I'm studying for a test and I came across the following question:
The particle on a ring is a useful model for the motion of electrons around the porphine ring. We may treat the group as a circular ring of radius 440pm, with 22 electrons in the conjugated system moving along the perimeter of the ring. Assume that in the ground state of the molecule quantized each level is occupied by two electrons. (A) calculate the energy and angular momentum of an electron in the highest occupied level.
In the solution of this problem, it's stated that:
There are 22 electrons in the conjugated system. There are two electrons in each lowest state. Therefore, the highest occupied state is ml = ±5
My question is: how did they find this quantum number? I can't for the life of me figure out what steps they took or what model was followed to arrive at this number.
Best Answer
For each energy level (above the ground state) you have two waves circulating in opposite directions, and each state can hold two electrons. That makes 4 electrons for each $m_l$. So the occupancy is:
$m_l$ = 0 - 2 electrons
$m_l$ = +1 and -1 - 4 electrons
$m_l$ = +2 and -2 - 4 electrons
$m_l$ = +3 and -3 - 4 electrons
$m_l$ = +4 and -4 - 4 electrons
$m_l$ = +5 and -1 - 4 electrons
To a grand total of 22 electrons as specified.