Electrons, energy levels and atomic orbitals

1. Overview

The electrons in an atom are arranged in shells, sub-shells and orbitals, and each sub-shell has its own energy. This section covers how many electrons each can hold, the order in which they fill, and how to write the electronic configuration of any atom or ion from hydrogen to krypton, in full, in shorthand and as electrons in boxes. It also covers the shapes of s and p orbitals and what makes a species a free radical. These ideas underpin ionisation energies, bonding and the chemistry of the transition elements.

Key Definitions

• Shell: a main energy level of an atom, made up of all the orbitals with the same principal quantum number, n; shell n can hold at most 2n² electrons (2, 8 and 18 for the first three shells).
• Principal quantum number (n): the whole number (1, 2, 3, 4 and so on) that labels a shell; in general, electrons in a shell with a larger n have more energy and are further from the nucleus on average.
• Sub-shell: a group of orbitals of the same type (s, p or d) within one shell; s, p and d sub-shells contain 1, 3 and 5 orbitals and can hold up to 2, 6 and 10 electrons.
• Atomic orbital: a region of space around the nucleus where there is a high probability of finding an electron; one orbital can hold at most two electrons, and they must have opposite spins.
• Ground state: the arrangement of electrons in an atom or ion that has the lowest possible energy; if an electron moves up to a higher-energy orbital, the atom is in an excited state instead.
• Electronic configuration: the arrangement of the electrons of an atom or ion in shells, sub-shells and orbitals, written in full (sodium: 1s² 2s² 2p⁶ 3s¹) or in shorthand with a noble gas core (sodium: [Ne] 3s¹).
• Spin pair: two electrons in the same orbital, which have opposite spins; they repel each other more strongly than two electrons in separate orbitals of the same sub-shell.
• Free radical: a species (an atom, molecule or ion) with one or more unpaired electrons.

Content

Shells, sub-shells and orbitals

Electrons are described at three levels, from largest to smallest.

1. Shells. Each shell is labelled by its principal quantum number, $n$ = 1, 2, 3, 4 and so on. In general, the larger $n$ is, the further the electrons are from the nucleus on average and the more energy they have.
2. Sub-shells. Shell $n$ is split into $n$ types of sub-shell: shell 1 has only 1s; shell 2 has 2s and 2p; shell 3 has 3s, 3p and 3d. Shell 4 has four types (4s, 4p, 4d and 4f), but you only need 4s and 4p.
3. Orbitals. Each sub-shell is made of orbitals. An orbital is a region of space where an electron is likely to be found, and it holds at most two electrons.

Sub-shell	Number of orbitals	Maximum number of electrons
s	1	2
p	3	6
d	5	10

Shell ($n$)	Sub-shells	Orbitals	Maximum electrons ($2n^2$)
1	1s	1	2
2	2s, 2p	1 + 3 = 4	8
3	3s, 3p, 3d	1 + 3 + 5 = 9	18

In a label such as 3p⁴, the 3 is the shell number ($n$), the p is the sub-shell and the superscript is the number of electrons in that sub-shell, shared between its three orbitals.

Each electron also has a property called spin, which has two possible states, drawn as ↑ and ↓. Two electrons in the same orbital must have opposite spins: they form a spin pair.

Ground state

The ground state is the arrangement of electrons with the lowest possible energy. If an atom absorbs energy, an electron can move up to a higher orbital, giving an excited state: sodium as 1s² 2s² 2p⁶ 3p¹ instead of its ground state, 1s² 2s² 2p⁶ 3s¹.

Every configuration you will write in this topic is a ground state, for hydrogen to krypton (proton number, Z, from 1 to 36). "Ground state" is about electrons only. It has nothing to do with the standard state of a substance.

The order of sub-shell energies

In order of increasing energy, the sub-shells up to 4p are:

$$\text{1s} < \text{2s} < \text{2p} < \text{3s} < \text{3p} < \text{4s} < \text{3d} < \text{4p}$$

• In any atom with more than one electron, s is lower than p within a shell, and p is lower than d.
• The energy gaps between shells get smaller as $n$ increases, so the third and fourth shells overlap in energy: the 4s sub-shell is lower in energy than 3d and fills first. After argon, the next two electrons go into 4s. The 3d sub-shell then fills from scandium (Z = 21) to zinc (Z = 30), and 4p from gallium (Z = 31) to krypton (Z = 36).

Filling the orbitals: energy and repulsion

Three rules give the ground state. Each keeps the total energy as low as possible.

