Section 02
The Idea, Step by Step
Strike a salt grain in a flame and it glows yellow; neon glows red; a sodium street lamp glows orange. Every element gives off its own fixed set of colours — a fingerprint. Why should an atom only ever emit those colours and never the ones in between? That single puzzle is what the atomic models below were built to explain.
Niels Bohr's answer (1913) was that an electron can only sit on certain allowed rungs of an energy ladder — not anywhere it likes, the way a marble can sit on stair steps but never float between them. Each rung $n$ has a fixed energy. For hydrogen, $E_n=-\dfrac{13.6\text{ eV}}{n^2}$: the ground rung ($n=1$) is deepest at $-13.6$ eV, and the rungs crowd together as you climb. When an electron drops from a high rung to a low one, the leftover energy leaves as a single particle of light — a photon — carrying exactly the gap: $\Delta E=E_i-E_f=hf$. Fixed rungs mean fixed gaps mean fixed colours.
Work one out. An electron falling from $n=3$ to $n=2$ in hydrogen releases $E_3-E_2=(-1.51)-(-3.40)=1.89$ eV. Turn that into a wavelength with $\lambda=1240/E_{\rm(eV)}$ nm and you get $\lambda\approx656$ nm — the deep-red H$\alpha$ line you can spot in starlight and in glowing nebulae. That is the second rung of the Balmer series, the only hydrogen series that lands in visible light.
The precise statement quantises angular momentum, $L=n\hbar$, which fixes both the rung energies and the orbit radii $r_n=n^2a_0/Z$ (with $a_0=52.9$ pm), and packages every line into the Rydberg formula. But Bohr's neat circles are only a stepping stone: Schrödinger's wave equation (1926) replaced fixed orbits with orbitals — fuzzy probability clouds $\psi_{nlm}$ where only $|\psi|^2$, the chance of finding the electron, is real. Three more quantum numbers join $n$: the shape $l$ (s, p, d, f), the orientation $m_l$, and the spin $m_s=\pm\tfrac12$. Pauli's rule that no two electrons share all four numbers is what stacks electrons into shells and builds the whole periodic table. In multi-electron atoms each outer electron feels a screened pull $Z_{\rm eff}=Z-\sigma$, not the bare nucleus.
The sliders map straight onto this story: n picks the starting rung, n_f sets which series (1 = Lyman/UV, 2 = Balmer/visible), Z deepens every level as $Z^2$, l switches the orbital shape, and σ sets the shielding.
Try this in the sim above. (1) Keep $n_f=2$ and step $n$ from 3 up to 6 — watch the emitted wavelength march through the Balmer colours toward the series limit. (2) Drag $Z$ up from 1 to 2 (He$^+$) and notice every energy jump fourfold, because energy scales as $Z^2$. (3) Switch to the Quantum Orbitals mode and sweep $l$ from 0 to 3 to morph a round s orbital into the dumbbell p, cloverleaf d, and lobed f shapes.
Section 03
Equations & Derivation
Bohr Model Energy Levels
$$E_n=-\frac{13.6\;\text{eV}}{n^2}Z^2,\quad r_n=n^2a_0/Z,\quad a_0=52.9\;\text{pm}$$
Rydberg Formula
$$\frac{1}{\lambda}=R_\infty Z^2\!\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right),\quad R_\infty=1.097\times10^7\;\text{m}^{-1}$$
Quantum Numbers
$$n=1,2,3\ldots;\quad l=0\ldots n-1;\quad m_l=-l\ldots l;\quad m_s=\pm\tfrac{1}{2}$$
Selection Rules & Effective Nuclear Charge
$$\Delta l=\pm1,\quad\Delta m_l=0,\pm1;\quad Z_{\rm eff}=Z-\sigma\;\text{(Slater\'s rules)}$$
Spectral Series
| Symbol | Quantity | Unit/Value |
|---|
| $\text{Lyman}$ | n_f=1 | UV |
| $\text{Balmer}$ | n_f=2 | Visible (Hα=656 nm, Hβ=486 nm) |
| $\text{Paschen}$ | n_f=3 | IR |
| $\text{Brackett}$ | n_f=4 | Far IR |
| $\text{Pfund}$ | n_f=5 | Far IR |
1
Bohr's key insight (1913). Electrons orbit in quantised circles where angular momentum $L=n\hbar$. This gives discrete energy levels $E_n=-13.6/n^2$ eV and radii $r_n=n^2a_0$. Transitions between levels emit/absorb photons of energy $\Delta E=hf$. Correctly predicted hydrogen spectrum, but fails for multi-electron atoms.
2
From Bohr to quantum mechanics. Schrödinger's wave equation (1926) replaced Bohr orbits with orbitals — probability clouds described by wavefunctions $\psi_{nlm}$. The Bohr energies survive but the "orbit" concept is replaced by $|\psi|^2$ probability density. This explains multi-electron atoms, chemical bonding, and the periodic table.
3
Quantum numbers. $n$ (shell): energy. $l$ (subshell): orbital shape (s/p/d/f for l=0,1,2,3). $m_l$ (orientation): 2l+1 values. $m_s$ (spin): ±½. Pauli exclusion: no two electrons in the same atom share all four quantum numbers → explains electron configuration and chemistry.
4
Effective nuclear charge. In multi-electron atoms, inner electrons shield outer electrons from the full nuclear charge $Z$. $Z_{\rm eff}=Z-\sigma$ (Slater's rules). This explains why $3s$ electrons are more tightly bound than $3p$: $s$ orbitals penetrate closer to the nucleus, experiencing higher $Z_{\rm eff}$.
Ref: Halliday, Resnick & Walker 10th Ed., Chs. 39–40; Griffiths — Introduction to Quantum Mechanics (2nd Ed.), Ch. 4; Atkins — Physical Chemistry (11th Ed.), Ch. 9.
Section 05
Common Misconceptions
❌ The Bohr model describes all atoms accurately.
✅ The Bohr model works exactly only for one-electron species (H, He⁺, Li²⁺). For multi-electron atoms, electron-electron repulsion shifts energy levels; the Bohr model predicts incorrect spectra. The full quantum mechanical treatment with Schrödinger's equation (and configuration interaction for correlation) is needed.
📖 HRW 10th Ed., §40-2; Levine — Physical Chemistry, Ch. 11.
❌ Electrons in atoms follow fixed circular orbits.
✅ Quantum mechanics describes electrons via wavefunctions $\psi_{nlm}$. Only $|\psi|^2$ is observable — the probability of finding the electron at a given position. Orbitals (s, p, d, f) are probability density distributions, not fixed tracks. An s orbital is spherically symmetric — no "orbit" in the classical sense.
📖 HRW 10th Ed., §40-1; Griffiths — Introduction to Quantum Mechanics, Ch. 4.
❌ Atoms with the same number of electrons have the same chemical properties.
✅ Chemical properties are determined by electron configuration, especially valence electrons. Isobars (same A, different Z) have different electron configurations and different chemistry. Isotopes (same Z, different N) have virtually the same chemistry because electrons, not neutrons, determine bonding (exception: isotope effects on reaction rates at low temperature).
📖 Atkins — Physical Chemistry, Ch. 9.
Misconception research: Driver et al. — Making Sense of Secondary Science; Taber (2005) Chem. Ed. Research.