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Overview

By the end of this chapter you should be able to describe Thomson’s and Rutherford’s atomic models; explain the limitations of Bohr’s model; explain the key modifications introduced by the modern atomic theory; define the electron cloud and the orbital; identify the four quantum numbers; relate the principal quantum number to sublevels and orbitals; write electron configurations using the Aufbau principle, Hund’s rule and Pauli’s exclusion principle; and appreciate the contribution of scientists to our understanding of the atom.
Lessons in this chapter:

Comparison aspect	Main idea	Keywords
Early atomic models	Thomson and Rutherford	ThomsonRutherford
•	The atomic emission spectrum and Bohr’s atomic model	atomicemissionspectrumBohr
The modern atomic theory	dual nature uncertainty principle wave mechanics	dualnatureuncertaintyprinciple
•	Quantum numbers	Quantumnumbers
Distribution of electrons in the atom	Pauli Aufbau Hund	PauliAufbauHund

🌳 Concept tree

Chapter 2 — Atomic Structure

├── Lesson 1→Early atomic models — Thomson and Rutherford

│ ├── 1.1 Thomson's atomic model (1897)

│ ├── 1.2 Rutherford's gold foil experiment

│ └── 1.3 Postulates of Rutherford's atomic model (1911)
├── Lesson 2 The atomic emission spectrum and Bohr's atomic model

│ ├── 2.1 Atomic emission (line) spectrum

│ ├── 2.2 Bohr's atomic model (1913)

│ ├── 2.3 The hydrogen spectrum explained

│ ├── 2.4 Success of Bohr's model

│ └── 2.5 Shortcomings of Bohr's model

├── Lesson 3→The modern atomic theory

│ ├── 3.1 Dual nature of the electron

│ ├── 3.2 Heisenberg's uncertainty principle

│ ├── 3.3 Wave-mechanical theory (Schrödinger
1926)

│ └── 3.4 Electron cloud and orbital

├── Lesson 4→Quantum numbers

│ ├── 4.1 Principal quantum number (n)

│ ├── 4.2 Subsidiary quantum number (ℓ)

│ ├── 4.3 Magnetic quantum number (mℓ)

│ ├── 4.4 Spin quantum number (ms)

│ ├── 4.5 Relations between n
sublevels
orbitals and electrons

│ └── 4.6 Worked examples

└── Lesson 5→Distribution of electrons in the atom

├── 5.1 Pauli's exclusion principle

├── 5.2 Aufbau (building-up) principle

├── 5.3 Hund's rule

└── 5.4 Putting it together — rules checklist

Chapter 2 — Atomic Structure · 1 / 20

Lesson 1: Early atomic models — Thomson and Rutherford

Scientists made several attempts to understand the structure of the atom.
The main stages were (1) Thomson’s atomic model, (2) Rutherford’s atomic model, (3) Bohr’s atomic model, and (4) the principles of the modern atomic theory.
Each model explained the known facts of its time and was then improved or replaced when new experiments revealed facts it could not explain.
This is how scientific knowledge grows.

Chapter 2 — Atomic Structure · 2 / 20

1.1 Thomson’s atomic model (1897)

After discovering the electron, J.
J.
Thomson proposed that the atom is a solid (not hollow), homogeneous sphere of uniform positive charge, with negative electrons embedded in it like seeds in a watermelon (the "plum pudding" picture).
The total negative charge of the electrons equals the positive charge of the sphere, so the atom as a whole is electrically neutral.
Features:

the positive charge is spread evenly through the whole volume of the atom;
the electrons are scattered inside the positive sphere;
the atom is solid, with no empty space.

Thomson’s model could explain neutrality but gave no explanation of how the positive charge and the mass were arranged.
An experiment with alpha particles tested it and showed it to be wrong.

Chapter 2 — Atomic Structure · 3 / 20

1.2 Rutherford’s gold foil experiment

Geiger and Marsden
working in Rutherford’s laboratory
carried out the experiment with the following apparatus:

a source of alpha particles (He nuclei, ⁴₂He²⁺, positively charged) inside a lead box with a narrow opening, so that a thin beam is emitted;

a very thin gold foil in the path of the beam;

a circular screen coated with zinc sulphide, which gives a tiny flash of light whenever an alpha particle hits it.
Steps:

Without the foil, the beam reaches the screen at one spot. Rutherford recorded the position and number of flashes.
A very thin gold foil was then placed in the path, and the flashes were observed all around the foil.

