What Are the Discrete Energy Levels in the Bohr Model of Hydrogen? (HL)

Direct Answer

In the Bohr model of hydrogen, the electron can occupy only specific, discrete energy levels given by

$$E_n=-\frac{13.6}{n^2}\ \mathrm{eV},$$

where $n=1,2,3,\ldots$ is the principal quantum number. This is Additional Higher Level (AHL) content in E.1 Structure of the atom.

Why the Energy Levels Are Discrete

Bohr proposed that an electron may orbit the proton only when its angular momentum satisfies the quantization condition

$$mvr=\frac{nh}{2\pi}.$$

Combining this condition with the electrostatic force providing the centripetal force allows only certain orbital radii and energies. Therefore, the electron cannot possess an energy between two allowed levels.

The allowed orbital radius increases as $n^2$, while the magnitude of the binding energy decreases as $1/n^2$. Because energy depends on an integer $n$, it is quantized rather than continuous.

For example:

• $E_1=-13.6\ \mathrm{eV}$, the ground state
• $E_2=-3.40\ \mathrm{eV}$

The negative sign means the electron is bound to the proton. Energy must be supplied to remove it to infinity, where the energy is defined as zero. As $n$ increases, adjacent levels become closer together.

An electron changes levels by absorbing or emitting a photon whose energy equals the difference between the levels:

$$\Delta E=hf.$$

For a transition from $n=3$ to $n=2$, the emitted photon has energy

$$\Delta E=(-1.51)-(-3.40)=1.89\ \mathrm{eV}.$$

Using $1.89\ \mathrm{eV}=3.02\times10^{-19}\ \mathrm{J}$, $f=\Delta E/h\approx4.56\times10^{14}\ \mathrm{Hz}$. A downward transition emits a photon; an upward transition requires absorption. This produces hydrogen's characteristic line spectrum.

The common misconception is that an electron gradually loses energy while remaining between levels. In the Bohr model, transitions occur between allowed levels, and the photon energy must match the energy difference exactly.

Exam Technique

In an IB Physics HL calculation, identify the initial and final values of $n$, calculate both energies, and use the positive magnitude of their difference for the photon energy. For Explain questions, connect quantized angular momentum to discrete energies and then connect transitions to line emission or absorption spectra.