1. Lowest energy first. Each electron enters the lowest-energy sub-shell that still has space.
2. Two per orbital, with opposite spins.
3. Singly before pairing. In an isolated atom, the orbitals of one p or d sub-shell all have the same energy. Electrons occupy them one at a time, with the same spin (parallel arrows), and pair up only when every orbital in the sub-shell has one electron.

Rule 3 comes from inter-electron repulsion. All electrons are negative, so they repel each other. Two electrons sharing an orbital occupy the same region of space, so on average they are closer together and repel more strongly than two electrons in separate orbitals. Nitrogen's three 2p electrons therefore occupy all three 2p orbitals singly. Oxygen's fourth 2p electron has no empty 2p orbital left, so it pairs: ↑↓ ↑ ↑. This extra repulsion in a spin pair reappears when you explain ionisation energies.

Writing electronic configurations

1. Count the electrons. An atom has Z electrons, where Z is the proton (atomic) number. A positive ion has fewer (Z minus the charge); a negative ion has more.
2. Fill the sub-shells in energy order, each up to its maximum.
3. Write them in shell order: 3d before 4s, even though 4s filled first. Iron (Z = 26) is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶ 4s², its full electronic configuration. Some books list 4s before 3d instead; it is the same configuration.
4. Check that the superscripts add up to the number of electrons.

The shorthand electronic configuration replaces the inner electrons with the noble gas that has the same arrangement, in square brackets:

Core	Stands for	Use it for
[He]	1s²	lithium to neon
[Ne]	1s² 2s² 2p⁶	sodium to argon
[Ar]	1s² 2s² 2p⁶ 3s² 3p⁶	potassium to krypton

So iron is [Ar] 3d⁶ 4s². The [Ar] core contains no 3d electrons: they are always written after it.

Electrons in boxes

Electrons in boxes shows every orbital, so paired and unpaired electrons are easy to see.

• Each box is one orbital. The boxes of a sub-shell sit side by side under its label: three for 2p, five for 3d.
• Each arrow is one electron; a spin pair is ↑↓ in one box.
• Put one arrow in each box of a sub-shell, all pointing the same way, before any box gets a second.
• Inner sub-shells can be replaced by the noble gas core.

For iron's 3d⁶, the first five electrons take one box each and the sixth pairs up in the first box:

Fe	3d	3d	3d	3d	3d	4s
[Ar]	↑↓	↑	↑	↑	↑	↑↓

Iron has four unpaired electrons, all in 3d. Which box holds the pair does not matter; filling from the left is the usual convention.

Worked example 1: configuration of a selenium atom

Question: Selenium has proton number 34. (a) Give its full and shorthand electronic configurations. (b) State the number of electrons in each shell. (c) Show the 4s and 4p sub-shells as electrons in boxes, and state the number of unpaired electrons in the atom.

Step 1: fill in energy order, keeping a running total up to 34 electrons.

Sub-shell	1s	2s	2p	3s	3p	4s	3d	4p
Electrons added	2	2	6	2	6	2	10	4
Running total	2	4	10	12	18	20	30	34

Only 4 electrons are left for 4p, so it is part-filled.

Step 2: write in shell order and check. 2 + 2 + 6 + 2 + 6 + 10 + 2 + 4 = 34.

Step 3: count each shell. Shell 1: 2. Shell 2: 2 + 6 = 8. Shell 3: 2 + 6 + 10 = 18. Shell 4: 2 + 4 = 6.

Step 4: boxes. The first three 4p electrons go in singly; the fourth must pair.

Se	4s	4p	4p	4p
[Ar] 3d¹⁰	↑↓	↑↓	↑	↑

Every inner orbital is full, so the only unpaired electrons are the two single 4p electrons.

Answer: (a) 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁴, or [Ar] 3d¹⁰ 4s² 4p⁴. (b) 2, 8, 18, 6. (c) 4s ↑↓; 4p ↑↓ ↑ ↑; two unpaired electrons.

Chromium and copper: the two exceptions

Element	Predicted by the rules	Actual ground state	Unpaired electrons
Chromium (Z = 24)	[Ar] 3d⁴ 4s²	[Ar] 3d⁵ 4s¹	6
Copper (Z = 29)	[Ar] 3d⁹ 4s²	[Ar] 3d¹⁰ 4s¹	1

These are the only two exceptions from hydrogen to krypton. The 3d and 4s sub-shells are so close in energy that a small saving decides which arrangement is lower.

