Bohr Model of the Hydrogen Atom

In 1913, Niels Bohr proposed a model of the hydrogen atom that, while ultimately superseded by quantum mechanics, remains one of the most important conceptual tools in physics and chemistry. Bohr's model explains why atoms emit light at only certain wavelengths (spectral lines) and correctly predicts the hydrogen spectrum to remarkable precision.

The problem Bohr solved

By 1913, physicists knew that hydrogen gas, when excited (by heating or electrical discharge), emits light at only specific wavelengths — discrete spectral lines. The Balmer series (visible lines: red at 656 nm, cyan at 486 nm, blue at 434 nm, violet at 410 nm) had been known since 1885, but nobody could explain why only these wavelengths appeared.

Rutherford had shown that the atom is mostly empty space with a tiny positive nucleus. But a classical electron orbiting a nucleus should continuously radiate energy (accelerating charges radiate), spiral inward, and crash into the nucleus in about $10^{-11}$ seconds. Atoms should be unstable — yet they clearly aren't.

Bohr's postulates

Bohr made three radical assumptions:

1. Quantized orbits: Electrons can only exist in certain discrete orbits at specific radii. These are labeled by a quantum number $n = 1, 2, 3, \ldots$ The radius of the $n$-th orbit is:

$r_{n} = n^{2} \cdot a_{0} = n^{2} \times 0.0529\text{ nm}$

where $a_{0} = 0.0529$ nm is the Bohr radius (the smallest orbit, $n = 1$).

2. No radiation in orbits: An electron in a stable orbit does NOT radiate energy, despite being accelerated. This contradicts classical electromagnetism and is simply postulated.

3. Photon emission on transitions: When an electron jumps from orbit $n_{i}$ to a lower orbit $n_{f}$, it emits a photon whose energy equals the energy difference between the orbits:

$E_{\text{photon}} = E_{n_{i}} - E_{n_{f}}$

The energy levels

The energy of an electron in the $n$-th orbit of hydrogen is:

$E_{n} = -\dfrac{13.6}{n^{2}} \text{ eV}$

The negative sign means the electron is bound — you'd need to add 13.6 eV to free it from $n = 1$ (the ground state). Higher $n$ means less tightly bound (closer to zero energy):

• $n = 1$ (ground state): $E_{1} = -13.6$ eV
• $n = 2$: $E_{2} = -3.4$ eV
• $n = 3$: $E_{3} = -1.51$ eV
• $n = 4$: $E_{4} = -0.85$ eV
• $n = 5$: $E_{5} = -0.544$ eV
• $n = 6$: $E_{6} = -0.378$ eV
• $n = \infty$ (free electron): $E_{\infty} = 0$ eV

The spectral series

Transitions are grouped by the lower level $n_{f}$:

Lyman series ($n_{f} = 1$, ultraviolet):

• $3 \to 1$: $E = 13.6(1 - 1/9) = 12.09$ eV, $\lambda = 102.6$ nm
• $2 \to 1$: $E = 13.6(1 - 1/4) = 10.2$ eV, $\lambda = 121.6$ nm

Balmer series ($n_{f} = 2$, visible):

• $3 \to 2$: $E = 13.6(1/4 - 1/9) = 1.89$ eV, $\lambda = 656$ nm (red, H-alpha)
• $4 \to 2$: $E = 13.6(1/4 - 1/16) = 2.55$ eV, $\lambda = 486$ nm (cyan, H-beta)
• $5 \to 2$: $E = 13.6(1/4 - 1/25) = 2.86$ eV, $\lambda = 434$ nm (blue, H-gamma)
• $6 \to 2$: $E = 13.6(1/4 - 1/36) = 3.02$ eV, $\lambda = 410$ nm (violet, H-delta)

Paschen series ($n_{f} = 3$, infrared):

• $4 \to 3$: $E = 13.6(1/9 - 1/16) = 0.66$ eV, $\lambda = 1875$ nm

Worked example: the H-alpha line

The most famous hydrogen line is H-alpha ($n = 3 \to 2$), the red line at 656 nm:

$E = 13.6\left(\dfrac{1}{2^{2}} - \dfrac{1}{3^{2}}\right) = 13.6\left(\dfrac{1}{4} - \dfrac{1}{9}\right) = 13.6 \times \dfrac{5}{36} = 1.889\text{ eV}$

Converting to wavelength:

$\lambda = \dfrac{hc}{E} = \dfrac{(6.626 \times 10^{-34})(3 \times 10^{8})}{1.889 \times 1.602 \times 10^{-19}} = \dfrac{1.988 \times 10^{-25}}{3.027 \times 10^{-19}} = 656.3\text{ nm}$

This is visible red light. When you see red nebulae in astronomical images (like the Orion Nebula), that red glow is H-alpha emission from hydrogen gas.

Limitations of the Bohr model

Bohr's model works perfectly for hydrogen (one electron) but fails for multi-electron atoms. It can't explain:

• Why some spectral lines are brighter than others (transition probabilities)
• Fine structure (splitting of lines in magnetic fields)
• Chemical bonding
• Atoms with more than one electron

These limitations led to the development of full quantum mechanics (Schrödinger, Heisenberg, Dirac) in the 1920s, which replaces discrete orbits with probability clouds (orbitals).

Real-world applications

• Astronomy: Identifying elements in stars by their spectral lines. The Balmer series is used extensively — H-alpha is the key diagnostic for hydrogen.
• Neon signs and gas discharge tubes: Each gas produces its characteristic spectral lines when excited.
• Laser operation: Lasers work by stimulating specific electron transitions — the wavelength of the laser is determined by the energy levels involved.
• Atomic clocks: The most precise clocks use specific electron transitions in cesium or rubidium atoms.

Common mistakes

• Forgetting the negative signs. Energy levels are negative (bound states). The energy of a photon emitted in a transition is the difference — always positive: $E = |E_{i}| - |E_{f}|$ where $|E_{f}| > |E_{i}|$ since the electron moves to a lower (more negative) energy.
• Confusing $n_{i}$ and $n_{f}$. For emission, the electron goes from high $n$ to low $n$. For absorption, low to high.
• Thinking electrons "orbit" like planets. In the full quantum model, electrons don't have well-defined trajectories. The Bohr model is a useful approximation, not a literal description.

Try it in the visualization

Click on different transitions to see the electron jump and the photon emitted. The photon's color matches its wavelength (UV appears as a dashed line since it's invisible). Switch between Lyman, Balmer, and Paschen series to see which transitions produce visible light. The energy level diagram on the right updates with each transition, showing the energy difference and resulting wavelength.