Observations and the conclusion drawn from each:

Observation	Conclusion
Most alpha particles passed straight through the foil and produced flashes at the original position.	Most of the atom is empty space the atom is not a solid sphere as Thomson thought
A very small fraction (about 1 in thousands) bounced back (was reflected) towards the source.	There is a part of the atom of very high density that occupies a very small space: the nucleus.
Some particles were deflected from the straight path to either side	The charge of the dense part (where most of the mass is concentrated) is positive, like the alpha particles, so it repels them.

A useful analogy firing bullets at a thin curtain in which a few small steel balls hang.
Almost all the bullets pass; one that hits a ball bounces back or is deflected.

✍️ Worked examples

Worked example 1.1 — interpreting a result.
In a repetition of the experiment, 99 % of the particles passed straight through.
What does this show?
Since nearly all particles were not disturbed, the volume occupied by the nucleus must be very small compared with the volume of the atom, and the atom is mainly empty space.
Worked example 1.2 — why would the alpha particles be reflected?
An alpha particle has a positive charge and a mass about 7000 times that of an electron.
Electrons cannot turn it back.
Only a heavy, positively charged, concentrated object (the nucleus), which repels it, can reverse it.
Therefore reflection proves the existence of a dense positive nucleus holding most of the mass.
Worked example 1.3 — if Thomson were right.
If the atom were a uniform positive sphere, the charge would be spread out, no strong repulsion would occur at any point, and every alpha particle would pass through with only a small deflection.
Backward reflection would never occur.
This is how the experiment disproved Thomson’s model.

Chapter 2 — Atomic Structure · 4 / 20

1.3 Postulates of Rutherford’s atomic model (1911)

From this and other experiments Rutherford proposed:

Comparison aspect	Main idea	Keywords
The atom	has an extremely small size but a complex structure resembling the solar system: a central nucleus (like the Sun) around the electrons revolve (like planets)	extremelysmallsizecomplex
The nucleus	is much smaller than the atom with a vast space between it and the electron orbits (so the atom is not solid). It is positively charged and most of the atom’s mass is concentrated in it	muchsmalleratomvast
The electrons	have a very small mass negligible compared with the nucleus their total negative charge equals the positive charge of the nucleus so the atom is electrically neutral orbit the nucleus at high speed in definite orbits. The attraction between nucleus and electrons is balanced by an equal and opposite (centrifugal) force so the electron does not fall into the nucleus	verysmallmassnegligible

Limitation.

Rutherford’s model did not explain the arrangement (system) in which the electrons move around the nucleus, nor the line spectra of elements.
This was solved in part by Bohr.
Summary table

Model	Year	Idea	Evidence/limit
Thomson	1897	Solid positive sphere with embedded electrons	Disproved by gold foil
Rutherford	1911	Small dense positive nucleus electrons orbit in empty space	Could not explain how electrons are arranged no spectra

Chapter 2 — Atomic Structure · 5 / 20

2.1 Atomic emission (line) spectrum

When the atoms of a pure element in the gaseous or vapour state are heated to a high temperature, or are subjected to an electric discharge at low pressure in a discharge tube, they emit radiation.
When this light is passed through a slit and a prism in a spectrometer, it does not give a continuous rainbow.
Instead it appears as a limited number of coloured lines, each of definite wavelength and frequency, separated by dark regions.
It is therefore called the line (emission) spectrum.

Experimentally, the line spectrum of every element is a characteristic property: no two elements have the same line spectrum, like a fingerprint.
For this reason spectra are used to identify elements (for example in stars and in flame tests).
The visible spectrum of hydrogen (discharge tube containing H₂ at low pressure) shows four lines: violet (410 nm), blue (434 nm), green (486 nm) and red (656 nm).