• Chromium: 3d⁵ 4s¹ puts one electron in each of six orbitals (five 3d and one 4s), so no orbital holds a pair. Avoiding the repulsion between two electrons sharing the 4s orbital saves more energy than it costs to move one electron up into 3d. The 3d sub-shell is half-filled: ↑ ↑ ↑ ↑ ↑, with ↑ in 4s.
• Copper: the arrangement with a completely filled 3d sub-shell, 3d¹⁰ 4s¹, has a lower total energy than 3d⁹ 4s², so one 4s electron moves into 3d.

Configurations of ions

Positive ions lose electrons from the shell with the highest $n$ first.

• For scandium to zinc, remove the 4s electrons before any 3d electrons, even though 4s filled first. Once 3d holds electrons, the two sub-shells are very close in energy, and the 4s electrons, further from the nucleus on average, are lost first. Fe²⁺ is [Ar] 3d⁶, not [Ar] 3d⁴ 4s².
• For gallium to krypton, remove 4p electrons, then 4s; the full 3d stays. Ga³⁺ is [Ar] 3d¹⁰.

Negative ions gain electrons into the next available orbitals in the normal energy order. Many end up with a full outer p sub-shell, like the next noble gas.

Worked example 2: configurations of ions

Question: Nickel has proton number 28. (a) Give the shorthand electronic configurations of a nickel atom and of a Ni²⁺ ion. (b) Show the 3d and 4s sub-shells of Ni²⁺ as electrons in boxes and state its number of unpaired electrons. (c) Give the shorthand configuration of the selenide ion, Se²⁻ (Z of Se = 34).

Step 1: the atom. Ni has 28 electrons, 10 more than [Ar]. In energy order, 4s takes 2 and 3d takes the other 8: [Ar] 3d⁸ 4s².

Step 2: the 2+ ion. Ni²⁺ has 28 − 2 = 26 electrons. Both lost electrons come from 4s: [Ar] 3d⁸, or in full 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁸.

Step 3: boxes for Ni²⁺. Five 3d electrons go in singly, then the next three pair up.

Ni²⁺	3d	3d	3d	3d	3d	4s
[Ar]	↑↓	↑↓	↑↓	↑	↑	(empty)

Step 4: the 2− ion. Se²⁻ has 34 + 2 = 36 electrons. Selenium is [Ar] 3d¹⁰ 4s² 4p⁴ (Worked example 1), and the two extra electrons fill 4p.

Answer: (a) Ni: [Ar] 3d⁸ 4s²; Ni²⁺: [Ar] 3d⁸. (b) 3d ↑↓ ↑↓ ↑↓ ↑ ↑ with 4s empty: two unpaired electrons. (c) Se²⁻: [Ar] 3d¹⁰ 4s² 4p⁶, the same configuration as a krypton atom.

Shapes of s and p orbitals

	s orbital	p orbital
Shape	a sphere centred on the nucleus	two equal lobes on opposite sides of the nucleus (a dumbbell)
Number in a sub-shell	1	3, at right angles to each other: $\text{p}_x$, $\text{p}_y$ and $\text{p}_z$
In higher shells	same shape, larger: 2s is bigger than 1s	same shape, larger: 3p is bigger than 2p
How to sketch	a circle with the nucleus as a dot at its centre	draw the axes with the nucleus at the origin; draw two lobes along one axis, meeting at the nucleus

The two lobes of a p orbital make up one orbital, not two: an electron in $\text{p}_x$ can be found in either lobe. $\text{p}_x$ lies along the x axis, $\text{p}_y$ along the y axis and $\text{p}_z$ along the z axis.

Free radicals

A free radical is a species with one or more unpaired electrons. An unpaired electron is alone in an orbital. A lone pair is not unpaired: it is two paired electrons.

• A radical is written with a dot for the unpaired electron, as in Cl• or •CH₃.
• Radicals can form when a covalent bond breaks evenly, each atom keeping one electron of the shared pair (homolytic fission): Cl₂ → 2Cl•.
• A species with two or more unpaired electrons is also a free radical.
• Quick test: a species with an odd number of electrons must have at least one unpaired electron. An even number does not settle it: check the configuration.

Worked example 3: which species are free radicals?

Question: Decide whether each species is a free radical: (a) a selenium atom; (b) a selenide ion, Se²⁻; (c) C₂H₅, formed when one C-H bond in ethane breaks evenly.

(a) 34 electrons, an even number, but the 4p sub-shell is ↑↓ ↑ ↑ (Worked example 1): two unpaired electrons.

