Question

How many electrons are in `n = 2`? What about `n = 4, l = 3`? What about `n = 6, l = 2, m_l = -1`?

Answer 1

Here's what I got.

Explanation:

Since the question is a bit ambiguous, I will assume that you're dealing with three distinct sets of quantum numbers.

In addition to this, I will also assume that you're fairly familiar with quantum numbers, so I won't go into too much details about what each represents.

figures.boundless.com

• `1^"st"` set `-> n=2`

The principal quantum number, `n`, tells you the energy level on which an electron resides. In order to be able to determine how many electrons can share this value of `n`, you need to determine exactly how many orbitals you have in this energy level.

The number of orbitals you get per energy level can be found using the equation

`color(blue)("no. of orbitals" = n^2)`

Since each orbital can hold amaximum of two electrons, it follows that as many as

`color(blue)("no. of electrons" = 2n^2)`

In this case, the second energy level holds a total of

`"no. of orbitals" = n^2 = 2^2 = 4`

orbitals. Therefore, a maximum of

`"no. of electrons" = 2 * 4 = 8`

electrons can share the quantum number `n=2`.

• `2^"nd"` set `-> n=4, l=3`

This time, you are given both the energy level, `n=4`, and the subshell, `l=3`, on which the electrons reside.

Now, the subshell is given by the angular momentum quantum number, `l`, which can take values ranging from `0` to `n-1`.

• `l=0 ->` the s-subshell
• `l=1 ->` the p-subshell
• `l=2 ->` the d-subshell
• `l=3 ->` the f-subshell

Now, the number of orbitals you get per subshell is given by the magnetic quantum number, `m_l`, which in this case can be

`m_l = -l, ..., -1, 0, 1, ..., +l`

`m_l = {-3; -2; -1; 0; 1; 2; 3}`

So, the f-subshell can hold total of seven orbitals, which means that you have a maximum of

`"no. of electrons" = 2 * 7 = 14`

electrons that can share these two quantum numbers, `n=4` and `l=3`.

• `3^"rd"` set `-> n=6, l=2, m_l = -1`

This time, you are given the energy level, `n=6`, the subshell, `l=2`, and the exact orbital, `m_l = 1`, in which the electrons reside.

Since you know the exact orbital, it follows that only two electrons can share these three quantum numbers, one having spin-up, `m_s = +1/2`, and the other having spin-down, `m_s = -1/2`.