# How many electrons have...?

## a) $n = 4$, $l = 2$ b) $n = 4$, $l = 2$, ${m}_{l} = + 2$ c) $n = 4$, $l = 2$, ${m}_{l} = + 2$, ${m}_{s} = + \frac{1}{2}$

Mar 10, 2015

The answers would be a) 10 electrons, b) 2 electrons, and c) 1 electron.

I assume you're familiar with quantum numbers, so I won't go into too much detail about them.

So, here are the four quantum numbers you have to work with

Let's start with point a). Since you already know that n = 4 and l = 2, you only have to decide the possible values for the magentic quantum number, ${m}_{l}$, and for the spin quantum number, ${m}_{s}$.

Since the magnetic quantum number can only go from $- l$ to $+ l$, you'll have these values for ${m}_{l}$

${m}_{l} = - 2 , - 1 , 0 , + 1 , + 2$

SInce each of these orbitals can hold a maximum of two electrons, one having spin-up and one having spin-down, a total of 10 electrons can share the quantum numbers n = 4 and l = 2

${m}_{l} = - 2 \implies {m}_{s} = \pm \frac{1}{2}$

${m}_{l} = - 1 \implies {m}_{s} = \pm \frac{1}{2}$

${m}_{l} = 0 \implies {m}_{s} = \pm \frac{1}{2}$

${m}_{l} = + 1 \implies {m}_{s} = \pm \frac{1}{2}$

${m}_{l} = + 2 \implies {m}_{s} = \pm \frac{1}{2}$

On to point b). Now you know that n = 4, l = 2, and ${m}_{l}$ = 2, which means that only 2 electrons, one having spin-up and one having spin-down, can share these three quantum number values.

${m}_{s} = \pm \frac{1}{2}$

FInally, point c). Because every electron has a specific set of quantum numbers that describes its position around the nucleus, the values given to you for all the four quantum numbers can only describe 1 electron.