(b) 36 electrons, [Ar] 3d¹⁰ 4s² 4p⁶. Every orbital is full: no unpaired electrons.

(c) 2 × 6 + 5 × 1 = 17 electrons, an odd number, so at least one is unpaired.

Answer: (a) a free radical; (b) not a free radical; (c) a free radical, written C₂H₅•. Parts (a) and (b) both have an even number of electrons, so only the configuration can decide.

Key Equations

Maximum number of electrons in shell $n$ (2, 8 and 18 for $n$ = 1, 2 and 3): $$\text{maximum electrons} = 2n^2$$

Maximum electrons in a sub-shell (s: 2, p: 6, d: 10): $$\text{maximum electrons} = 2 \times \text{number of orbitals}$$

Number of electrons in a species ($Z$ = proton number; $q$ = charge with its sign, so +2 for Ni²⁺ and −2 for Se²⁻): $$\text{number of electrons} = Z - q$$

Unpaired electrons in one sub-shell, ground state ($k$ = number of orbitals in the sub-shell; $e$ = number of electrons in it): $$\text{unpaired} = e \ \text{ if } e \le k \qquad \text{unpaired} = 2k - e \ \text{ if } e > k$$ For 4p⁴: $k = 3$ and $e = 4$, so $2(3) - 4 = 2$. For 3d⁸: $2(5) - 8 = 2$.

Filling order (lowest energy first; chromium and copper are the exceptions): $$\text{1s} \to \text{2s} \to \text{2p} \to \text{3s} \to \text{3p} \to \text{4s} \to \text{3d} \to \text{4p}$$

Common Mistakes to Avoid

• Mistake: taking electrons out of 3d before 4s when forming a d-block ion. Fix: 4s empties first. Ni²⁺ is [Ar] 3d⁸, not [Ar] 3d⁶ 4s².
• Mistake: writing chromium and copper by the filling rules. Fix: learn the two exceptions: Cr is [Ar] 3d⁵ 4s¹, not 3d⁴ 4s², and Cu is [Ar] 3d¹⁰ 4s¹, not 3d⁹ 4s².
• Mistake: pairing electrons before every orbital of the sub-shell has one. Fix: selenium's 4p⁴ is ↑↓ ↑ ↑ with two unpaired electrons, not ↑↓ ↑↓ with an empty box and none unpaired.
• Mistake: putting the 19th and 20th electrons into 3d. Fix: 4s is lower in energy than 3d, so it fills first. The first 3d electron arrives at scandium (Z = 21), [Ar] 3d¹ 4s².
• Mistake: mixing up orbitals and sub-shells when counting. Fix: 3d is one sub-shell of five orbitals. The eight 3d electrons of Ni²⁺ fill three orbitals with pairs and two with one electron each.
• Mistake: confusing the ground state with a physical or standard state. Fix: the ground state is the electron arrangement with the lowest energy. Sodium's ground state is 1s² 2s² 2p⁶ 3s¹; 1s² 2s² 2p⁶ 3p¹ is an excited state.
• Mistake: defining a free radical as a species with a lone pair, or with a "free" or "unshared" electron. Fix: say "a species with one or more unpaired electrons". The hydroxyl radical, OH•, has 9 electrons and one is unpaired; the hydroxide ion, OH⁻, has 10 electrons, all in pairs.
• Mistake: sketching a p orbital off the axis, or with lobes that do not meet at the nucleus. Fix: put the nucleus at the origin and draw both lobes of $\text{p}_y$ along the y axis, touching at the nucleus.

Exam Tips

• Give a full configuration with every sub-shell from 1s and no noble gas core. Add up the superscripts and check they equal Z for an atom, or Z minus the charge for an ion.
• Complete an electrons-in-boxes diagram with one arrow per electron, ↑↓ for a pair, and single arrows all the same way. Use the boxes you are given and leave empty orbitals empty.
• Explain a configuration in terms of energy and repulsion: electrons enter the lowest-energy orbital available, and they occupy separate orbitals of a sub-shell before pairing because paired electrons in one orbital repel more.
• Sketch orbitals with labelled axes and the nucleus marked: an s orbital as a circle centred on the nucleus, a p orbital as two lobes along the named axis.
• Define a free radical as "a species with one or more unpaired electrons", and the ground state as the lowest-energy arrangement of the electrons. Two unpaired electrons count too, as in a selenium atom.
• Deduce the number of unpaired electrons from the boxes (or the rule in Key Equations), not by reading the superscript: 4p⁴ and 4p² both have two unpaired electrons.