Atomic Spectra and Bohr Model

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Line Spectra — Fingerprints of Elements:

When an atomic gas is excited (by heat, electric discharge, or radiation), it emits light at specific wavelengths — not a continuous rainbow of colours, but discrete coloured lines. This is the atomic emission spectrum. Each element has a unique emission spectrum, like a fingerprint. Conversely, when white light passes through a cool gas, the gas absorbs at the same specific wavelengths it would emit — this is the absorption spectrum.

This discreteness cannot be explained by classical physics (which predicted continuous spectra). Bohr’s model explained it by proposing quantised energy levels.

Bohr’s Model of the Hydrogen Atom:

Bohr proposed three revolutionary postulates for hydrogen:

1. Stationary States: Electrons revolve in specific circular orbits without emitting radiant energy. These orbits have fixed radii and correspond to discrete energy levels.

2. Quantisation of Angular Momentum: The angular momentum of the electron in orbit is quantised: mvr = nh/(2π), where n = 1, 2, 3, … is the principal quantum number (orbit number). This means only specific orbits are allowed.

3. Energy Emission/Absorption: An electron can jump between stationary states by emitting or absorbing a photon of exactly the right frequency: E_photon = E_final - E_initial = hf = hc/λ.

Key Results for Hydrogen:

Bohr calculated the allowed orbital radii: r_n = n²a₀, where a₀ = 0.529 Å (Bohr radius) is the radius for n = 1.

The energy of the electron in orbit n: E_n = -13.6 eV/n² (in electronvolts, relative to ionisation = 0 eV). For n = 1 (ground state): E₁ = -13.6 eV. For n = 2: E₂ = -3.4 eV. For n = 3: E₃ = -1.51 eV.

The ionisation energy of hydrogen from ground state = 13.6 eV (the energy needed to remove the electron completely).

⚡ ECAT Tip: The Balmer series (visible hydrogen lines) corresponds to transitions where the final orbit is n_f = 2 (e.g., n_i = 3 → 2, 4 → 2, 5 → 2, etc.). The Lyman series (ultraviolet) ends at n_f = 1. The Paschen series (infrared) ends at n_f = 3.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Derivation of Bohr Radius and Energy:

Starting from Coulomb’s law: centripetal force on electron = electrostatic attraction: mv²/r = (1/(4πε₀)) × (e × e)/r² = ke²/r², where k = 1/(4πε₀).

From Bohr quantisation of angular momentum: mvr = nh/(2π) → v = nh/(2πmr).

Substituting v into the force equation: m × (n²h²/(4π²m²r²)) / r = ke²/r² → n²h²/(4π²m r³) = ke²/r² → r_n = (n²h²)/(4π²mke²) = n² × (h²/(4π²mke²)).

Evaluating the constant: h = 6.626 × 10⁻³⁴ J·s, m = 9.11 × 10⁻³¹ kg, k = 8.99 × 10⁹ N·m²/C², e = 1.6 × 10⁻¹⁹ C. This gives r₁ = 0.529 × 10⁻¹⁰ m = 0.529 Å. Good.

Total energy E = KE + PE = ½mv² - ke²/r = ½(ke²/r) - ke²/r = -ke²/(2r). Substituting r_n: E_n = -ke²/(2r_n) = -ke² × (4π²mke²)/(2n²h²) = -(2π²mk²e⁴)/(n²h²). Numerically: E_n = -(13.6 eV)/n².

Hydrogen Spectral Lines — Wavelength Calculation:

The wavelength of emitted/absorbed photon during transition from n_i to n_f: ΔE = hc/λ = E_lower - E_higher = -13.6/n_f² + 13.6/n_i² (taking E_n = -13.6/n² eV).

So 1/λ = R_H × (1/n_f² - 1/n_i²), where R_H = Rydberg constant for hydrogen = 1.097 × 10⁷ m⁻¹ (approximately).

⚡ ECAT Tip: For the Balmer series (visible, n_f = 2), the longest wavelength (lowest energy, smallest n_i difference) is for n_i = 3 → n_f = 2: 1/λ = R_H(1/4 - 1/9) = R_H × (5/36). The shortest wavelength in Balmer is as n_i → ∞: 1/λ = R_H/4, λ = 4/R_H = 364.6 nm (the series limit).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Limitations of Bohr Model:

Bohr’s model successfully explained hydrogen’s spectrum but failed for:

1. Multi-electron atoms: Bohr could not explain the spectra of helium, lithium, or any atom with more than one electron, because electron-electron repulsions make the problem unsolvable with this model.
2. Fine structure: Spectral lines split further under high resolution due to relativistic effects and spin-orbit coupling — Bohr model has no concept of electron spin.
3. Zeeman effect: External magnetic fields split spectral lines (anomalous Zeeman effect) — Bohr model only explains the normal Zeeman effect.
4. Intensity of spectral lines: Bohr model gives no framework for calculating transition probabilities (why some transitions are bright and others dim).
5. Heisenberg uncertainty: Bohr’s precise circular orbits violate the uncertainty principle — we cannot simultaneously know exact r and p.

The Bohr model was superseded by the Schrödinger equation (quantum mechanics), which gives orbital shapes (s, p, d, f orbitals), energy levels that depend on both n and l (azimuthal quantum number), electron spin, and fine structure.

Quantum Numbers in Modern Atomic Theory:

Four quantum numbers describe each electron in an atom:

1. Principal quantum number n: determines the energy level (shell) and approximate distance from nucleus (n = 1, 2, 3, 4… correspond to K, L, M, N shells)
2. Azimuthal quantum number l: determines orbital shape (0 = s, 1 = p, 2 = d, 3 = f). For a given n, l ranges from 0 to n-1.
3. Magnetic quantum number m_l: determines orbital orientation in space. For a given l, m_l ranges from -l to +l (2l+1 values).
4. Spin quantum number m_s: electron spin (intrinsic angular momentum). m_s = +½ or -½ (spin up or spin down).

Pauli exclusion principle: no two electrons in an atom can have the same set of all four quantum numbers. This is why electrons fill shells and subshells in a specific order, leading to the periodic table’s structure.

⚡ ECAT Pattern: ECAT frequently tests: (1) calculating the wavelength of emitted light from hydrogen using the Rydberg formula 1/λ = R_H(1/n_f² - 1/n_i²); (2) identifying which series (Lyman, Balmer, Paschen) a given wavelength belongs to; (3) Bohr radius calculation; (4) energy level diagrams; and (5) ionisation energy calculations. A typical ECAT question: “Calculate the wavelength of the photon emitted when an electron in hydrogen jumps from n = 4 to n = 2.” 1/λ = R_H(1/4 - 1/16) = R_H × 3/16 → λ = 16/(3R_H) = 16/(3 × 1.097 × 10⁷) = 486 nm (this is the blue-green line of the Balmer series, known as H-β).