Electrons and Shells: UNIZOR.COM - Physics4Teens - Atoms - Electronic St...

Notes to a video lecture on http://www.unizor.com

Electrons and Shells

First of all, there are some very complicated theories of the electronic structure of an atom. They are experimentally confirmed and, therefore, are considered as true representation of how electrons are arranged around a nucleus.
These theories are above the level of this course, but certain facts based on them we will mention without any proof.

Recall the Bohr's model of an atom and electrons that are supposed to be on stationary orbits (or, rather, within stationary shells), where they do not emit any energy.

Shells of a larger radius can hold more electrons, and electrons in those higher orbit shells have higher level of energy.

Shells are numbered in order of increasing radius as shell #1, shell #2, ...shell #N.

Any shell can have one or more subshells. The number of subshells in each shell corresponds to this shell's number, that is:
shell #1 has 1 subshell,
shell #2 has 2 subshell,
shell #3 has 3 subshell,
shell #4 has 4 subshell,
shell #5 has 5 subshell,
shell #6 has 6 subshell,
shell #7 has 7 subshell,
etc.

For historical reasons subshells within each shell are not enumerated, but rather assigned a letter. The first 4 subshells are called s, p, d and f, then the letter is assigned in alphabetical order.
So, the first few subshell names are:
s for subshell #1,
p for subshell #2,
d for subshell #3,
f for subshell #4,
g for subshell #5,
h for subshell #6,
i for subshell #7,
etc.

To bring a bit of math into this structure, let's use letter N for a shell's number and letter m for a subshell number within a shell.
Using these symbols, we can state the following:
shell #N has N subshells:
#1, #2,...,#N.

Each subshells has its own capacity to hold electrons. There are theoretical reasons for this based on Quantum Theory. We just state the result of this theory confirmed by experiments:
subshell #m has 4·m−2 electrons.

So,
subshell #1 (s) has 2 electrons,
subshell #2 (p) has 6 electrons,
subshell #3 (d) has 10 electrons,
subshell #4 (f) has 14 electrons,
subshell #5 (g) has 18 electrons,
subshell #6 (h) has 22 electrons,
subshell #7 (i) has 26 electrons,
etc.

Now let's calculate the maximum number of electrons in each shell.

Shell #1 has 1 subshell #1(s) and, therefore, can hold no more than 2 electrons.

Shell #2 has 2 subshells #1(s), #2(p) and, therefore, can hold no more than 2+6=8 electrons.

Shell #3 has 3 subshells #1(s), #2(p), #3(d) and, therefore, can hold no more than 2+6+10=18 electrons.

These calculations can be generalized in a formula for the maximum number of electrons in shell #N:
Σ_m∈[1,N](4m−2) = 2·N²

This formula can be easily proven by induction.
Indeed, it's correct for N=1 because
4·1−2 = 2 = 2·1²
Assuming the formula is correct for some number N, let's check it for N+1.
Σ_m∈[1,N+1](4m−2) =
=Σ_m∈[1,N](4m−2)+4(N+1)−2 =
= 2·N² + 4·(N+1)−2 =
= 2·N² + 4·N + 2 =
= 2·(N² + 2·N + 1) =
= 2·(N+1)²
which is the same formula, but for N+1.