Chapter 2 — Atomic Structure · 6 / 20

2.2 Bohr’s atomic model (1913)

The Danish scientist Niels Bohr explained the hydrogen spectrum in 1913 (Nobel Prize in Physics, 1922).
He kept three ideas of Rutherford (1) a positive nucleus at the centre; (2) the number of electrons equals the number of positive charges in the nucleus; (3) a centrifugal force balances the attraction of the nucleus.
Then he added the following postulates:

Electrons move rapidly around the nucleus without losing or gaining energy while they stay in a given orbit.
Electrons move in a number of definite energy levels; the spaces between them are forbidden regions for electron motion.
An electron has a definite energy depending on the distance of its level from the nucleus; the energy increases as the radius of the level increases. Each level is labelled by an integer, the principal quantum number n (1, 2, 3, …).
Normally the electron stays in the lowest possible energy level (ground state, stable). If it gains a quantum of energy (by heating or electric discharge) the atom becomes excited and the electron jumps to a higher level corresponding to the quantum absorbed. This state is unstable; the electron soon returns, losing the same quantum as light of a characteristic wavelength and frequency — producing a spectral line.
Many atoms absorb different quanta at the same time and many emit others, so a set of lines appears, indicating the energy levels between which the electrons jump. This explains the hydrogen lines.

A quantum is the amount of energy gained or emitted when an electron moves from one energy level to another.

Chapter 2 — Atomic Structure · 7 / 20

2.3 The hydrogen spectrum explained

When the excited electron falls to the second level (n = 2) from n = 3, 4, 5, 6, the energy released is visible light: n = 3→2 gives the red line (656 nm); 4→2 gives the green (486 nm); 5→2 blue (434 nm); 6→2 violet (410 nm). These make up the visible line spectrum of hydrogen.
When the electron returns to the first level (n = 1), the energy released is greater and the lines are in the ultraviolet region, invisible to the eye.
Transitions to levels n = 3 and higher give lines in the infrared region, also invisible.
Important observations of Bohr’s calculations:

The energy differences between neighbouring levels are not equal; they decrease as we move away from the nucleus. Hence the quantum needed to move an electron between different levels is not constant, and each transition gives a different line.
The electron never stays between levels; it moves in definite jumps from one level to another.
The larger the energy gap, the shorter the wavelength (higher frequency) of the emitted light, so 6→2 (violet) has a larger gap than 3→2 (red).

Worked example 2.1.→Which transition of hydrogen gives the line of the greatest energy among the four visible lines? The violet line, 410 nm (the shortest wavelength), arising from the electron falling from n = 6 to n = 2. The red line (3→2, 656 nm) has the smallest energy.

✍️ Worked examples

Worked example 2.2.
Explain why the electron in a hydrogen atom, heated strongly, gives several lines even though there is only one electron.
Because a large number of atoms are present in the sample; different atoms have their electrons excited to different levels and fall back by different routes, so all the possible transitions are emitted simultaneously.
Worked example 2.3. The electron of hydrogen is raised from n = 1 to n = 3 and falls directly back to n = 1. What type of radiation is emitted? Ultraviolet, since the end level is n = 1.

Chapter 2 — Atomic Structure · 8 / 20

2.4 Success of Bohr’s model

(a) It explained correctly the line spectrum of the hydrogen atom.
(b) It was the first theory to introduce the quantum concept into the determination of electron energy in the different levels.

Chapter 2 — Atomic Structure · 9 / 20

2.5 Shortcomings of Bohr’s model

Many quantitative results did not agree with experiment. The most important shortcomings:

It dealt mainly with hydrogen, the simplest atom. It explained only its spectrum, and failed to explain the spectrum of any other element, even helium with only two electrons.
The electron was treated purely as a negative material particle; its wave properties were ignored.
It assumed that the position and the speed of the electron can be determined precisely at the same time, which is practically impossible (Heisenberg).
Bohr’s equations treat the electron as moving in a circular plane, so the hydrogen atom would be flat (planar); atoms are known to be three-dimensional.

These shortcomings led scientists to continue the search that produced the modern atomic theory.

