Question

I'm having trouble with knowing the relations between quantum numbers n, l, m(sub l), and m(sub s). Could you please explain?

Answer 1

Remember this. It will do it all.
n = principle quantum number. May take any whole number beginning with 1.
l = azimuthal quantum number. May be any whole number beginning with 0 with a maximum of n-1.
m(l) = magnetic quantum number. May take any whole value from -l(ell) to +l(ell) including zero.
m(s) = spin quantum number. May be either +1/2 or -1/2.
Example: for n = 1
l may be 0 (nothing else)
m(l) = 0 (nothing else)
m(s) = +1/2 or -1/2
So for the single electron in H,(1s1),
n = 1
l = 0
m(l) = 0
m(s) = +1/2

For He (1s2),
One electron has the above as in H and other one is exactly the same EXCEPT it has m(s) = -1/2

For n = 2, then
l may be 0 or 1
m(l) may be 0 for the l = 0 and it may be -1, 0, +1 for the l = 1
and m(s) may be +1/2 or -1/2 for each of the m(l) values. Notice that there are 4 m(l) values and two electrons in each which allows 8 electrons in n=2 but you already knew there were 8 electrons for n =2 didn't you.