Question

Question #10924

Answer 1

Here's what I got.

Explanation:

I'm guessing that you want to figure out a possible set of quantum numbers that can describe an electron located in the `5f` subshell.

For starters, you should know that we can use a total of `4` quantum numbers to describe the location and spin of an electron inside an atom.

Now, the subshell in which an electron resides is given by the angular momentum quantum number, `l`, which can take on of the following values

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

and so on. So in your case, all the electrons that can reside in the `5f` subshell will have

`l = 3`

Notice that the value of the angular momentum quantum number depends on the value of the principal quantum number, `n`.

This means that in order for your electrons to have access to the `color(red)(5)f` subshell, they need to be located on the fifth energy level, so

`n = color(red)(5)`

The `5f` subshell contains a total of `7` orbitals, all described by a distinct value of the magnetic quantum number, `m_l`.

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

The spin quantum number, `m_s`, which describes the spin of the electron inside its orbital, can only take two possible values

`m_s = { -1/2, + 1/2}`

Since each orbital can hold a maximum of `2` electrons of opposite spins, i.e. of opposite `m_2` values--think Pauli's Exclusion Principle here--you can say that the `5f` subshell can hold a maximum of

`7 color(red)(cancel(color(black)("orbitals"))) * "2 e"^(-)/(1color(red)(cancel(color(black)("orbital")))) = "14 e"^(-)`

This implies that you can write a total of `14` distinct quantum number sets to describe one of the `14` electrons that can share the `5f` subshell.

For example, you can have

• `n =5 , l= 3 , m_l = -3, m_s = +1/2`
• `n = 5, l = 3, m_l = 0, m_s = -1/2`
• `n = 5, l = 3, m_l = 2, m_s = +1/2`
• `n = 5, l = 3, m_l = -1, m_s = -1/2`

and so on. Notice that all the `14` sets share the principal quantum number and the angular momentum quantum number, which is what you should expect for electrons that share a subshell--the `f` subshell-- located on a specific energy level--the fifth energy level.