Chapter 2 — Atomic Structure · 10 / 20

Lesson 3: The modern atomic theory

The modern theory is based on fundamental modifications of Bohr’s model.
The most important are (a) the dual nature of the electron, (b) Heisenberg’s uncertainty principle, (c) the wave-mechanical theory of the atom (Schrödinger).

Earlier models treated the electron only as a tiny negatively charged particle.
Experiments proved that the electron has a dual nature it is a material particle which also has wave properties.
(Beams of electrons can be diffracted like light.)

Bohr assumed we can know exactly both the position and the speed of the electron at the same moment.
Heisenberg, using quantum mechanics, concluded that it is practically impossible to determine the position and the velocity of the electron simultaneously.
We can only say that the electron is probably (more or less) present in a certain place.
Speaking in terms of probability is the more accurate approach.

The reason to "see" an electron we must hit it with radiation, and this disturbs its position or speed.
The very small mass of the electron makes this disturbance significant.

The Austrian scientist Schrödinger set up the wave equation that describes the motion of an electron in the atom.
Its solution gives the permitted energy levels and the regions around the nucleus where the probability of finding the electron is high in each level.
This changed the concept of electron motion the electron does not move in fixed orbits with forbidden gaps between them (Bohr), but is found with a certain probability in regions of space around the nucleus.

The electron cloud expresses the region of space around the nucleus in which the electron is likely to be found, in all directions and dimensions. It is denser where the probability is higher.
Within the cloud there are regions where the probability of finding the electron is greatest; each such region is called an orbital.
The mathematical solution of Schrödinger’s equation gives four numbers, the quantum numbers, that describe each electron in a multi-electron atom.

✍️ Worked examples

Worked example 3.1.
Why is the picture of "an electron running on a circle like a planet" no longer accepted?
Because we cannot know the exact position and speed together (Heisenberg); we can only give a probability map, the orbital.
Orbits (fixed paths) were replaced by orbitals (regions of probability).
Worked example 3.2.
Compare "orbit" and "orbital".
An orbit (Bohr) is a definite circular path at a fixed distance, in a plane.
An orbital (wave mechanics) is a three-dimensional region around the nucleus where the electron is most likely to be found, with a characteristic shape.
Worked example 3.3. Match each scientist to the idea: Thomson — positive sphere with embedded electrons; Rutherford — nucleus; Bohr — quantised energy levels; Heisenberg — uncertainty principle; Schrödinger — wave equation and orbitals.

Chapter 2 — Atomic Structure · 11 / 20

Lesson 4: Quantum numbers

To describe the energy of electrons in multi-electron atoms we need four quantum numbers:

Quantum number	Symbol	What it describes
Principal	n	Distance of the electron from the nucleus (main energy level)
Subsidiary (secondary, azimuthal)	ℓ	Shape of the electron cloud in the sublevel the sublevels
Magnetic	mℓ	Orientation of the orbital in space number of orbitals
Spin	m_s	The spin motion of the electron about its own axis

Chapter 2 — Atomic Structure · 12 / 20

4.1 Principal quantum number (n)

Used already by Bohr
it:

determines the order of the principal energy levels (shells) and their number: in the heaviest known atom in the ground state there are seven shells (n = 1 to 7: K, L, M, N, O, P, Q);
determines the maximum number of electrons that fill a shell = 2n²:
- K (n = 1): 2(1)² = 2 electrons;
- L (n = 2): 2(2)² = 8;
- M (n = 3): 2(3)² = 18;
- N (n = 4): 2(4)² = 32.

The rule does not apply to levels beyond the fourth (an atom becomes unstable if a level has more than 32 electrons).
n is always a positive integer 1, 2, 3, …; it can never be zero or fractional.
A larger n means a larger and higher-energy shell.

Chapter 2 — Atomic Structure · 13 / 20

4.2 Subsidiary quantum number (ℓ)

Each main level is divided into sublevels (subshells) whose energy is determined by ℓ.

The number of sublevels in a main level = its principal quantum number n.
The values of ℓ run from 0 to (n − 1).
The sublevels have the symbols s, p, d, f:

ℓ	0	1	2	3
Sublevel	s	p	d	f

So level 1 has 1s only; level 2 has 2s, 2p; level 3 has 3s, 3p, 3d; level 4 has 4s, 4p, 4d, 4f.
Within the same main level the sublevels differ slightly in energy: f > d > p > s.

