Bohr's Atomic Model and Spectral Lines

The Mystery: Why Do Atoms Emit Specific Colors?

Heat a gas (like hydrogen) and it glows. But unlike a hot iron that emits a continuous spectrum of colors, gases emit only specific wavelengths — discrete colored lines against a dark background.

This was deeply puzzling in the early 1900s. Classical physics predicted that electrons orbiting a nucleus should:

1. Emit radiation continuously (accelerating charges radiate)
2. Spiral into the nucleus in about 10⁻¹¹ seconds
3. Produce a continuous spectrum, not discrete lines

Atoms should not exist, and yet they do. And they emit these mysterious spectral lines.

What Are Spectral Lines?

Emission vs Absorption

When you pass white light through a cool gas, specific wavelengths are absorbed, leaving dark lines (absorption spectrum). When you heat a gas, it emits light at those same wavelengths (emission spectrum).

EMISSION SPECTRUM (hot gas):
────────────────────────────────────────────
     |    |        |   |
     λ₁   λ₂       λ₃  λ₄

     Bright lines on dark background


ABSORPTION SPECTRUM (cool gas + white light):
████████████████████████████████████████████
     ▼    ▼        ▼   ▼
     λ₁   λ₂       λ₃  λ₄

     Dark lines on bright background

The Hydrogen Spectrum

Hydrogen, the simplest atom, shows a remarkably regular pattern. In 1885, Johann Balmer discovered an empirical formula for the visible lines:

\lambda = B \cdot \frac{n^2}{n^2 - 4}, \quad n = 3, 4, 5, \ldots

where B = 364.56 nm.

This gives the Balmer series (visible light):

• n = 3: λ = 656.3 nm (red, “H-alpha”)
• n = 4: λ = 486.1 nm (cyan, “H-beta”)
• n = 5: λ = 434.0 nm (blue, “H-gamma”)
• n = 6: λ = 410.2 nm (violet, “H-delta”)

The Rydberg Formula

In 1888, Johannes Rydberg found a more general formula:

\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

where:

• R_H \approx 1.097 \times 10^7 \text{ m}^{-1} is the Rydberg constant
• n_1, n_2 are positive integers with n_2 > n_1

Different values of n_1 give different series:

But why did these formulas work? What physical mechanism produces exactly these wavelengths? This remained a mystery until Bohr.

Bohr’s Revolutionary Idea (1913)

Niels Bohr made three radical postulates:

Postulate 1: Quantized Orbits

Electrons can only orbit the nucleus at specific radii where the angular momentum is quantized:

L = n\hbar, \quad n = 1, 2, 3, \ldots

where \hbar = h/2\pi is the reduced Planck constant.

This was revolutionary — there was no classical justification for it. Bohr simply postulated it because it worked.

Postulate 2: Stationary States

In these allowed orbits, electrons do not radiate energy, despite being accelerated. They remain in stable “stationary states.”

This directly contradicted classical electrodynamics, which says accelerating charges must radiate.

Postulate 3: Quantum Jumps

Electrons can jump between orbits by absorbing or emitting a photon. The photon energy equals the energy difference:

E_\gamma = h\nu = E_{n_2} - E_{n_1}

where \nu is the photon frequency.

Bohr’s Calculation

Step 1: Force Balance

An electron in a circular orbit experiences:

• Centripetal acceleration: a = v^2/r
• Coulomb attraction: F = \frac{e^2}{4\pi\varepsilon_0 r^2}

Setting F = ma:

\frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{m_e v^2}{r}

This gives:

v^2 = \frac{e^2}{4\pi\varepsilon_0 m_e r}

Step 2: Quantization Condition

Angular momentum is quantized:

L = m_e v r = n\hbar

So:

v = \frac{n\hbar}{m_e r}

Step 3: Solving for the Radius

Substituting the velocity into the force equation:

\left(\frac{n\hbar}{m_e r}\right)^2 = \frac{e^2}{4\pi\varepsilon_0 m_e r}

Solving for r:

r_n = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2} \cdot n^2 = a_0 \cdot n^2

where a_0 is the Bohr radius:

a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2} \approx 0.529 \text{ Å} = 5.29 \times 10^{-11} \text{ m}

This is the size of a hydrogen atom in its ground state!

