15.3 Emission and Absorption Spectra

Atoms absorb or emit a photon only when the photon's energy exactly matches the gap between two allowed energy levels, which is why each element produces its own set of spectral lines. You use the relationship $E = hf = \frac{hc}{\lambda}$ to connect a transition's energy difference to the frequency and wavelength of the photon, and you read energy level diagrams for single-electron atoms like hydrogen.

Why This Matters for the AP Physics 2 Exam

Emission and absorption spectra tie together quantization, photon energy, and conservation of energy, so this topic shows up in both multiple-choice reasoning and free-response analysis. The first free-response question, the Mathematical Routines question, asks you to create and use mathematical models, build or interpret representations, and write a clear explanation backed by evidence. Energy level diagrams and spectral line problems are good practice for that kind of work because they require you to connect a calculation ($\Delta E$ to wavelength) with a conceptual claim (why a specific line appears).

You should be ready to:

• Read an energy level diagram and predict which transitions are possible.
• Calculate photon energy, frequency, or wavelength for a transition.
• Explain why emission and absorption spectra identify elements.
• Justify claims using conservation of energy and quantized energy states.

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Key Takeaways

• An atom absorbs or emits a photon only when the photon energy equals the difference between two allowed energy states.
• One transition produces or absorbs exactly one frequency and one wavelength, which appears as a single spectral line.
• Photon energy connects to frequency and wavelength through $E = hf = \frac{hc}{\lambda}$.
• Each element has a unique set of energy levels, so its spectrum acts like a fingerprint for identifying elements.
• Emission spectra show bright lines from downward transitions; absorption spectra show dark lines from upward transitions in a continuous background.
• Binding energy is the energy needed to remove an electron; an atom in the ground state needs the most energy to ionize.
• In AP Physics 2, energy level diagrams are limited to single-electron atoms such as hydrogen.

Photon Emission and Absorption

In AP Physics 2, atomic energy level diagrams are limited to single-electron atoms such as hydrogen. When you read diagrams and run calculations in this topic, keep that single-electron context in mind.

Energy Transfer via Photons

Photons carry electromagnetic energy in atomic processes, and they transfer that energy when they interact with an atom. This is the foundation of spectroscopy.

• An atom is modeled here as a system of a nucleus and an electron.
• When a photon is absorbed, its energy goes into the atom, and the electron moves to a higher energy state.
• During emission, the atom releases a photon as the electron drops to a lower energy state, turning internal energy into electromagnetic radiation.

Energy Differences Between Atomic States

Atoms exist only in specific discrete energy states, not in a continuous range like macroscopic objects. This quantization is a core idea of quantum theory.

• Energy is absorbed or emitted only in amounts that exactly match the difference between two allowed energy states.
• An atom in a lower state can absorb a photon whose energy equals the gap and jump to a higher state.
• An excited atom can spontaneously emit a photon and drop to a lower state, with the photon carrying away exactly the energy difference.
• These differences reflect changes in the electron-nucleus interaction:
  • Higher energy states correspond to weaker electron-nucleus interaction (electron less tightly bound).
  • Lower energy states correspond to stronger electron-nucleus interaction (electron more tightly bound).

Atomic Transitions and Photon Energy

Photon energy is directly related to its frequency and wavelength:

$E = hf = \frac{hc}{\lambda}$

where $h$ is Planck's constant, $f$ is frequency, $c$ is the speed of light, and $\lambda$ is wavelength.

• For a transition between two states with energy difference $\Delta E$, the photon energy is $E_{photon} = \Delta E = E_2 - E_1$.
• The frequency is $f = \frac{\Delta E}{h}$.
• The wavelength is $\lambda = \frac{hc}{\Delta E}$.
• One specific pair of energy states gives one photon energy, so it produces or absorbs light of one frequency and one wavelength. Each allowed transition appears as a single spectral line.
• Larger energy transitions produce higher frequency (shorter wavelength) photons.
• Smaller energy transitions produce lower frequency (longer wavelength) photons.

Unique Atomic Spectra

Each element has its own set of allowed energy levels because of its specific nuclear and electronic structure. That means each element also has its own set of absorption and emission frequencies, which make up its characteristic spectrum. These act like spectral fingerprints for identifying elements.

• Emission spectra show bright lines where atoms emit photons as electrons drop from higher to lower states.
  • They appear as bright colored lines against a dark background.
  • Scientists use emission spectra to identify elements in distant stars and nebulae.
  • The line pattern is unique to each element.
  • An emission spectrum can be used to determine the elements in a source of light.
• Absorption spectra show dark lines where atoms absorb photons.
  • They appear as dark lines in an otherwise continuous spectrum.
  • When light passes through a gas, the atoms absorb specific frequencies.
  • The dark lines reveal the composition of the gas.
  • An absorption spectrum can be used to determine the elements composing a substance by observing what light it absorbed.
  • Sunlight passing through the Sun's outer layers creates the solar absorption spectrum (an application of this idea).
• Energy level diagrams visually represent:
  • The allowed energy states of an atom (horizontal lines).
  • The possible transitions between states (vertical arrows).
  • The energy of photons emitted or absorbed during transitions.
  • In AP Physics 2, these diagrams are limited to single-electron atoms such as hydrogen.

Binding Energy and Ionization

Binding energy is the energy that holds an electron to the nucleus. Supply enough energy to overcome it and the atom ionizes, leaving a free electron and a positive ion.

