Question

Question #f0e05

Answer 1

Here's my take on this.

Explanation:

I assume that you're interested in finding out how many electrons can share those two quantum numbers

`n=4" "` and `" "m_s = -1/2`

The idea here is that you can use the fact that the number of orbitals each energy level can hold is given by the equation

`color(blue)(|bar(ul(color(white)(a/a)"no. of orbitals" = n^2color(white)(a/a)|)))`

Here

`n` - the principal quantum number, the quantum number that gives you the energy level on which the electron resides

In your case, the fourth energy level will have a total of

`"no. of orbitals" = 4^2 = 16`

This tells you that the fourth energy level can hold a total of `16` orbitals. Since each orbital can hold a maximum of `2` electrons, it follows that the maximum number of electrons that can share the quantum number `n=4` is

`"no. of e"^(-) = 2 xx "16 orbitals" = "32 e"^(-)`

Now, half of these electrons will have spin-up, which is given by `m_s = +1/2`, and the other half will have spin-down, which is given by `m_s = -1/2`.

Since you're interested in the number of electrons that have `n=4` and `m_s = -1/2`, you will get

`"no. of electrons" = "32 e"^(-)/2 = color(green)(|bar(ul(color(white)(a/a)color(black)("16 e"^(-))color(white)(a/a)|)))`