Difficulty: Intermediate to Advanced
Prerequisites: Part 1 of these notes (Electromagnetic Radiation and Energy Quantisation). You should be comfortable with E = hν = hc/λ and the concept of quantised energy before proceeding.
This section takes the quantisation ideas from Part 1 and applies them to the structure of the atom itself. The Bohr model explains hydrogen's line spectrum using quantised energy levels, but it has serious limitations. The quantum mechanical model (Schrodinger) replaces fixed orbits with probability distributions described by wave functions. The four quantum numbers that come out of this model are the "address system" for every electron in every atom. This is the material your instructor considers the most important in the chapter.
Bohr showed that hydrogen's electron can only occupy specific energy levels, producing discrete line spectra. The quantum mechanical model replaced Bohr's fixed orbits with orbitals (probability distributions described by wave functions). Four quantum numbers (n, ℓ, mℓ, ms) specify the energy, shape, orientation, and spin of every electron.
Emission spectrum
The spectrum of bright lines or bands produced when a substance loses energy and its electrons return to lower energy states. Each element has a unique emission spectrum.
In simple terms, it is the specific colours of light an element gives off when heated or excited.
Absorption spectrum
A spectrum showing which wavelengths of light a substance absorbs. Appears as dark lines against a continuous background, at the same wavelengths as the emission lines.
Line spectrum
A spectrum consisting of discrete lines at specific frequencies, produced by passing light through a slit and then a prism or diffraction grating. Hydrogen's visible line spectrum has lines at 410, 434, 486, and 656 nm.
Ground state
The lowest energy state of an electron in an atom. For hydrogen, this is n = 1. An electron remains in the ground state unless energy is added.
Bohr model
A model of the hydrogen atom in which the electron occupies fixed circular orbits at specific energy levels. It correctly predicts hydrogen's line spectrum but fails for multi-electron atoms and incorrectly assumes fixed orbital paths.
Think of it as a useful first approximation that gets the energy levels right but the geometry wrong.
Wave function (ψ, psi)
A mathematical solution to the Schrodinger equation that describes the behaviour of an electron as a matter-wave. Each allowed wave function corresponds to a specific energy and orbital shape.
Orbital
A region of space around the nucleus where there is a high probability (typically 90%) of finding an electron. Orbitals are the allowed energy states described by wave functions. They are not fixed paths.
In simple terms, an orbital is a 3D probability map showing where an electron is likely to be.
Electron density / electron probability
The probability of finding an electron at a given point in space. The square of the wave function (ψ²) gives this probability. Electron density maps, electron clouds, and orbitals all refer to the same concept.
Heisenberg uncertainty principle
You cannot simultaneously know both the exact position and exact momentum of an electron. If you know the energy precisely, the position becomes uncertain, and vice versa.
Principal quantum number (n)
An integer (1, 2, 3, ...) that determines the energy level and general size of an orbital. Larger n means higher energy and greater average distance from the nucleus. A shell with principal quantum number n can hold 2n² electrons.
Angular momentum quantum number (ℓ)
An integer from 0 to n−1 that determines the shape (subshell type) of an orbital. ℓ = 0 is s, ℓ = 1 is p, ℓ = 2 is d, ℓ = 3 is f.
In simple terms, ℓ tells you whether you are looking at a sphere (s), a dumbbell (p), a cloverleaf (d), or a more complex shape (f).
Magnetic quantum number (mℓ)
An integer from −ℓ to +ℓ that specifies the orientation of an orbital within its subshell. For a p subshell (ℓ = 1), mℓ can be −1, 0, or +1, giving three p orbitals.
Spin quantum number (ms)
Either +1/2 or −1/2. Describes the intrinsic spin of an electron. The first electron placed in an orbital is assigned +1/2, the second is −1/2.
Nodal plane
A plane through the nucleus where the probability of finding the electron is exactly zero. p orbitals have one nodal plane, d orbitals have two, f orbitals have three.
Each element produces a unique set of spectral lines (its "fingerprint").
Johann Balmer worked out a mathematical relationship for the three longest-wavelength visible lines of hydrogen (red at 656 nm, blue-green at 486 nm, violet at 434 nm).
Niels Bohr connected spectra to quantised energy: the single electron of hydrogen can only occupy certain energy states (stationary states).
"The Mother of all Assumptions": an electron remains in its lowest energy state (ground state) unless energy is added.
n = 1 is the ground state for hydrogen.
An electron with n = 1 has the most negative energy and is most strongly attracted to the nucleus.
Higher values of n mean less negative energy and weaker attraction to the nucleus.
Transitions between energy levels:
Higher to lower n: a photon is emitted (emission line observed).
Lower to higher n: a photon is absorbed (absorption line observed).
The energy of the photon exactly equals the energy difference between the two levels: E = hc/λ.
