Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Bohr Model and Hydrogen Spectra Worked Examples cheat sheet - grade 11-12

Click image to open full size

Physics Grade 11-12

Bohr Model and Hydrogen Spectra Worked Examples Cheat Sheet

A printable reference covering Bohr orbits, hydrogen energy levels, photon energy, wavelength, and spectral series for grades 11-12.

TopicsQuantum PhysicsAtomsEnergyLightWavesPhysicsStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

The Bohr model explains the hydrogen atom by placing the electron in specific allowed energy levels instead of any possible orbit. This cheat sheet helps students connect atomic structure to the bright-line spectrum of hydrogen. It is useful for solving worked examples involving electron transitions, emitted photons, absorbed photons, and wavelengths.

Grade 11-12 physics students need these tools for atomic physics, quantum ideas, and spectroscopy problems.

The most important idea is that each hydrogen energy level has energy En=−13.6 eVn2E_n = \frac{-13.6\,\text{eV}}{n^2}, where nn is a positive integer. A photon is emitted when an electron falls to a lower level and absorbed when it jumps to a higher level. The photon energy is the magnitude of the energy change, Ephoton=∣ΔE∣=hf=hcλE_{\text{photon}} = |\Delta E| = hf = \frac{hc}{\lambda}.

The Rydberg equation, 1λ=RH(1nf2−1ni2)\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), gives hydrogen spectral wavelengths for transitions with ni>nfn_i > n_f.

Key Facts

  • Hydrogen energy levels in the Bohr model are given by En=−13.6 eVn2E_n = \frac{-13.6\,\text{eV}}{n^2} for n=1,2,3,…n = 1, 2, 3, \ldots.
  • The energy change for a transition is ΔE=Ef−Ei\Delta E = E_f - E_i, and an emitted photon has energy Ephoton=∣ΔE∣E_{\text{photon}} = |\Delta E|.
  • Photon energy is related to frequency and wavelength by E=hf=hcλE = hf = \frac{hc}{\lambda}.
  • Use 1 eV=1.602×10−19 J1\,\text{eV} = 1.602 \times 10^{-19}\,\text{J} when converting photon energy between electron volts and joules.
  • The Rydberg equation for hydrogen emission is 1λ=RH(1nf2−1ni2)\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), where RH=1.097×107 m−1R_H = 1.097 \times 10^7\,\text{m}^{-1} and ni>nfn_i > n_f.
  • Transitions ending at nf=1n_f = 1 form the Lyman series, transitions ending at nf=2n_f = 2 form the Balmer series, and transitions ending at nf=3n_f = 3 form the Paschen series.
  • A transition from a higher level to a lower level emits light, while a transition from a lower level to a higher level absorbs light.
  • The ionization energy of ground-state hydrogen is 13.6 eV13.6\,\text{eV} because raising the electron from n=1n = 1 to n=∞n = \infty changes the energy from −13.6 eV-13.6\,\text{eV} to 0 eV0\,\text{eV}.

Vocabulary

Bohr Model
A model of hydrogen in which the electron can occupy only certain allowed energy levels around the nucleus.
Energy Level
A permitted electron state with energy En=−13.6 eVn2E_n = \frac{-13.6\,\text{eV}}{n^2} for hydrogen.
Photon
A packet of electromagnetic energy with energy E=hf=hcλE = hf = \frac{hc}{\lambda}.
Emission Spectrum
A pattern of bright spectral lines produced when atoms emit photons during transitions to lower energy levels.
Absorption Spectrum
A pattern of missing dark lines produced when atoms absorb specific photon energies and electrons jump to higher levels.
Rydberg Constant
The hydrogen spectral constant RH=1.097×107 m−1R_H = 1.097 \times 10^7\,\text{m}^{-1} used in the Rydberg wavelength equation.

Common Mistakes to Avoid

  • Using En=13.6 eV/n2E_n = 13.6\,\text{eV}/n^2 with a positive sign is wrong because bound hydrogen energy levels are negative relative to the ionized state.
  • Forgetting to take the absolute value of ΔE\Delta E for emitted photon energy is wrong because photon energy must be positive even when ΔE<0\Delta E < 0 for the atom.
  • Putting nin_i and nfn_f in the wrong places in 1λ=RH(1nf2−1ni2)\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) is wrong because emission requires ni>nfn_i > n_f and a positive value of 1λ\frac{1}{\lambda}.
  • Mixing electron volts and joules in E=hfE = hf is wrong because Planck's constant h=6.626×10−34 J⋅sh = 6.626 \times 10^{-34}\,\text{J}\cdot\text{s} requires energy in joules.
  • Assuming larger nn means more negative energy is wrong because EnE_n approaches 0 eV0\,\text{eV} as nn increases, so the electron is less tightly bound.

Practice Questions

  1. 1 Find the energy of the hydrogen level n=3n = 3 in electron volts using En=−13.6 eVn2E_n = \frac{-13.6\,\text{eV}}{n^2}.
  2. 2 A hydrogen electron falls from ni=4n_i = 4 to nf=2n_f = 2. Calculate the photon energy in electron volts and identify the spectral series.
  3. 3 Use 1λ=RH(122−132)\frac{1}{\lambda} = R_H\left(\frac{1}{2^2} - \frac{1}{3^2}\right) to find the wavelength of the ni=3n_i = 3 to nf=2n_f = 2 Balmer transition.
  4. 4 Explain why the hydrogen spectrum contains separate lines instead of a continuous rainbow.

Understanding Bohr Model and Hydrogen Spectra Worked Examples

An energy-level diagram makes the pattern easier to see. The lowest level is far below zero energy because the electron is tightly bound to the proton. Higher levels are still below zero, but they get closer together.

This crowding is important. A jump between low levels releases a large amount of energy, while jumps between very high levels release smaller amounts. Zero energy is the point where the electron has escaped completely.

The electron is no longer part of the atom at that point. This explains why ionization needs a minimum energy input and why an electron can absorb extra energy after it has already reached an excited state.

For a worked problem, begin by finding the initial and final energy separately. Keep the signs during this step. Subtract the initial energy from the final energy to find the change in energy.

A downward jump gives a negative change because the atom loses energy. The photon energy is the positive size of that change. An upward jump gives a positive change because energy enters the atom.

Convert electron volts to joules only when a question uses Planck's constant in joule seconds or asks for an answer in joules. Then use photon energy equals Planck's constant times frequency, or photon energy equals Planck's constant times the speed of light divided by wavelength. A larger photon energy means a higher frequency and a shorter wavelength.

Spectral lines are evidence that atomic energies come in fixed steps. When light from excited hydrogen passes through a prism or diffraction grating, it separates into distinct colored lines instead of a smooth rainbow. Each line matches one possible transition.

Lines that end at the second level are visible or near visible light, so they are especially common in school experiments. Lines ending at the first level are ultraviolet, which human eyes cannot see. Lines ending at the third level are infrared.

Astronomers use these patterns to identify hydrogen in stars and nebulae. The wavelengths can reveal motion too, because movement changes the observed wavelength through the Doppler effect.

Pay close attention to what the question gives and what it asks for. The Rydberg method is usually quickest when the starting and ending levels are known and the wavelength is needed. The energy-level method is often clearer when a question asks for energy, frequency, or whether light is emitted or absorbed.

Do not reverse the order of levels in an emission calculation, since this can produce a negative wavelength. Check units carefully, especially metres, nanometres, electron volts, and joules.

The Bohr model works well for hydrogen because it has one electron. It does not accurately describe most multi-electron atoms, but it remains useful because it shows the core quantum idea that atoms exchange energy in separate packets.