Why are there only 8 electrons in the second energy level of an atom?

Answer 1

The rules that electrons must follow when populating energy levels are governed by 4 quantum numbers. These numbers, and their relationships to each other, can be derived through the use of quantum mechanics, but that is beyond the scope of this answer. Instead, I'll list the numbers and their corresponding rules and then explicitly show why the second energy level can only have 8 electrons.

The quantum numbers are:

n, where n ≥ 1,

l, where n - 1 ≥ l ≥ 0,

ml, where l ≥ ml ≥ -l, and

ms, where ms = ±½.

n corresponds to the energy level of an atom, thus n = 2 corresponds to the second energy level.

For n = 2:

2 - 1 ≥ l ≥ 0 = 1 ≥ l ≥ 0, so l can be only 0 or 1.

For l = 0:

0 ≥ ml ≥ -0 = 0 ≥ ml ≥ 0, so ml = 0.

For l = 1:

1 ≥ ml ≥ -1, so ml can be -1, 0, or 1.

So far, then, we have 4 unique sets of quantum numbers, which I'll list below using the format n, l, ml.

2, 0, 0,

2, 1, -1,

2, 1, 0,

2, 1, 1.

The final step is to add the quantum number ms, which can be either ½ or -½, to each of those 4 sets of numbers above. This quantum number corresponds to the fact that electrons can have an intrinsic spin value of ±½. This now gives us the 8 unique sets of quantum numbers, corresponding to the 8 possible states that an electron can occupy in an atom's second energy level, that we were looking for. I'll list them below.

2, 0, 0, ½,

2, 0, 0, -½,

2, 1, -1, ½,

2, 1, -1, -½,

2, 1, 0, ½,

2, 1, 0, -½,

2, 1, 1, ½,

2, 1, 1, -½.

Answer 2

The second energy level of an atom can hold a maximum of 8 electrons due to the formula 2n^2, where n is the principal quantum number. This is based on the allowable quantum states within that energy level.