shell model explained
what is included in the shell model of the atom?
energy shells
number of the principal
energy shell, n,
energy and distance
from the nucleus
increases. principal
energy shell 1 is closest
to the nucleus and has
the lowest energy.
number of
electrons
each
energy shell can
hold a maximum
number of
electrons, 2n².
subshells each principal
energy shell has subshells:
s, p, d, and f. each subshell
is made up of orbitals,
shown by a □. each orbital
can hold two electrons.
each shell has a certain
number of subshells.
principal shell 1 2 3 4
subshells 1s 2s, 2p 3s, 3p, 3d 4s, 4p, 4d, 4f
each subshell has a specific
number of orbitals.
subshell s p d f
number of
orbitals 1 3 5 7
25 sep use models why does the number of electrons in each principal energy
shell increase as the number of the shell increases?

Question

Accepted Answer

# Brief Explanations:
To determine why the number of electrons in each principal energy shell increases with the shell number, we analyze the shell model:

1. **Subshells per Shell**: As the principal shell number ($n$) increases, the number of subshells ($s, p, d, f, \dots$) in the shell increases. For example, shell $n = 1$ has 1 subshell ($1s$), $n = 2$ has 2 subshells ($2s, 2p$), $n = 3$ has 3 subshells ($3s, 3p, 3d$), and $n = 4$ has 4 subshells ($4s, 4p, 4d, 4f$).

2. **Orbitals per Subshell**: Each subshell has a fixed number of orbitals: $s = 1$ orbital, $p = 3$ orbitals, $d = 5$ orbitals, $f = 7$ orbitals, etc.

3. **Electrons per Orbital**: Each orbital holds a maximum of 2 electrons (Pauli exclusion principle).

4. **Total Electrons per Shell**: The total number of electrons in a principal shell is given by $2n^2$ (derived from summing electrons in all subshells of the shell). As $n$ increases, $n^2$ increases, so $2n^2$ (total electrons) increases. For example:
   - $n = 1$: $2(1)^2 = 2$ electrons (1 subshell, 1 orbital, $1 \times 2 = 2$).
   - $n = 2$: $2(2)^2 = 8$ electrons (2 subshells: $2s$ (1 orbital, 2 e⁻) + $2p$ (3 orbitals, $3 \times 2 = 6$ e⁻); total $2 + 6 = 8$).
   - $n = 3$: $2(3)^2 = 18$ electrons (3 subshells: $3s$ (2 e⁻) + $3p$ (6 e⁻) + $3d$ (10 e⁻); total $2 + 6 + 10 = 18$).

In summary, as the principal shell number ($n$) increases, the shell contains more subshells, each with more orbitals (or a greater total number of orbitals), and thus can hold more electrons (since each orbital holds 2 electrons). The formula $2n^2$ reflects this, as $n$ (and $n^2$) grows with the shell number.

# Answer:
As the principal energy shell number ($n$) increases, the shell contains more subshells (e.g., $n = 1$ has 1 subshell, $n = 2$ has 2, $n = 3$ has 3, etc.). Each subshell has orbitals (e.g., $s = 1$, $p = 3$, $d = 5$, $f = 7$ orbitals), and each orbital holds 2 electrons. The total electrons per shell follow $2n^2$, so as $n$ increases, $n^2$ (and thus total electrons) increases. More subshells → more orbitals → more electrons per shell.

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