Energy Shells & Subshells: How Electrons Are Organised in Atoms

What Are Energy Shells?

In a multi-electron atom, electrons do not all have the same energy. They are organised into distinct energy shells (also called principal energy levels), numbered 1 through 7 and labelled with the letters K through Q.

Key principle: Electrons in shells closer to the nucleus have lower energy and are held more tightly. Removing them requires more energy (higher ionisation energy).

Learning Goals: By the end of this guide, you should be able to:

1. State the number of energy shells and their letter designations.
2. List the subshells present in each shell.
3. Calculate the number of orbitals and maximum electrons for any shell or subshell.
4. Apply the $2n^2$ formula and explain its derivation.

What Are Subshells?

Each energy shell is further divided into subshells (also called sub-levels), labelled $s$, $p$, $d$, and $f$. The number of subshells in a shell equals its shell number $n$:

The letters originate from early spectroscopy: sharp, principal, diffuse, fundamental. They correspond to the angular momentum quantum number $l = 0, 1, 2, 3$.

Within a given shell, subshells are ordered by energy: $s < p < d < f$.

Orbitals and Electron Capacity

An orbital is a region of space where there is a high probability of finding an electron. Each orbital can hold a maximum of 2 electrons with opposite spins (a consequence of the Pauli exclusion principle).

Orbitals per Subshell

The number of orbitals in a subshell follows the formula $2l + 1$:

The $2n^2$ Rule

The maximum number of electrons in shell $n$ is given by:

Deriving the Formula

Shell $n$ contains $n$ subshells. The orbital counts form the odd number sequence: $1, 3, 5, \ldots, (2n-1)$.

The sum of the first $n$ odd numbers equals $n^2$:

Since each orbital holds 2 electrons: total capacity $= 2n^2$.

Energy Level Ordering and Crossover

Within a single shell, the energy order is straightforward: $s < p < d < f$. However, between shells, the energies of subshells can overlap. This is called energy level crossover:

Key crossover: The $4s$ orbital is lower in energy than $3d$ in neutral atoms. This is why potassium ($Z = 19$) has the configuration $[Ar]\ 4s^1$ rather than $[Ar]\ 3d^1$.

This crossover is a consequence of penetration and shielding effects — the $4s$ orbital penetrates closer to the nucleus than $3d$, experiencing a higher effective nuclear charge.

Important: The $2n^2$ rule gives the maximum capacity of each shell, not the actual number of electrons present. The actual filling order follows the Aufbau principle, which accounts for energy crossover.

Shapes of Orbitals (Qualitative)

At A-Level, you should know the general shapes:

• $s$ orbitals: Spherical. Each successive $s$ orbital ($1s$, $2s$, $3s$, …) is larger.
• $p$ orbitals: Dumbbell-shaped, oriented along the $x$, $y$, and $z$ axes ($p_x$, $p_y$, $p_z$).
• $d$ orbitals: More complex, with four of the five having a cloverleaf shape. $d_{z^2}$ has a unique lobe-and-ring structure.
• $f$ orbitals: Even more complex shapes (beyond A-Level examination scope).

Note: Orbital shapes describe probability distributions, not fixed paths. The electron is most likely found within these regions but can be anywhere.

Worked Examples

Example 1: Electron Capacity of Shell 3

Shell $n = 3$ contains subshells $3s$, $3p$, $3d$.

Orbitals: $1 + 3 + 5 = 9$

Maximum electrons: $2 \times 9 = 18$ or $2 \times 3^2 = 18$ ✓

Example 2: Number of Orbitals in Shell 4

Shell $n = 4$ contains $4s$, $4p$, $4d$, $4f$.

Orbitals: $1 + 3 + 5 + 7 = 16 = 4^2$ ✓

Maximum electrons: $2 \times 16 = 32$

Example 3: Why Can't Shell 1 Have a $p$ Subshell?

Shell $n = 1$ can only have $n = 1$ subshell. The only possibility is $l = 0$ (the $s$ subshell). Since $l$ ranges from $0$ to $n-1$, and $n-1 = 0$, there is no $p$ subshell ($l = 1$ is not allowed).

Common Mistakes

1. Confusing shells with subshells — Shell 3 contains three subshells ($3s$, $3p$, $3d$), not three orbitals. The total number of orbitals is 9.

2. Assuming $2n^2$ gives the actual electron count — It gives the maximum capacity. Outer shells are rarely full in ground-state atoms.

3. Forgetting energy crossover — Students often assume $3d$ fills before $4s$ because $n = 3 < 4$. In reality, $4s$ is lower in energy for neutral atoms.

4. Mixing up orbital count and electron count — A $p$ subshell has 3 orbitals but holds 6 electrons. A $d$ subshell has 5 orbitals but holds 10 electrons.

Exam Tips (A-Level / AP / IB / 高考)

• Memorise the table: $s$ = 1 orbital (2e), $p$ = 3 (6e), $d$ = 5 (10e), $f$ = 7 (14e).
• For $2n^2$ questions, show your working: list all subshells, count orbitals, multiply by 2.
• When asked "how many subshells does shell $n$ have?", the answer is simply $n$.
• Energy crossover ($4s$ before $3d$) explains the position of the transition metals in Period 4.

Frequently Asked Questions

Why are there only 4 types of subshells (s, p, d, f)?

For the elements currently known, only shells 1–7 are occupied. Shell 7 would theoretically contain a $g$ subshell ($l = 4$), but no known element has electrons in $5g$ or higher in its ground state.

What determines the shape of an orbital?

The angular momentum quantum number $l$ determines the orbital shape: $l = 0$ (spherical), $l = 1$ (dumbbell), $l = 2$ (cloverleaf), $l = 3$ (complex). The shapes represent regions of highest electron probability density.

Is the $2n^2$ rule universally valid?

Yes, it correctly predicts the maximum capacity of any shell. However, for shells $n \geq 3$, the outer subshells may not fully fill before electrons start occupying the next shell — this is governed by the filling order (Aufbau principle), not by shell capacity.

Related Topics

• Electron Configuration Rules — Learn how electrons fill orbitals using the Aufbau, Pauli, and Hund rules.
• Atomic Models — Trace the evolution from Bohr's shells to quantum mechanical orbitals.
• Periodic Trends — How shell structure drives patterns in ionisation energy and atomic radius.