Two major limitations of Bohr's model:
It only works for atoms or ions with one electron (H, He⁺, Li²⁺).
The electron does not orbit the nucleus in a fixed circular path.
After Bohr, physicists pursued two approaches: Bohr's particle-based model and Schrodinger's wave-based model.
Erwin Schrodinger built on de Broglie's equation, treating the electron as a standing wave. His approach was more successful than Bohr's.
Solutions to the Schrodinger equation are wave functions (ψ). Each ψ corresponds to an allowed energy.
ψ² gives the probability of finding the electron at a given point in space. This is what we mean by "electron density" or "electron cloud."
The Heisenberg uncertainty principle: you cannot simultaneously know both the position and momentum of an electron with perfect precision. We can only calculate the probability of finding an electron in a given region.
Orbitals are the allowed energy states (the matter-waves described by ψ). An orbital is a probability distribution, not a fixed path.
The radial probability graph shows the total probability of finding the electron at a given distance from the nucleus. The peak of this curve represents the most probable distance.
Every electron in an atom is described by a unique set of four quantum numbers.
n (principal quantum number)
Values: 1, 2, 3, ... to infinity.
Determines the energy level and size of the orbital.
The number of sublevels in a shell equals n (e.g. n = 3 has 3 sublevels: s, p, d).
Maximum electrons per shell: 2n².
ℓ (angular momentum quantum number)
Values: 0, 1, 2, ... (n−1).
Determines the shape of the orbital (subshell type).
ℓ = 0 → s (1 orbital), ℓ = 1 → p (3 orbitals), ℓ = 2 → d (5 orbitals), ℓ = 3 → f (7 orbitals).
The letters stand for: sharp, principal, diffuse, fundamental (from early spectroscopy).
mℓ (magnetic quantum number)
Values: integers from −ℓ to +ℓ, including zero.
Determines the orientation of the orbital in space.
Number of orbitals in a subshell = 2ℓ + 1.
Example: for ℓ = 2 (d subshell), mℓ = −2, −1, 0, +1, +2, giving 5 d orbitals.
ms (spin quantum number)
Values: +1/2 or −1/2.
The first electron placed in an orbital gets +1/2, the second gets −1/2.
Accounts for the behaviour of electrons in magnetic fields.
s orbitals: Spherical. Size increases with n. The number of radial nodes equals n−1. There is no sharp boundary; orbitals represent regions of high (90%) probability.
p orbitals: Dumbbell-shaped (two lobes). One nodal plane passes through the nucleus. Three orientations: pₓ, pᵧ, pᵨ.
d orbitals: Cloverleaf shapes (four lobes) with two nodal planes, except dz² which has a unique shape (a dumbbell with a torus). Five orientations.
f orbitals: Complex shapes with three nodal planes. Seven orientations. Eight lobed regions.
Bohr energy equation (hydrogen atom)
E = −2.178 × 10⁻¹⁸ J × (Z²/n²)
where Z = nuclear charge (Z = 1 for hydrogen), n = principal quantum number (integer). The negative sign means the electron bound to the nucleus has lower energy than a free electron (at n = ∞, E = 0).
Energy change between levels
ΔE = E(final) − E(initial)
If ΔE is negative, energy was released (photon emitted). If positive, energy was absorbed.
Photon energy from a transition
E(photon) = hν = hc/λ = |ΔE|
Exercise 4: Energy to excite hydrogen from n = 1 to n = 2
E(1) = −2.178 × 10⁻¹⁸ × (1/1²) = −2.178 × 10⁻¹⁸ J.
E(2) = −2.178 × 10⁻¹⁸ × (1/2²) = −5.445 × 10⁻¹⁹ J.
ΔE = E(2) − E(1) = (−5.445 × 10⁻¹⁹) − (−2.178 × 10⁻¹⁸) = +1.633 × 10⁻¹⁸ J.
Positive ΔE confirms energy is absorbed.
Wavelength of light absorbed: λ = hc/ΔE = (6.626 × 10⁻³⁴)(3.00 × 10⁸) / (1.633 × 10⁻¹⁸) = 1.216 × 10⁻⁷ m (UV region).
Exercise 5: Ionisation energy of hydrogen (removing the electron from n = 1)
E(∞) = 0 (electron completely removed).
E(1) = −2.178 × 10⁻¹⁸ J.
ΔE = 0 − (−2.178 × 10⁻¹⁸) = +2.178 × 10⁻¹⁸ J.
This is the ionisation energy for one hydrogen atom.
Exercise 6: Subshells for n = 5
ℓ ranges from 0 to n−1, so ℓ = 0, 1, 2, 3, 4.
Designations: 5s, 5p, 5d, 5f, 5g.