Chapter 2 — Atomic Structure · 14 / 20

4.3 Magnetic quantum number (mℓ)

It indicates the number of orbitals in a sublevel and their directions in space.
Its values are integers from −ℓ to +ℓ including zero. So there are (2ℓ + 1) orbitals in a sublevel.

n	ℓ	sublevel	mℓ values	orbitals
1	0	1s	0	1
2	0	2s	0	1
2	1	2p	−1, 0, +1	3
3	0	3s	0	1
3	1	3p	−1, 0, +1	3
3	2	3d	−2, −1, 0, +1, +2	5
4	0, 1, 2, 3	4s, 4p, 4d, 4f	… and for 4f: −3 … +3	1, 3, 5, 7

Shapes:

An s orbital has the shape of a sphere, symmetrical around the nucleus.
A p sublevel has three orbitals along the three axes x, y, z, named p_x, p_y, p_z, perpendicular to each other. Each is dumbbell-shaped: two lobes meeting at the nucleus where the electron density is zero.
The d sublevel has 5 orbitals, the f sublevel 7.

Chapter 2 — Atomic Structure · 15 / 20

4.4 Spin quantum number (ms)

Any orbital can hold at most two electrons.
Each electron spins about its own axis (like the Earth) while moving around the nucleus.
Two electrons in one orbital are both negative and might be expected to repel; however, their spins are in opposite directions, creating opposing magnetic fields that attract each other and lower the repulsion.
They are said to be spin-paired (↑↓).
The spin quantum number determines the direction of spin clockwise gives ms = +½ (↑), anticlockwise ms = −½ (↓).

Chapter 2 — Atomic Structure · 16 / 20

4.5 Relations between n, sublevels, orbitals and electrons

Comparison aspect	Main idea	Keywords
•	The number of sublevels in a main level = n (level 1 has one level 2 two	numbersublevelsmainlevel
The number of orbitals in a main level = n² (level 2	2s + 2px 2py 2pz = 4 level 3: 1 + 3 + 5 = 9)
The number of orbitals in a sublevel = (2ℓ + 1)	p has 2(1) + 1 = 3
•	The maximum number of electrons in a main level = 2n² (twice the orbitals)	maximumnumberelectronsmain
Maximum electrons in a sublevel = 2(2ℓ + 1)	s 2 p 6 d 10 f 14. This explains why the p sublevel takes six electrons and d takes ten	explainstakes

Summary up to the third level:

Level	n	Sublevels (ℓ)	Orbitals	Max electrons
K	1	1s	1	2
L	2	2s, 2p	1 + 3 = 4	8
M	3	3s, 3p, 3d	1 + 3 + 5 = 9	18
N	4	4s, 4p, 4d, 4f	1 + 3 + 5 + 7 = 16	32

Chapter 2 — Atomic Structure · 17 / 20

4.6 Worked examples

Comparison aspect	Main idea	Keywords
Worked example 4.1 — allowed values.	For n = 3 ℓ may be 0 2 (sublevels 3s 3d). For ℓ = 2 mℓ = −2 +2 (5 orbitals)	sublevelsorbitals
Worked example 4.2 — for n = 4 count everything.	Sublevels: 4. Orbitals: 4² = 16. Maximum electrons: 2 × 16 = 32. The f sublevel (ℓ = 3) has mℓ = −3 +3 (7 orbitals 14 electrons)	SublevelsOrbitalsMaximumelectrons
Worked example 4.3 — check a set of quantum numbers.	n = 2, ℓ = 2, mℓ = 0, m_s = +½. Invalid, because ℓ must be less than n (maximum n − 1 = 1). n = 3, ℓ = 1, mℓ = −1, m_s = −½: valid (a 3p electron).	Invalidbecausemustless
Worked example 4.4 — identify the sublevel.	An electron has n = 4 and ℓ = 3. It is in the 4f sublevel which can hold 14 electrons	electronholdelectrons

