Atoms: Bohr Model & Hydrogen Spectrum

Rutherford's Nuclear Model

Alpha particle scattering from gold foil showed: (1) Most $\alpha$ particles pass straight through (atom is mostly empty space). (2) Some are deflected at large angles (dense, positive nucleus). (3) Very few bounce back (nucleus is very small but contains most of the mass).

Impact parameter (b) and scattering angle (θ): b = (Ze²cotθ/2)/(4πε₀ × ½mv²). Larger b → smaller θ.

Distance of closest approach: r₀ = 2kZe²/(½mv²) = kZe²/KE, where KE is the $\alpha$ particle's kinetic energy.

Limitations: Cannot explain atomic stability (accelerating electrons should radiate and spiral in) or discrete emission spectra.

Bohr Model of Hydrogen Atom

Bohr's postulates for hydrogen-like atoms (single electron, nuclear charge Ze):

1. Quantized orbits: Electrons revolve in specific circular orbits without radiating.
2. Angular momentum quantization: L = mvr = nℏ = nh/(2π), where n = 1, 2, 3, ...
3. Energy transitions: Photon emitted/absorbed when electron jumps between orbits: hf = $E_{i}$ - $E_{f}$.

Bohr Model Results (Hydrogen-like Atoms)

Radius of nth orbit: $r_n = \frac{n^2 a_0}{Z}$ where a₀ = 0.529 Å (Bohr radius).

Velocity in nth orbit: $v_n = \frac{Z \cdot v_0}{n}$ where v₀ = 2.18 × 10⁶ m/s = c/137.

Energy of nth orbit: $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$

Time period: $T_{n}$ = 2π$\frac{r_{n}}{v_{n}}$ ∝ n³/Z²

Frequency of revolution: $f_{n}$ ∝ Z²/n³

Current due to orbiting electron: $I_{n}$ = ef_n = ev_n/(2π$r_{n}$) ∝ Z²/n³

Hydrogen Spectrum

When an electron transitions from $n_{i}$ to $n_{f}$ ($n_{i}$ > $n_{f}$), a photon is emitted with:

$\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

where R = 1.097 × 10⁷ m⁻¹ is the Rydberg constant.

Spectral Series:

• Lyman series ($n_{f}$ = 1): UV region. λ_max: 1→2 (121.6 nm), series limit: n→1 (91.2 nm)
• Balmer series ($n_{f}$ = 2): Visible region. λ_max: 3→2 (656.3 nm, Hα), series limit: n→2 (364.6 nm)
• Paschen series ($n_{f}$ = 3): Near IR. λ_max: 4→3 (1875 nm)
• Brackett series ($n_{f}$ = 4): IR
• Pfund series ($n_{f}$ = 5): Far IR

Number of Spectral Lines

From level n, the maximum number of spectral lines = n(n-1)/2.

Excitation and Ionization

Ionization energy of hydrogen: 13.6 eV (ground state). For nth level: 13.6/n² eV.

Excitation energy from ground state to nth level: 13.6(1 - 1/n²) eV.

First excitation energy (n=1 to n=2): 13.6(1 - $\frac{1}{4}$) = 10.2 eV.

When hydrogen atoms are excited to level n by electron bombardment or photon absorption, they can emit photons corresponding to all possible transitions down to the ground state.

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