Bohr’s Model for Hydrogen Atom

Niels Bohr (1913) explained hydrogen’s structure and spectrum using Planck’s energy quantization idea. Though simplified, his model helps us understand atomic structure! 🎯

Key Postulates

  1. 💫 Electrons orbit the nucleus in fixed circular paths called orbits (stationary states). Energy doesn’t change in an orbit.
  2. ⚡ Energy is absorbed when electrons jump to higher orbits (e.g., n=1 → n=2), and emitted when falling to lower orbits (e.g., n=3 → n=1). Changes happen instantly!
  3. 🌈 Frequency of radiation during jumps: \[\nu = \frac{\Delta E}{h} = \frac{E_2 – E_1}{h}\]
  4. 🌀 Angular momentum is quantized (only specific values allowed): \[m_e v r = n \cdot \frac{h}{2\pi} \quad (n=1,2,3,…)\]

Important Formulas

  • Orbit radius (for hydrogen): \[r_n = n^2 a_0 \quad (a_0 = 52.9 \text{pm})\] Example: n=1 orbit = 52.9 pm (smallest orbit).
  • Electron energy (for hydrogen): \[E_n = -R_H \left( \frac{1}{n^2} \right) \quad (R_H = 2.18 \times 10^{-18} \text{J})\] Why negative? ⚠️ Negative energy means the electron is bound to the atom. Ground state (n=1) is most stable: \(E_1 = -2.18 \times 10^{-18} \text{J}\).

Hydrogen-like Ions (He⁺, Li²⁺, etc.)

  • Energy: \[E_n = -2.18 \times 10^{-18} \left( \frac{Z^2}{n^2} \right) \text{J}\]
  • Radius: \[r_n = \frac{52.9(n^2)}{Z} \text{pm}\] Note: Higher Z = smaller radius + more negative energy (tighter binding).

Explaining Hydrogen’s Spectrum

Light is emitted when electrons fall from higher orbits (nᵢ) to lower orbits (nf):

  • Energy change: \[\Delta E = 2.18 \times 10^{-18} \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right) \text{J}\]
  • Photon frequency: \[\nu = 3.29 \times 10^{15} \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right) \text{Hz}\]

Series names:
– nf=1: Lyman (UV) ☀️
– nf=2: Balmer (visible) 🌈
– nf=3: Paschen (IR) 🔥

Example Calculation

Transition from n=5 → n=2 in hydrogen:

\[\Delta E = 2.18 \times 10^{-18} \left( \frac{1}{2^2} – \frac{1}{5^2} \right) = -4.58 \times 10^{-19} \text{J}\] \[\nu = \frac{|\Delta E|}{h} = 6.91 \times 10^{14} \text{Hz}\] \[\lambda = \frac{c}{\nu} = 434 \text{nm}\] Result: Blue light in Balmer series! 💙

Limitations of Bohr’s Model

  • ❌ Can’t explain fine spectral lines (doublets) or atoms beyond hydrogen (e.g., helium).
  • ❌ Fails with magnetic (Zeeman effect) or electric fields (Stark effect).
  • ❌ Doesn’t explain chemical bonding.

NEET Focus Areas

  1. 📌 Energy & radius formulas for hydrogen/hydrogen-like atoms
  2. 📌 Transition calculations (ΔE, frequency, wavelength)
  3. 📌 Spectral series (Lyman, Balmer, Paschen)
  4. 📌 Quantization of angular momentum
  5. 📌 Limitations of Bohr’s model