Comparison aspect	Main idea	Keywords
Worked example 4.5 — find an error.	n = 2, ℓ = 1, mℓ = +2, m_s = ½. mℓ must be between −1 and +1, so +2 is not allowed (it would need ℓ ≥ 2).	mustbetweenallowedwould
Worked example 4.6 — quantum numbers of the last electron.	Convention used: the p orbitals are filled in the order mℓ = −1, 0, +1, first singly with m_s = +½, then pairing with m_s = −½ in the same order (other textbooks may reverse the order; the key is to follow Hund’s rule consistently).	Conventionusedorbitalsfilled

Comparison aspect	Main idea	Keywords
Boron (₅B)	1s² 2s² 2p¹. The last electron is the first 2p electron: n = 2 ℓ = 1 mℓ = −1 ms = +½	lastelectronfirst
Fluorine (₉F)	1s² 2s² 2p⁵. Electrons 1 to 3 of 2p occupy mℓ = −1 +1 (all +½) the 4th pairs in mℓ = −1 (−½) the 5th pairs in mℓ = 0 (−½). So the last electron has n = 2 ℓ = 1 mℓ = 0 ms = −½	Electronsoccupypairslast
Sodium (₁₁Na)	1s² 2s² 2p⁶ 3s¹. The last electron: n = 3 ℓ = 0 mℓ = 0 ms = +½	lastelectron

Chapter 2 — Atomic Structure · 18 / 20

Lesson 5: Distribution of electrons in the atom

Three principles govern how electrons fill the sublevels and orbitals: the Aufbau principle, Pauli’s exclusion principle, and Hund’s rule.

5.1 Pauli’s exclusion principle

No two electrons in the same atom can have the same set of four quantum numbers.
Consequence an orbital (same n, ℓ, mℓ) can hold only two electrons, and these must differ in ms (+½ and −½).
Example, the two electrons of the 3s sublevel:

Electron	n	ℓ	mℓ	m_s
first	3	0	0	+½
second	3	0	0	−½

5.2 Aufbau (building-up) principle

Each main level contains sublevels differing slightly in energy, so the real energy order follows the order of sublevels, not of main levels.
The Aufbau principle states electrons fill the lower-energy sublevels first, then the higher.
The order of increasing energy:
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
A useful rule compare (n + ℓ); the lower value has lower energy.
If two sublevels have the same (n + ℓ), the one with the lower n has lower energy.
s 4 + 0 = 4; 3d: 3 + 2 = 5 → 4s is filled before 3d.
p 3 + 1 = 4 and 4s: 4 + 0 = 4 are equal; the lower n (3p) is first, so 3p < 4s.
p (5) and 5s (5) equal; lower n is 4p, so 4p < 5s.
(In the diagonal arrows method, write the sublevels in rows 1s; 2s 2p; 3s 3p 3d; 4s 4p 4d 4f; … and follow diagonal arrows from the upper right to the lower left.)
Maximum occupancy s² , p⁶ , d¹⁰ , f¹⁴.

✍️ Worked examples

Comparison aspect	Main idea	Keywords
•	Worked example 5.1 (book examples)	Workedexamplebookexamples
₁₁Na	1s² 2s² 2p⁶ 3s¹
₂₀Ca	1s² 2s² 2p⁶ 3s² 3p⁶ 4s²
₃₀Zn	1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰	3d¹⁰
•	Worked example 5.2 — new atoms	Workedexampleatoms
₁₇Cl	1s² 2s² 2p⁶ 3s² 3p⁵ (2 + 2 + 6 + 2 + 5 = 17)
₂₆Fe	1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶ (total 26)	total
₃₅Br	1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁵ (total 35)	3d¹⁰total
₁₀Ne	1s² 2s² 2p⁶ ₁₈Ar: …3s² 3p⁶	₁₈Ar
Worked example 5.3 — number of occupied levels and valence electrons.	Phosphorus ₁₅P: 1s² 2s² 2p⁶ 3s² 3p³. Three main levels are occupied valence shell (n = 3) has 5 electrons	PhosphorusThreemainshell
Notes on special cases.	The stability of half-filled (d⁵) and completely filled (d¹⁰) sublevels makes chromium ₂₄Cr: [Ar] 4s¹ 3d⁵ and copper ₂₉Cu: [Ar] 4s¹ 3d¹⁰ as shown in the periodic table data of the next chapter (the 4s sublevel gives one electron to complete 3d⁵ or 3d¹⁰)	stabilityhalffilledcompletely