Step 4: Energy Levels

The total energy is kinetic plus potential:

E = \frac{1}{2}m_e v^2 - \frac{e^2}{4\pi\varepsilon_0 r}

Using the force balance relation m_e v^2 = e^2/(4\pi\varepsilon_0 r):

E = \frac{1}{2} \cdot \frac{e^2}{4\pi\varepsilon_0 r} - \frac{e^2}{4\pi\varepsilon_0 r} = -\frac{e^2}{8\pi\varepsilon_0 r}

Substituting r_n = a_0 n^2:

E_n = -\frac{e^2}{8\pi\varepsilon_0 a_0} \cdot \frac{1}{n^2} = -\frac{13.6 \text{ eV}}{n^2}

The energy levels are:

Step 5: Deriving the Rydberg Formula

When an electron jumps from level n_2 to n_1, it emits a photon:

E_\gamma = E_{n_2} - E_{n_1} = 13.6 \text{ eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

Using E_\gamma = h\nu = hc/\lambda:

\frac{1}{\lambda} = \frac{13.6 \text{ eV}}{hc} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

This is exactly the Rydberg formula! And Bohr could calculate the Rydberg constant from fundamental constants:

R_H = \frac{m_e e^4}{8\varepsilon_0^2 h^3 c} \approx 1.097 \times 10^7 \text{ m}^{-1}

This matched the experimental value to high precision — a triumph!

What Bohr Explained

                    n = ∞  ─────────────────  E = 0 (ionized)

                    n = 5  ─────────────────  E = -0.54 eV
                    n = 4  ─────────────────  E = -0.85 eV
                    n = 3  ─────────────────  E = -1.51 eV
                              │  │  │
                              │  │  └── Paschen series (IR)
                              │  │
                    n = 2  ───┴──┴──────────  E = -3.4 eV
                              │  │  │  │
                              │  │  │  └── Balmer series (visible)
                              │  │  │
                              │  │  │
                    n = 1  ───┴──┴──┴───────  E = -13.6 eV
                              │  │  │  │  │
                              └──┴──┴──┴──┴── Lyman series (UV)

                    GROUND STATE

Each arrow represents a photon emission. The energy (and thus wavelength) is determined by the difference between levels.

The Deeper Meaning

Bohr’s calculation revealed something profound: the Rydberg constant is not arbitrary. It can be derived from:

• m_e — electron mass
• e — electron charge
• \hbar — Planck’s constant
• \varepsilon_0 — permittivity of free space
• c — speed of light

The formula:

R_H = \frac{m_e e^4}{8\varepsilon_0^2 h^3 c}

This showed that atomic spectra are determined by fundamental constants of nature. The discrete spectral lines are a direct window into the quantum structure of matter.

Limitations of Bohr’s Model

Despite its success, the Bohr model has serious limitations:

The Real Issue

Bohr’s quantization rule L = n\hbar was ad hoc — it had no deeper justification. The question remained: why is angular momentum quantized?

The answer came from:

1. de Broglie (1924): Electrons are waves with wavelength \lambda = h/p
2. Schrödinger (1926): The wave equation for electrons
3. Born (1926): The wave function gives probabilities

In the full quantum theory, electrons don’t orbit at all — they exist as standing wave patterns (orbitals) around the nucleus. The quantization emerges naturally from the wave equation.

See also: The Schrödinger Equation

Summary

Bohr’s model was wrong in detail but right in spirit. Electrons don’t orbit like planets, but energy quantization is real. The model bridged classical and quantum physics and pointed the way to the full quantum theory.

References

• N. Bohr, “On the Constitution of Atoms and Molecules,” Philosophical Magazine 26, 1–24 (1913) — Original paper
• D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018), Chapter 4
• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed. (Pearson, 2003)
• R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, 1985), Chapters 4–5
• NIST Atomic Spectra Database — Experimental spectral line data