Achieve Homework Week 2 #21

[Isabella Perez Dis3L]

Hello, for this question (#21) "How many electrons in an atom could have these sets of quantum numbers?" I understood the first part of the question, "n=3," but for the second ("n=5,ℓ=2") and third ("n=6,ℓ=2,mℓ=−1"), I keep getting it wrong. Could someone explain how to do the second and third part?

[Ethan Cheng 2k]

Hi,
For the second part, l = 2 means that ml can be -2,-1,0,1,2 and each electron with a certain magnetic quantum number can have an ms of +1/2 or -1/2. So there are 5*2 = 10 electrons that could have that set of quantum numbers.
For the third part, since ml is given, there can be 2 electrons with that specific magnetic quantum number in the 6d subshell because ms can be +1/2 or -1/2.

[Macy_Anderson_2F]

When the problem gives you n = 5 and ℓ=2, it is important to look at the value of ℓ. When ℓ=2 we know that there are 5 possible values of mℓ (-2, -1, 0, 1, 2). For each value of mℓ, there can be two electrons.

When the problem gives you n=6,ℓ=2,mℓ=−1, it is important to look at the value of mℓ. As I stated above, for every one value of mℓ, there can be two electrons. Because the value of mℓ is given (one value), we know that there must be two electrons.

[Ayaan_Ekram_2J]

Hello!

For this problem, we have to remember what n, ℓ, and mℓ correspond to. The n is the principal quantum number, which reflects the energy level of the electron. This is also the number we see at the beginning of each shell when we write out electron configurations. The ℓ tells us the shape of the orbitals, and mℓ tells us the orientation. So, for the n, we simply write the number given as the first part of the shell. Then, we use the ℓ to find which shape we are using. In class we learned that ℓ=0 corresponds to s, ℓ=1 corresponds to p, ℓ=2 corresponds to d, and so on. So now we know the orbital. Based on the shape, we can say how many electrons can have these quantum numbers (s can have 2, p can have 6, d can have 10, f can have 14, etc). When we are given the value of mℓ, we are told that there is a single orientation we are looking at. For each orientation, we can only have a maximum of two electrons, one spin up (ms=+1/2) and one spin down (ms=-1/2). I hope this helps!

[Rio Gagnon 1G]

For the second part, ("n=5,ℓ=2") means that the electron is in the 5d orbital (ℓ=2 corresponds to the d orbital) and we know that each d orbital can hold a maximum of 10 electrons.
For the third part, ("n=6,ℓ=2,mℓ=−1") the electron in in the 6d orbital and for every value of mℓ, there can be two electrons (with opposite spin), so the answer would be 2.