Number of orbitals: 1 + 3 + 5 + 7 + 9 = 25 orbitals total.
Line spectra are the basis of flame tests in chemistry labs: dip a metal salt in a flame, observe the colour, identify the element. Astronomers use the same principle to determine the composition of distant stars by analysing their emission and absorption spectra.
The quantum mechanical model underpins all of modern chemistry, from explaining why the periodic table is shaped the way it is, to predicting how atoms bond and what shapes molecules take.
Students often think electrons orbit the nucleus in fixed circular paths like planets around the Sun. They do not. Orbitals are probability distributions; we can only say where an electron is likely to be, not where it is at any given moment.
Students confuse "orbital" with "orbit." An orbit is a fixed path (Bohr model). An orbital is a three-dimensional probability cloud (quantum mechanical model). These are fundamentally different concepts.
Students sometimes think the Bohr model works for all atoms. It only works for one-electron species (H, He⁺, Li²⁺). For everything else, you need the quantum mechanical model.
Students often forget that n limits ℓ, and ℓ limits mℓ. A 2d orbital does not exist (n = 2 only allows ℓ = 0 or 1, so only 2s and 2p).
⚠️ The four quantum numbers and their allowed values are heavily tested. Know the rules: n ≥ 1, ℓ from 0 to n−1, mℓ from −ℓ to +ℓ, ms = ±1/2.
⚠️ Be able to identify invalid sets of quantum numbers (e.g. n = 2, ℓ = 2 is not allowed).
⚠️ Know the shapes: s = sphere, p = dumbbell (3 orientations), d = cloverleaf (5 orientations).
⚠️ Free-response questions often ask you to calculate ΔE for a hydrogen transition and then find the wavelength of the photon.
⚠️ Understand that the negative sign in Bohr's equation means the bound electron is lower in energy than a free electron. Negative ΔE = energy emitted; positive ΔE = energy absorbed.
⚠️ Radial probability diagrams appear on the AP exam. Know that the peak represents the most probable distance from the nucleus, and that s orbitals "penetrate" closer to the nucleus than p orbitals of the same n.
True or false: The Bohr model correctly predicts the line spectrum of helium. (False. It only works for one-electron species.)
Fill in the blank: For n = 4, the allowed values of ℓ are ________. (0, 1, 2, 3)
True or false: An orbital is a fixed path that an electron follows around the nucleus. (False. An orbital is a probability distribution.)
How many orbitals are in a d subshell? (5)
Fill in the blank: When an electron transitions from n = 3 to n = 1, a photon is ________. (emitted)
Q: What set of quantum numbers describes the outermost electron in sulfur (Z = 16)?
A: Sulfur's electron configuration is 1s² 2s² 2p⁶ 3s² 3p⁴. The 16th electron is the 4th electron in the 3p subshell, placed in the mℓ = −1 orbital as the second electron (down arrow). So: n = 3, ℓ = 1, mℓ = −1, ms = −1/2.
Q: Why does the energy of an electron in a hydrogen atom depend only on n, while in multi-electron atoms it depends on both n and ℓ?
A: In hydrogen, there is only one electron, so there are no electron-electron repulsions. All subshells within the same n have identical energy. In multi-electron atoms, inner (core) electrons shield outer electrons from the full nuclear charge, and different subshells penetrate to the nucleus to different extents (s > p > d > f), raising the energy of subshells with higher ℓ.
Q: Is the following set of quantum numbers valid: n = 3, ℓ = 3, mℓ = 0, ms = +1/2? Explain.
A: No. For n = 3, the maximum value of ℓ is n−1 = 2. ℓ = 3 is not allowed at the third energy level.
Q: Calculate the energy of the photon emitted when a hydrogen electron falls from n = 4 to n = 2.
A: E(4) = −2.178 × 10⁻¹⁸ / 16 = −1.361 × 10⁻¹⁹ J. E(2) = −2.178 × 10⁻¹⁸ / 4 = −5.445 × 10⁻¹⁹ J. ΔE = E(2) − E(4) = −4.084 × 10⁻¹⁹ J. The photon energy is 4.084 × 10⁻¹⁹ J (the negative sign indicates emission).
Q: How many orbitals exist in the n = 3 shell, and what are their designations?
A: n = 3 allows ℓ = 0, 1, 2, giving 3s (1 orbital), 3p (3 orbitals), 3d (5 orbitals). Total: 9 orbitals.
The quantum numbers and orbital shapes from this section are the foundation for electron configurations and the Aufbau principle, covered in Part 3 of these notes.
The concepts of shielding and penetration (s penetrates more than p, which penetrates more than d) directly explain periodic trends in ionisation energy, atomic radius, and electronegativity.
The Bohr model's energy equation for hydrogen reappears in discussions of ionisation energy and photoelectron spectroscopy (PES).
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