Atomic Structure & Quantized Energy Levels – Comprehensive Study Notes

Scope of Study – Part (i)

• 5 sub-topics students must master:
– Thomson’s Model of the atom
– Rutherford’s Model
– Bohr Model of the Hydrogen Atom
– Bohr’s Postulates
– Emission & Absorption Line Spectra of Hydrogen gas

Thomson’s “Plum-Pudding” Model (1898)

• Proposed by Joseph John Thomson (discoverer of the electron, 1856-1940).
• Atom imagined as a uniformly distributed positive “pudding” in which tiny negative electrons (the “plums”) are embedded.
• Total positive charge balances the total negative charge ⇒ electrically neutral atom.
• No distinct nucleus; mass and charge spread more or less homogeneously.

Rutherford’s Model (1911)

• Devised by Ernest Rutherford after the gold-foil α-scattering experiment.
• Experimental layout:
– Source containing radon emits $\alpha$ particles.
– Narrow beam directed at ultra-thin metal (gold) foil.
– Fluorescent ZnS viewing screen detects scattered $\alpha$’s.
• Key observations & inferences:
– Most $\alpha$ particles pass straight through ⇒ atom is mostly empty space.
– A few scatter through large angles; some rebound ⇒ existence of a small, dense, positively-charged nucleus.
• Hypotheses quantified:

1. >99.9 % of atomic mass concentrated in the nucleus.

2. Electrons orbit the nucleus (planetary picture).

3. If electrons were stationary they would spiral into the nucleus by electrostatic attraction; hence motion is essential.

4. Nuclear radius order $10^{-15}\text{ m}$ to $10^{-14}\text{ m}$.
   • Model limitations: could not explain atomic stability or discrete spectra.

Bohr Model of the Hydrogen Atom (1913–1915)

• Formulated by Niels Bohr (Nobel, 1922); often termed the Rutherford–Bohr model.
• Incorporated emerging quantum ideas to correct Rutherford’s classical instability.

Core Quantum Ideas

• Electron energies are quantized ⇒ only certain stationary orbits with specific radii/energies are allowed.
• Photon emission/absorption occurs when an electron makes a transition between two allowed levels:
$hf = E<em>U - E</em>L$
where h = 6.626\times10^{-34}\,\text{J·s}.
• Angular momentum quantisation postulate:
$L = mvr<em>n = n\hbar = \frac{nh}{2\pi},\qquad n = 1,2,3,\ldots$ • Centripetal balance with Coulomb attraction: $F = \frac{1}{4\pi\varepsilon</em>0}\frac{Ze^2}{r^2}$
(for hydrogen, $Z = 1$).

Derived Orbit Radii

• Smallest (ground) orbit radius (Bohr radius):
$a<em>0 \equiv r</em>1 = 0.529\times10^{-10}\,\text{m}$.
• General orbit:
$r<em>n = a</em>0\,\frac{n^2}{Z}$.

Allowed Energies (Hydrogen, $Z=1$)

• General formula:
$E<em>n = -\frac{13.6\,\text{eV}}{n^2}$. • Sample values: – $n=1$ (ground): $E</em>1 = -13.6\,\text{eV}$.
– $n=2$ (first excited): $E<em>2 = -3.40\,\text{eV}$. – $n=3$: $E</em>3 = -1.51\,\text{eV}$.
• Negative sign: energy referenced so that $E=0$ when electron is free at $r\to\infty$.
• Binding (ionization) energy from ground state: $13.6\,\text{eV}$.

Bohr’s Four Postulates (explicit)

1. Electrons move in circular orbits about the nucleus, but only certain discrete orbits are permitted.

2. While in a particular orbit, an electron does not radiate energy.

3. Radiation (photon) is emitted or absorbed only when the electron jumps between stationary states; energy is conserved.

4. Orbital angular momentum is quantized: $L = n h / 2\pi$.

Emission & Absorption Line Spectra

• Emission spectrum: an excited, low-pressure gas emits light at specific wavelengths; appears as bright lines through a spectrometer slit.
– Requires high T, low P, low density.
• Absorption spectrum: when continuous light traverses rarefied gas, dark lines (missing wavelengths) appear; gas absorbs same frequencies it would emit.
– Observed in sunlight (Fraunhofer lines), heated solids behind cooler gases, etc.
• Hydrogen series labels (by $n<em>L$ → $n</em>U$):
– Lyman ($n<em>L = 1$, UV), Balmer ($n</em>L = 2$, visible), Paschen ($n_L = 3$, IR), etc.
• Spectral “fingerprints” corroborate quantized energy levels.

