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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 15 · Topic 15.3

Connecting Spectral Lines to Energy Transitions

Quantization, probability, nuclear processes, and relativity extend classical models at atomic scales.

1. Topic Lens

Connecting Spectral Lines to Energy Transitions is studied through modern physics. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[E=hf\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. A photon carries discrete energy hf.
  2. The material requires work function φ to release an electron.
  3. Energy conservation leaves the remainder as the maximum electron kinetic energy.
\[K_{\text{max}}=hf-\phi\]

3. Detailed Visual Model

Pixel diagram for Connecting Spectral Lines to Energy TransitionsOriginal schematic connecting Connecting Spectral Lines to Energy Transitions to Modern Physics.
Connecting Spectral Lines to Energy Transitions: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

4. Worked Example and Lab Link

A 5 eV photon strikes a surface with work function 2 eV. Find maximum electron kinetic energy.

Answer: Kmax=5-2=3 eV.

Investigation idea: Analyze original stopping-potential data versus frequency and infer threshold frequency and Planck-slope meaning.

Common trap: Light intensity changes photon count; below threshold frequency it does not release electrons in the ideal model.

Expanded Topic 15.3 Lesson

5. Learning Targets

  • Translate an atomic energy-level diagram into emission or absorption wavelengths.
  • Explain why the same energy gap gives matching emission and absorption line positions.
  • Use photon energy, wavelength, frequency, wavenumber, and the Rydberg relation consistently.
  • Recognize Lyman, Balmer, and Paschen series and calculate each series limit.
  • Separate what a line's position, intensity, width, and splitting reveal about a source.

6. Three Spectra, Three Physical Stories

ContinuousA broad range of wavelengths with no isolated gaps; a hot dense source is a common example.
Emission linesExcited atoms emit photons at discrete wavelengths when they move to lower energy states.
Absorption linesA cooler gas removes selected wavelengths when photons raise atoms to available higher states.

A spectrum is not a picture of electron paths. It is a record of photon energies exchanged by an ensemble of atoms.

7. One Conservation Law Connects Every Line

Let ΔEatom = Ef − Ei. For absorption the atom gains energy, while for emission the photon carries away the atom's lost energy:

\[\text{absorption:}\quad hf=E_f-E_i>0\]
\[\text{emission:}\quad E_\gamma=E_i-E_f>0\]
\[\boxed{E_\gamma=|E_f-E_i|=hf=\frac{hc}{\lambda}}\]

Always make photon energy positive. The direction of the arrow tells whether the atom absorbs or emits.

8. Original Energy-Level-to-Spectrum Map

Hydrogen transitions mapped to visible Balmer spectral lines Downward transitions from levels three, four, and five to level two map to red, blue-green, and violet lines at approximately 656, 486, and 434 nanometres. energyn = 1n = 2n = 3n = 4n = 53→24→25→2visible wavelength increases →434 nm486 nm656 nmline position ↔ energy gapline brightness ↔ number and probability of events
Each downward arrow creates one possible photon energy. The horizontal placement of a spectral line represents wavelength, not the physical position of an electron.

9. Fast Conversion Toolkit

Wavelength ↔ frequency\(f=c/\lambda\)
Photon energy\(E_\gamma=hf=hc/\lambda\)
Useful shortcut\(E_\gamma(\mathrm{eV})\approx1240/\lambda(\mathrm{nm})\)
Wavenumber\(\widetilde\nu=1/\lambda\), commonly in \(\mathrm{m^{-1}}\) or \(\mathrm{cm^{-1}}\)

Convert units before substitution: \(1\,\mathrm{nm}=10^{-9}\,\mathrm m\) and \(1\,\mathrm{eV}=1.602\times10^{-19}\,\mathrm J\). Larger photon energy means higher frequency and shorter wavelength.

10. Worked Example: Hydrogen \(3\to2\)

For hydrogen, \(E_3=-1.51\,\mathrm{eV}\) and \(E_2=-3.40\,\mathrm{eV}\). A downward transition emits

\[E_\gamma=E_3-E_2=(-1.51)-(-3.40)=1.89\,\mathrm{eV}\]
\[\lambda\approx\frac{1240\,\mathrm{eV\,nm}}{1.89\,\mathrm{eV}}=656\,\mathrm{nm}\]

This is the red Hα line. Reversing the arrow, \(2\to3\), requires a 656 nm photon in the ideal isolated atom.

