Quantum levels of beta carotene

The textbook states that: "There are 22 C atoms in the conjugated chain; each contributes one p electron to the levels, so each level up to n=11 is occupied by two electrons"

I don't quite understand this reasoning, can anyone explain it in more detail?

For reference, b-carotene:
https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2Fswh-826d.kxcdn.com%2Fwp-content%2Fuploads%2F2011%2F05%2FBeta-carotene-1.png&f=1&nofb=1

Original post by Animadvertō

The textbook states that: "There are 22 C atoms in the conjugated chain; each contributes one p electron to the levels, so each level up to n=11 is occupied by two electrons"

I don't quite understand this reasoning, can anyone explain it in more detail?

For reference, b-carotene:
https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2Fswh-826d.kxcdn.com%2Fwp-content%2Fuploads%2F2011%2F05%2FBeta-carotene-1.png&f=1&nofb=1

Electrons exist in molecular energy levels. The number of molecular levels = the sum of the atomic levels of the combined atoms.

In a conjugated system (alternating single and double bonds) the pi electrons are delocalised and located in molecular orbitals. There are a sum of pi delocalised orbitals equal to the number of carbon atoms in the conjugated chain (each carbon contributes one electron). Hence, in a conjugated chain of 22 carbon atoms there are 22 electrons available that need to be located in the delocalised orbitals. Two electrons are "allowed" per orbital, hence there are 11 occupied molecular orbitals, from n=1 to n=11.

Reply 2

Animadvertō

OP

4

Original post by charco

Electrons exist in molecular energy levels. The number of molecular levels = the sum of the atomic levels of the combined atoms.

In a conjugated system (alternating single and double bonds) the pi electrons are delocalised and located in molecular orbitals. There are a sum of pi delocalised orbitals equal to the number of carbon atoms in the conjugated chain (each carbon contributes one electron). Hence, in a conjugated chain of 22 carbon atoms there are 22 electrons available that need to be located in the delocalised orbitals. Two electrons are "allowed" per orbital, hence there are 11 occupied molecular orbitals, from n=1 to n=11.