5.3 Hund’s rule

Electrons do not pair in a sublevel until each orbital of the sublevel contains one electron (orbitals are filled singly first, with parallel spins).
Why?
The orbitals of one sublevel have the same energy.
Two electrons placed in the same orbital repel each other, increasing the energy of the atom; electrons spread over separate orbitals with parallel spins give the atom the maximum stability.
Nitrogen ₇N 1s² 2s² 2p³. The three 2p electrons occupy three different orbitals: 2px¹ 2py¹ 2pz¹ (all spins parallel).
Oxygen ₈O 2p⁴. Three electrons go singly into px, py, pz; the fourth could either pair in a p orbital or go to the higher sublevel 3s. Pairing in the 2p orbital costs less energy than moving to the higher 3s, so it pairs: 2px² 2py¹ 2pz¹.
Orbital box diagrams for the last level:

N: 2s [↑↓]; 2p [↑][↑][↑]

O: 2s [↑↓]; 2p [↑↓][↑][↑]

F: 2s [↑↓]; 2p [↑↓][↑↓][↑]

Ne→2s [↑↓]; 2p [↑↓][↑↓][↑↓]

✍️ Worked examples

Comparison aspect	Main idea	Keywords
•	Worked example 5.4 — unpaired electrons	Workedexampleunpairedelectrons
Carbon ₆C	1s² 2s² 2px¹ 2py¹: two unpaired electrons	2px¹2py¹unpairedelectrons
Oxygen	two unpaired (2py 2pz)	unpaired
Fluorine	one unpaired	unpaired
Neon	none	none
Phosphorus ₁₅P (3p³)	three unpaired electrons	threeunpairedelectrons
•	Worked example 5.5 — diagram is correct for oxygen?	Workedexamplediagramcorrect
•	2p must show one pair and two single electrons with parallel spins: [↑↓][↑][↑]	mustshowpairsingle
•	A diagram showing [↑↓][↑↓][ ] (two pairs one empty orbital) violates Hund’s rule	diagramshowingpairsempty
Worked example 5.6 — find the atomic number.	An atom has the configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹. The sum of the electrons is 19, so Z = 19 (potassium). In its ground state, its last electron has n = 4, ℓ = 0, mℓ = 0, m_s = +½.	configurationelectronspotassiumground

Fill sublevels in the order of the Aufbau principle.
Never put more than two electrons in an orbital, and then with opposite spins (Pauli).
In a sublevel with several orbitals, fill each orbital singly first (Hund).
Check that the sum of electrons equals the atomic number.

Chapter 2 — Atomic Structure · 19 / 20

Part 20

📖

📌 Comprehensive Scientific Terminology & Definitions:

Comparison aspect	Main idea	Keywords
Rutherford Alpha-Scattering Experiment	Landmark experiment proving that the atom is mostly empty space with a dense positively charged central core called the nucleus disproving the Thomson plum-pudding model	Landmarkprovingatommostly
Bohr Model of the Atom	Postulated that electrons orbit the nucleus in fixed quantized circular energy levels without radiating energy and only emit or absorb discrete packets of energy (photons) when jumping between levels	Postulatedelectronsorbitnucleus
The Four Quantum Numbers	(1) Principal (n): energy level and size. (2) Azimuthal/Angular Momentum (l): orbital shape (s, p, d, f). (3) Magnetic (m_l): orbital spatial orientation. (4) Spin (m_s): electron intrinsic spin (+1/2 or -1/2).	Principalenergylevelsize
Aufbau Principle	Electrons occupy atomic orbitals of the lowest available energy levels before occupying higher levels: 1s < 2s < 2p < 3s < 3p < 4s < 3d	Electronsoccupyatomicorbitals
Hund’s Rule	Every orbital in a subshell is singly occupied with electrons having parallel spins before any orbital is doubly occupied minimizing electrostatic repulsion	Everyorbitalsubshellsingly

