Unit 15 · Topic 15.3

15.3 Emission and Absorption Spectra

An atom can absorb or emit a photon only if the photon's energy exactly matches the gap between two of its energy levels. Each jump between two levels gives one frequency, and one wavelength, of light. Because each element has its own set of levels, its spectrum identifies it like a fingerprint.

Key terms

energy-level diagram
ground state
excited state
emission spectrum
absorption spectrum
ionization energy

Energy-level diagrams

An energy-level diagram shows an atom's allowed energies as horizontal lines, with higher energy higher up. The lowest level is the ground state. Every level above it is an excited state.

Energies are usually negative, measured from zero at the top. Zero means the electron has just barely escaped; the atom is ionized. A negative value means the electron is bound, and more negative means more tightly bound. Hydrogen's levels, for example, are −13.6, −3.40, −1.51 and −0.85 eV for the first four, crowding closer together toward 0.

An energy level is the energy of the whole electron-plus-nucleus system: the electric potential energy between them plus the electron's kinetic energy. So when the atom changes level, what changes is the interaction between the electron and the nucleus. That's why the atom is modeled as a system of an electron and a nucleus, not as an electron on its own.

Absorbing and emitting photons

Emission: an atom in an excited state can drop to a lower level on its own, giving off one photon that carries away the energy difference.

Absorption: an atom can absorb a photon and jump to a higher level, but only if the photon's energy exactly equals the gap. A photon with slightly too much or too little energy just passes by.

In both cases $\lvert \Delta E \rvert = hf = \frac{hc}{\lambda}$ . A big jump gives a high-energy, short-wavelength photon; a small jump gives a long-wavelength one.

One exception: a photon with more than enough energy to free the electron entirely (more than the ionization energy) can be absorbed. The leftover energy becomes the freed electron's kinetic energy.

Spectra as fingerprints

Every element has its own unique set of energy levels, so it has its own unique set of photon frequencies.

An emission spectrum comes from a hot, glowing gas. Passed through a grating or prism, the light shows bright lines on a dark background. Matching those lines to known patterns tells you which elements are in the gas.

An absorption spectrum comes from shining white light (all colors) through a cooler gas. Atoms absorb exactly their own frequencies, leaving dark lines in a rainbow. The dark lines sit at the same wavelengths as that element's bright emission lines. Astronomers use the dark lines in starlight to find out what stars are made of.

A cool gas has its atoms mostly in the ground state, so its absorption lines mostly come from jumps up from the ground state. That's why an absorption spectrum usually has fewer lines than the matching emission spectrum.

Counting lines and ionization

An atom excited to level n can fall back in many ways, one step at a time or in big jumps. The number of different possible emission lines from level n down is $\frac{n(n-1)}{2}$ . From n = 4 there are 6.

The ionization energy, also called binding energy here, is the energy needed to pull the electron completely free. From the ground state it's the largest, because the ground state is the most tightly bound. For hydrogen it's 13.6 eV.

