Table of Contents
Overview#
In 1913, Niels Bohr proposed a revolutionary model of the atom that explained the discrete spectral lines of hydrogen. He introduced the concept of quantized electron orbits.
The Problem#
Classical Atomic Model Failure#
Rutherford’s nuclear model:
- Electrons orbit nucleus
- Classical physics: accelerating charges radiate
- Predicted: electron spirals into nucleus
This didn’t match reality—atoms are stable!
Hydrogen Spectrum Mystery#
Hydrogen emits light at specific wavelengths:
$$ \frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) $$Where \(R_H = 1.097 \times 10^7\) m⁻¹ (Rydberg constant).
Why only these wavelengths?
Bohr’s Postulates#
1. Quantized Orbits#
Electrons can only occupy specific orbits where angular momentum is quantized:
$$ L = mvr = n\hbar = n\frac{h}{2\pi} $$Where \(n = 1, 2, 3, …\) (principal quantum number).
2. Stationary States#
In allowed orbits:
- Electrons don’t radiate energy
- Atoms are stable
- Classical electromagnetism doesn’t apply
3. Quantum Jumps#
Energy is emitted/absorbed only during transitions:
$$ \Delta E = E_{n_2} - E_{n_1} = h\nu $$Derivation of Hydrogen Energy Levels#
Force Balance#
Coulomb force = Centripetal force:
$$ \frac{ke^2}{r^2} = \frac{mv^2}{r} $$Quantization Condition#
$$ mvr = n\hbar $$Solving for Radius#
$$ r_n = \frac{n^2\hbar^2}{mke^2} = n^2 a_0 $$Where Bohr radius:
$$ a_0 = \frac{\hbar^2}{mke^2} = 0.529 \text{ Å} $$Energy Levels#
$$ E_n = -\frac{mk^2e^4}{2\hbar^2} \cdot \frac{1}{n^2} = -\frac{13.6 \text{ eV}}{n^2} $$Energy Level Diagram#
n = ∞ ────────────── 0 eV (ionization)
n = 4 ────────────── -0.85 eV
n = 3 ────────────── -1.51 eV
n = 2 ────────────── -3.40 eV
n = 1 ────────────── -13.6 eV (ground state)Spectral Series#
Lyman Series (UV)#
Transitions to \(n = 1\):
$$ \frac{1}{\lambda} = R_H\left(1 - \frac{1}{n^2}\right), \quad n = 2, 3, 4, ... $$Balmer Series (Visible)#
Transitions to \(n = 2\):
$$ \frac{1}{\lambda} = R_H\left(\frac{1}{4} - \frac{1}{n^2}\right), \quad n = 3, 4, 5, ... $$Paschen Series (IR)#
Transitions to \(n = 3\):
$$ \frac{1}{\lambda} = R_H\left(\frac{1}{9} - \frac{1}{n^2}\right), \quad n = 4, 5, 6, ... $$Predictions Confirmed#
| Prediction | Experimental Value | Bohr Value |
|---|---|---|
| Rydberg constant | 1.097 × 10⁷ m⁻¹ | 1.097 × 10⁷ m⁻¹ |
| Bohr radius | 0.529 Å | 0.529 Å |
| H-alpha wavelength | 656.3 nm | 656.3 nm |
Remarkable agreement!
Limitations#
| Limitation | Description |
|---|---|
| Only works for H | Multi-electron atoms fail |
| No fine structure | Misses spectral line splitting |
| Arbitrary quantization | Why is L quantized? |
| No chemical bonding | Can’t explain molecules |
| Incorrect angular momentum | Ground state has L=0, not ℏ |
Correspondence Principle#
At large quantum numbers, quantum results approach classical:
$$ \lim_{n \to \infty} (\text{quantum}) = \text{classical} $$Bohr used this to develop his theory.
Legacy#
What Bohr Got Right#
- Energy quantization
- Discrete spectral lines
- Stability of atoms
- Photon emission/absorption
Foundation for#
- Wave mechanics (Schrödinger)
- Matrix mechanics (Heisenberg)
- Quantum numbers
- Atomic structure
Bohr received the Nobel Prize in Physics in 1922.