⚖️ Governing Scientific Laws, Principles & Mathematical Formulas:

Comparison aspect	Main idea	Keywords
Photon Energy	ΔE = E₂ – E₁ = h ν = h c / λ
•	Maximum electron capacity of principal energy level = 2n² (valid only for n ≤ 4)	Maximumelectroncapacityprincipal
Subshell values	l = 0 (s) 1 (p) 2 (d) 3 (f) with range 0 to (n – 1)	range
•	Number of orbitals per subshell = 2l + 1 \| Magnetic quantum number m_l from -l to +l	NumberorbitalssubshellMagnetic
De Broglie Wavelength	λ = h / (m v) (wave-particle duality)	particleduality

🔍 In-Depth Explanations, Deductions & Core Concepts:

Comparison aspect	Main idea	Keywords
Why does the 4s subshell fill before the 3d subshell?	Because its energy is lower according to the (n + l) rule: for 4s (4 + 0 = 4) for 3d (3 + 2 = 5). Lower (n + l) fills first	Becauseenergyloweraccording
Why is the 2n² rule restricted to n ≤ 4?	Beyond the 4th level (n > 4) atoms become unstable if a principal energy level holds more than 32 electrons due to excessive electrostatic repulsion	Beyondlevelatomsbecome

Subshell	Azimuthal (l)	Number of Orbitals	Max Electrons	Spatial Geometry
s subshell	0	1 orbital	2 electrons	Spherical symmetrical
p subshell	1	3 orbitals (px, py, pz)	6 electrons	Dumbbell-shaped along x, y, z
d subshell	2	5 orbitals	10 electrons	Complex cloverleaf patterns
f subshell	3	7 orbitals	14 electrons	Multi-lobed complex geometry

💡 High-Yield Exam Traps & Key Takeaways:

Pauli Exclusion Principle No two electrons in the same atom can have the exact same four quantum numbers.
Subshells are most stable when empty, half-filled (d⁵, p³), or completely filled (d¹⁰, p⁶).

Chapter 2 — Atomic Structure · 20 / 20

Curriculum

Chemistry — Second Secondary

Chemistry — Second Secondary

Chemistry — Second Secondary 2026/2027

Chapter 2 — Atomic Structure

🌳 Concept tree

Chapter 2 — Atomic Structure

│ ├── 1.1 Thomson's atomic model (1897)

│ ├── 1.2 Rutherford's gold foil experiment

│ ├── 2.1 Atomic emission (line) spectrum

│ ├── 2.2 Bohr's atomic model (1913)

│ ├── 2.3 The hydrogen spectrum explained

│ ├── 2.4 Success of Bohr's model

│ └── 2.5 Shortcomings of Bohr's model

│ ├── 3.1 Dual nature of the electron

│ ├── 3.2 Heisenberg's uncertainty principle

│ └── 3.4 Electron cloud and orbital

│ ├── 4.1 Principal quantum number (n)

│ ├── 4.2 Subsidiary quantum number (ℓ)

│ ├── 4.3 Magnetic quantum number (mℓ)

│ ├── 4.4 Spin quantum number (ms)

│ └── 4.6 Worked examples

├── 5.1 Pauli's exclusion principle

├── 5.2 Aufbau (building-up) principle

├── 5.3 Hund's rule

└── 5.4 Putting it together — rules checklist

a very thin gold foil in the path of the beam;

✍️ Worked examples

✍️ Worked examples

✍️ Worked examples

5.1 Pauli’s exclusion principle

5.2 Aufbau (building-up) principle

✍️ Worked examples

5.3 Hund’s rule

N: 2s [↑↓]; 2p [↑][↑][↑]

O: 2s [↑↓]; 2p [↑↓][↑][↑]

F: 2s [↑↓]; 2p [↑↓][↑↓][↑]

✍️ Worked examples

📌 Comprehensive Scientific Terminology & Definitions:

⚖️ Governing Scientific Laws, Principles & Mathematical Formulas:

🔍 In-Depth Explanations, Deductions & Core Concepts:

💡 High-Yield Exam Traps & Key Takeaways:

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