In this course, energy-level diagrams are only for single-electron atoms, like hydrogen or a made-up atom a question gives you.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Wavelength of a transition
Hydrogen has levels at −13.6 eV (n = 1), −3.40 eV (n = 2) and −1.51 eV (n = 3). Find the wavelength of the photon emitted when the electron drops from n = 3 to n = 2. What color is it?
Show the solutionHide the solution
1. Step 1: Energy of the photon: $\Delta E = -1.51 - (-3.40) = 1.89$ eV.
2. Step 2: Wavelength: $\lambda = \dfrac{hc}{\Delta E} = \dfrac{1240\text{ eV·nm}}{1.89\text{ eV}} = 656$ nm.
3. Step 3: 656 nm is in the red part of the visible spectrum; it's the bright red line of glowing hydrogen.
Answer: 656 nm, red
Example 2Calculator allowed
All possible lines from n = 4
A hydrogen atom is excited to n = 4 (−0.85 eV). Using the levels −13.6, −3.40, −1.51 and −0.85 eV, how many different photon energies could it emit on its way down, and which transitions give the longest and shortest wavelengths?
Show the solutionHide the solution
1. Step 1: List the jumps: 4→3, 4→2, 4→1, 3→2, 3→1, 2→1. That's 6, matching $\frac{4(3)}{2} = 6$ .
2. Step 2: Longest wavelength means smallest energy gap: 4→3, ΔE = −0.85 − (−1.51) = 0.66 eV, $\lambda = \frac{1240}{0.66} \approx 1880$ nm (infrared; with unrounded levels, 1876 nm).
3. Step 3: Shortest wavelength means largest gap: 4→1, ΔE = 12.75 eV, $\lambda = \frac{1240}{12.75} = 97.3$ nm (ultraviolet).
Answer: 6 lines; longest from 4→3 (about 1880 nm), shortest from 4→1 (97.3 nm).
Example 3Calculator allowed
Which photons get absorbed? (trap)
Hydrogen atoms in the ground state (−13.6 eV) are hit by photons of 10.2 eV, 11.0 eV and 15.0 eV. The next level up is −3.40 eV. Which photons are absorbed, and what happens?
Show the solutionHide the solution
1. Step 1: 10.2 eV: exactly matches −3.40 − (−13.6) = 10.2 eV, so it's absorbed and the atom jumps to n = 2.
2. Step 2: 11.0 eV: more than 10.2 eV, but it doesn't match any gap (the next gap, to n = 3, is 12.1 eV). It is not absorbed. The trap is thinking the atom takes 10.2 eV and leaves 0.8 eV over.
3. Step 3: 15.0 eV: more than the 13.6 eV ionization energy, so it can be absorbed and free the electron, which leaves with 15.0 − 13.6 = 1.4 eV of kinetic energy.
Answer: 10.2 eV absorbed (to n = 2); 11.0 eV not absorbed; 15.0 eV ionizes the atom, leaving the electron with 1.4 eV.

Common mistakes

Thinking an atom can absorb part of a photon's energy. Below the ionization energy, the photon's energy must exactly match a gap or it isn't absorbed.
Using a level's energy instead of the difference between two levels. The photon's energy is the gap $\lvert \Delta E \rvert$ .
Linking a large energy jump to a long wavelength. Bigger jumps mean higher frequency and shorter wavelength.
Forgetting that emission lines and absorption lines of the same element appear at the same wavelengths.

On the exam

Expect energy-level diagrams: draw the arrows for the possible transitions, rank the photons by energy, frequency or wavelength, or find which level a photon came from.
Spectrum questions may ask how to identify an unknown gas. Compare its lines with known elements' spectra and explain that each element has its own unique levels.

Connected topics

Videos

Atomic spectra | Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Emission and Absorption Spectra
Bozeman ScienceWatch on YouTube (opens in a new tab)
High School Physics - Energy Level Diagrams
Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)
Bohr Model of the Hydrogen Atom, Electron Transitions, Atomic Energy Levels, Lyman & Balmer Series
The Organic Chemistry TutorWatch on YouTube (opens in a new tab)
High School Physics - Atomic Spectra
Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)

Check yourself

5 questions on 15.3 Emission and Absorption Spectra. Pick an answer to see if you got it, and why.

Question 1 of 5Calculator allowed

Hydrogen's energy levels include E₂ = −3.40 eV and E₃ = −1.51 eV. What is the wavelength of the photon emitted when an electron drops from n = 3 to n = 2? (Use hc = 1240 eV·nm.)

Question 2 of 5Calculator allowed

A gas of hydrogen atoms has electrons excited to the n = 4 level. As they return to the ground state, by any route, how many different photon energies can be emitted?

Level	Energy (eV)
n = 4	−3.0
n = 3	−4.0
n = 2	−6.0
n = 1 (ground state)	−10.0

Hypothetical data. The table lists all the bound energy levels of the single electron in an imaginary atom. Use hc = 1240 eV·nm.

Question 3 of 5Calculator allowed

Which transition emits the photon with the longest wavelength?

Question 4 of 5Calculator allowed

Atoms in the ground state are lit with photons of four energies. Which photon energy can NOT be absorbed?

Question 5 of 5Calculator allowed

What is the ionization energy of this atom from its ground state?

0 of 5 answered