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E.1.10 (HL)—Bohr hydrogen levels

Syllabus
First assessment 2025
Objective
—
Level
HL

Use Bohr Energy Levels

HL only

Use the hydrogen levels

In the Bohr model for hydrogen, n=1,2,3,…n=1,2,3,\ldots is the principal quantum number. Bound-state energies are negative and approach zero as nn increases; the equation is not the general spectrum formula for multi-electron atoms.

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

Worked example — fifth level

For n=5n=5, E5=−13.6/52=−0.544 eVE_5=-13.6/5^2=-0.544\,\mathrm{eV}. In joules this is (−0.544)(1.60×10−19)=−8.70×10−20 J(-0.544)(1.60\times10^{-19})=-8.70\times10^{-20}\,\mathrm{J}. Keep the negative sign for the bound level; use a positive energy difference for a photon.

Find a transition energy

For a transition between levels, calculate ΔE=∣Ei−Ef∣\Delta E=|E_i-E_f|. Emission occurs for a downward transition and absorption for an upward transition; the photon then obeys Eγ=hf=hc/λE_\gamma=hf=hc/\lambda.

Compare levels

The gaps are not equally spaced. A transition involving low nn can have a larger energy difference than one involving high nn, so compare the actual level values rather than relying on the visual spacing of an unscaled sketch.

Common trap

Do not omit the negative sign while identifying the level, but do use the positive magnitude of the difference when calculating photon energy.

E.1.10 (HL) Exam Analysis

HL only

Assessment in practice

1–2 marks
How it is assessed

Questions compare photon wavelengths or absorbed energies for transitions shown on a hydrogen energy-level diagram.

Command terms

Determine / Compare

What earns marks

Calculate the relevant level differences, compare photon energy before converting to wavelength, and remember that wavelength is inversely proportional to the gap.

Watch for

Comparing wavelength in the same direction as energy, or using the level label n instead of the actual energy difference.

Retrieve the HL Atomic Model

HL only

Retrieve the HL extensions

Use R=R0A1/3R=R_0A^{1/3} for nuclear scale, recognise when high-energy scattering exceeds the electrostatic model, and use energy conservation for head-on closest approach.

Retrieve the Bohr model

Hydrogen levels obey En=−13.6/n2 eVE_n=-13.6/n^2\,\mathrm{eV}, and allowed angular momentum mvr=nh/(2π)mvr=nh/(2\pi) produces discrete orbits and energies.