Bohr Model of the Atom

The Bohr model says an electron in hydrogen can occupy only certain allowed energy levels, not any energy it wants. Because only specific energy gaps exist, hydrogen absorbs and emits only specific wavelengths of light. The model is not the modern picture of the atom, but it makes quantized energy easy to calculate and see.

The Formula And Its Symbols

For hydrogen, the Bohr energy levels are

where $n$ is the principal quantum number (the level, $n = 1, 2, 3, \dots$), $E_n$ is the energy of that level in electronvolts, and the constant $13.6\ \text{eV}$ is fixed for hydrogen. The negative sign means a bound electron has lower energy than a free one. This formula is for hydrogen in the basic Bohr model; do not treat it as a general formula for all atoms.

When an electron jumps between levels, the photon energy matches the energy gap:

If the electron moves to a lower level, the atom emits a photon. If it absorbs exactly the right energy, it moves to a higher level.

Why The Levels Are Quantized

Why only certain wavelengths instead of a continuous rainbow? Hydrogen does not produce every possible wavelength; it produces distinct spectral lines. The Bohr model explains that pattern by allowing the electron to jump only between specific energy levels, so only specific energy changes, and therefore only specific photon energies, are possible. That single idea, allowed levels rather than a continuum, is the heart of the model and is what makes the $1/n^2$ spacing meaningful.

Worked Example: Hydrogen From $n = 3$ To $n = 2$

Compute each level first.

For $n = 3$:

For $n = 2$:

Now the energy change:

The negative sign shows the electron ended in a lower-energy state, so the atom emits a photon with energy $1.89\ \text{eV}$. One allowed jump gives one specific photon energy, not a continuous range.

Your Turn

Try the jump from $n = 2$ to $n = 1$ in hydrogen. Compute both levels, find the gap, and decide whether the atom emits or absorbs. Check your work: $E_1 = -13.6\ \text{eV}$ and $E_2 = -3.40\ \text{eV}$, so $\Delta E = E_1 - E_2 = -10.2\ \text{eV}$, meaning the atom emits a $10.2\ \text{eV}$ photon, much more energetic than the $n=3\to2$ jump.

Calculation And Concept Traps

• Sign errors. $\Delta E = E_{\text{final}} - E_{\text{initial}}$. A negative result means emission; mixing up which level is final flips the sign.
• Treating the formula as general. The $13.6\ \text{eV}$ value is for hydrogen (and hydrogen-like one-electron species). It is not valid for multi-electron atoms.
• Treating Bohr orbits as modern orbitals. Orbitals describe probability distributions, not fixed circular paths.
• Forgetting the hydrogen condition. Statements about the Bohr model are safest for hydrogen; outside that, the model becomes much less reliable.

Where The Model Stops Working

The Bohr model works best for hydrogen and hydrogen-like one-electron species. In multi-electron atoms, electron-electron interactions are too important for the simple orbit picture to stay accurate, and it treats electrons as moving on fixed circular paths, which modern quantum mechanics replaces with orbitals.

You still reach for the Bohr model to introduce quantized energy levels, explain the hydrogen emission spectrum, connect atomic structure to photon absorption and emission, and build intuition before learning orbitals and quantum numbers. Electron configuration is the natural next step, moving from fixed orbits to shells, subshells, and orbitals.