• Binding energy is the minimum energy needed to fully remove an electron from an atom.
• For an atom in the ground state (lowest energy level):
  • The electron is most tightly bound.
  • Maximum energy is required for ionization.
  • This maximum is the ionization energy.
• For an atom in an excited state:
  • The electron is already partially lifted away from the nucleus.
  • Less additional energy is needed to remove it.
  • Binding energy decreases as excitation level increases.
• Focus on how binding energy depends on the atom's allowed energy states and on the attraction between the nucleus and the electron in a single-electron atom.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify the two energy states involved and find $\Delta E = E_2 - E_1$. The photon energy equals the absolute value of that difference.
• Use $E = hf = \frac{hc}{\lambda}$ to switch between energy, frequency, and wavelength. Watch your units: convert eV to joules with $1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$ before using SI values of $h$ and $c$.
• For emission, the electron moves to a lower state and a photon leaves. For absorption, the electron moves up and a photon is taken in.
• Compare transition sizes: a bigger energy gap means higher frequency and shorter wavelength.

Free Response

• When explaining why a specific line appears, tie the calculation to a conceptual reason: the photon energy matches a specific energy gap, so only that frequency is emitted or absorbed.
• Use conservation of energy in your justification. The energy lost by the atom equals the energy carried by the emitted photon.
• Keep reasoning organized and sequential, since the Mathematical Routines question rewards clear, evidence-based explanation.

Common Trap

• An atom can only absorb a photon whose energy exactly matches an available energy gap. If the photon energy is less than the binding energy but does not match a gap, no transition happens.
• If a photon's energy is greater than the binding energy, the extra energy becomes kinetic energy of the freed electron, not another photon.

Practice Problem 1: Photon Energy in Transitions

Solution

First, find the energy difference between the two states: $\Delta E = E_1 - E_3 = -13.6 \text{ eV} - (-1.51 \text{ eV}) = -12.09 \text{ eV}$

The negative sign shows energy is released. The photon energy equals the magnitude of this difference: $E_{photon} = |\Delta E| = 12.09 \text{ eV}$

Convert to joules: $E_{photon} = 12.09 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 1.94 \times 10^{-18} \text{ J}$

Find the frequency: $f = \frac{E_{photon}}{h} = \frac{1.94 \times 10^{-18} \text{ J}}{6.63 \times 10^{-34} \text{ J\cdot s}} = 2.93 \times 10^{15} \text{ Hz}$

Find the wavelength: $\lambda = \frac{c}{f} = \frac{3.00 \times 10^8 \text{ m/s}}{2.93 \times 10^{15} \text{ Hz}} = 1.02 \times 10^{-7} \text{ m} = 102 \text{ nm}$

This photon is in the ultraviolet region of the electromagnetic spectrum.

Practice Problem 2: Identifying Elements from Spectra

Solution

The first set of wavelengths (410 nm, 434 nm, 486 nm, and 656 nm) matches known visible hydrogen emission lines, which occur when electrons transition from higher energy levels to the n=2 energy level.

The second set of wavelengths (589 nm, 615 nm, and 498 nm) does not match hydrogen's pattern. Since each element has a unique set of energy levels, it produces a unique emission spectrum, like a fingerprint.

The scientist can conclude that:

1. The unknown element is not hydrogen.
2. The unknown element must be a different element with its own energy level structure.
3. By comparing these wavelengths to known emission spectra, the scientist could identify the specific element.

This shows how emission spectroscopy lets scientists identify elements in distant stars and other light sources without physically sampling them.

Practice Problem 3: Binding Energy and Ionization

Solution

For the 12.1 eV photon: The binding energy is 13.6 eV, and the photon energy (12.1 eV) is less, so the electron cannot be completely removed. Instead, the electron is excited to a higher energy level.

Using a hydrogen energy-level diagram or table, a 12.1 eV absorption from the ground state matches the energy gap from n=1 to n=3, so the electron is excited to the n=3 state.

For the 15.0 eV photon: The photon energy (15.0 eV) exceeds the binding energy (13.6 eV), so the electron is completely removed and the atom is ionized. The excess energy (15.0 eV - 13.6 eV = 1.4 eV) becomes kinetic energy of the freed electron.

Common Misconceptions

• "Atoms can absorb any photon." An atom can only absorb a photon whose energy matches an available energy gap. A photon with slightly too little or too much energy is not absorbed for that transition.
• "One element can make any spectral pattern." Each element has a fixed set of energy levels, so its spectrum is fixed and unique. The line pattern is what identifies the element.
• "Emission and absorption lines are unrelated." For the same element, the frequencies of emission and absorption lines line up, because they come from the same set of energy gaps.
• "Higher energy levels mean the electron is more tightly bound." It is the opposite. Higher energy states are less tightly bound, so less energy is needed to ionize from an excited state than from the ground state.
• "Bigger transitions give longer wavelengths." A bigger energy gap gives a higher frequency and a shorter wavelength, not a longer one.
• "Energy level diagrams in this course work for any atom." In AP Physics 2, energy level diagrams are limited to single-electron atoms such as hydrogen.

Related AP Physics 2 Guides

• 15.1 Quantum Theory and Wave-Particle Duality
• 15.2 The Bohr Model of Atomic Structure
• 15.6 Compton Scattering
• 15.7 Fission, Fusion, and Nuclear Decay
• 15.8 Types of Radioactive Decay
• 15.4 Blackbody Radiation