pi*).
This means that for beta-carotene, with its extended conjugated pi system, we need 22 molecular orbitals to represent it fully.