Scope of Study – Part (ii) (Energy Levels)

• 7 additional learning objectives:

1. Evidence of Quantized Energy Levels.

2. Radius of the Bohr Orbit.

3. Energy of quantum state $n$ in hydrogen.

4. Energy of quantum state $n$ in a general atom.

5. Construction & interpretation of Energy Level Diagrams.

6. Qualitative grasp of the Franck–Hertz experiment.

7. Concepts of Excitation & Ionization.

Evidence for Quantized Energy

• Planck (1900): electromagnetic energy emitted/absorbed in discrete quanta: $E = h\nu$.
• Heating atoms: electrons absorb specific quanta, jump to higher levels, then fall back emitting identical quanta ⇒ line spectra.
• Experiments measure discrete absorption/emission wavelengths → match energy gaps $\Delta E = h\nu$.
• Energy may be specified by frequency $\nu$ (Hz) or wavelength $\lambda$ (m): $E = h\nu = \frac{hc}{\lambda}$.

General Atomic Energies (Beyond Hydrogen)

• For an electron in the $n^{\text{th}}$ orbit (nucleus charge $+Ze$):
– Kinetic: $K = \frac{1}{2} m v^2 = \frac{m Z^2 e^4}{8\varepsilon<em>0^2 h^2 n^2}$. – Potential: $U = -\frac{Ze^2}{4\pi\varepsilon</em>0 r} = -\frac{m Z^2 e^4}{4\varepsilon<em>0^2 h^2 n^2}$. – Total: $E = K + U = -\frac{m Z^2 e^4}{8\varepsilon</em>0^2 h^2 n^2}$ (reduces to Bohr formula for $Z=1$ when converted to eV).

Energy-Level Diagrams

• Levels converge toward $E=0$ (ionization limit) as $n \to \infty$.
• Series produced by downward transitions terminate at specific lower levels.
• Diagrammatic conventions:
– Horizontal lines labelled 1s, 2s/2p, 3s/3p/3d, etc.
– Vertical arrows show allowed transitions; line length ∝ photon energy.
• The Balmer (visible), Lyman (UV), and Paschen (IR) series illustrated on typical hydrogen diagram; numerical spacings: $-13.6$, $-3.40$, $-1.51$, $-0.85$ eV, …

Franck–Hertz Experiment (1914)

• Performed by James Franck & Gustav Hertz (Nobel 1925) using mercury vapour.
• Electrons accelerated through Hg; collected current versus accelerating voltage showed periodic drops every $\approx 4.9\,\text{V}$.
• Interpretation: electrons lose discrete energy (4.9 eV) exciting Hg atoms; strong direct evidence for quantized excited states, corroborating Bohr/quantum theory.

Concept of Excitation

• Any atomic state with energy higher than the ground state.
• Achieved via photon absorption or inelastic collisions.
• Photon condition: $E = h\nu = \frac{hc}{\lambda}$.
• De-excitation emits photons of identical energy; underlies fluorescence, lasers, nebular emission lines, etc.

Concept of Ionization

• Removal (or addition) of one/more electrons → atom acquires net charge.
• Ionization energy (first IE) = minimum energy to strip one electron completely:
$E<em>{ion} = E</em>{\infty} - E<em>{\text{initial}} = -E</em>{\text{initial}}$ (since $E_{\infty}=0$ by convention).
• Notation for degrees of ionization (astronomy/chemistry):
– I = neutral (H I), II = singly-ionized (H II), III = doubly-ionized, etc.
• Periodic trends (first 20 elements): IE generally increases across a period, decreases down a group; modulated by nuclear charge, electron distance, and shielding.
• Factors affecting IE magnitude:

1. Nuclear charge (protons) ↑ ⇒ stronger attraction ⇒ IE ↑.

2. Electron–nucleus distance ↑ (higher $n$) ⇒ attraction ↓ ⇒ IE ↓.

3. Inner-shell electron shielding ↑ ⇒ effective nuclear charge ↓ ⇒ IE ↓.

Practical / Philosophical Notes

• Discrete spectra underpin technologies: fluorescent lamps, lasers, sodium streetlights, spectroscopy for chemical analysis, astrophysical diagnostics.
• Understanding nuclear/atomic structure paved the way for quantum mechanics, semiconductor physics, and nuclear energy.
• Quip (Aristotle Onassis): “The secret of success is to know something nobody else knows” – mirrors scientific discovery of hidden atomic structure.