11. Why Emission and Absorption Lines Align

The same pair of stationary states has one energy difference. Therefore \(3\to2\) emission and \(2\to3\) absorption have the same ideal wavelength.

Emission needspopulation in the upper state and a route to a lower state.
Absorption needspopulation in the lower state plus an incident photon matching the gap.

Matching energy is necessary, but transition probability also depends on quantum selection rules. A level difference alone does not guarantee a strong observable line.

12. Position, Intensity, Width, and Splitting Mean Different Things

FeaturePrimary informationDo not assume
Line positionPhoton energy and separation of two atomic statesA brighter line has greater photon energy
Line intensityHow often photons from that transition reach the detectorIntensity is set by the gap alone
Line widthMotion, collisions, finite lifetime, and instrument resolution may contributeEvery source produces an infinitely thin line
Line splittingClosely spaced sublevels, often revealed by external fields or internal interactionsEvery nearby line is a different chemical element

Intensity can depend on upper-state population, transition probability, number of atoms, path length, detector response, and geometry. It is experimental evidence, not a direct label for energy.

13. From Hydrogen-Like Levels to the Rydberg Relation

For one-electron ions, \(E_n=-13.6Z^2/n^2\,\mathrm{eV}\). For emission from \(n_i\) to \(n_f\), where \(n_i>n_f\),

\[E_\gamma=13.6Z^2\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)\mathrm{eV}\]
\[\boxed{\frac{1}{\lambda}=RZ^2\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)},\qquad R\approx1.097\times10^7\,\mathrm{m^{-1}}\]

This is the usual ideal-model form. Precision spectroscopy uses a nucleus-dependent reduced-mass correction, so the measured constant is not exactly identical for every isotope or ion.

14. Spectral Series Organize a Forest of Lines

SeriesFinal levelMain region for HSeries limit
Lyman\(n_f=1\)Ultravioletabout 91.2 nm
Balmer\(n_f=2\)Visible and near ultravioletabout 364.6 nm
Paschen\(n_f=3\)Infraredabout 820 nm

Lines within one series share the same final level. As \(n_i\) increases, their wavelengths approach a limiting value and the lines crowd together.

15. Original Series-Convergence Model

Hydrogen Balmer lines converging at the series limit Balmer lines from increasingly high initial quantum numbers crowd together toward the ultraviolet limit at about 364.6 nanometres, while the transitions all end at level two. shorter wavelength · greater photon energylonger wavelength →364.6 nm limit410434486 nm656 nmnᵢ = ∞, 8, 7 …6 → 25 → 24 → 23 → 2
The drawing emphasizes convergence rather than exact scale. More initial levels fit into a shrinking energy interval near \(E=0\), so their spectral lines also bunch together.

16. Deriving a Series Limit

The limit occurs when the initial state approaches the ionization boundary, \(n_i\to\infty\), so \(1/n_i^2\to0\):

\[\frac{1}{\lambda_{\mathrm{limit}}}=RZ^2\frac{1}{n_f^2}\]

For hydrogen's Balmer series, \(\lambda_{\mathrm{limit}}=4/R\approx364.6\,\mathrm{nm}\). This is the shortest wavelength and greatest photon energy in the series, not its longest wavelength.

17. Worked Example: Hydrogen \(4\to2\)

\[E_\gamma=13.6\left(\frac14-\frac1{16}\right)=2.55\,\mathrm{eV}\]
\[\lambda\approx\frac{1240}{2.55}=486\,\mathrm{nm}\]

This blue-green Hβ line has a shorter wavelength than Hα because the \(4\to2\) energy gap is larger than the \(3\to2\) gap.

18. Nuclear-Charge Scaling: He\(^+\)

For the same \(4\to2\) transition in He\(^+\), \(Z=2\), so the ideal energy gap is four times the hydrogen gap:

\[E_\gamma=4(2.55)=10.2\,\mathrm{eV},\qquad \lambda\approx\frac{1240}{10.2}=121.6\,\mathrm{nm}\]

The wavelength is one fourth as large. Do not use this \(Z^2\) scaling for a multi-electron neutral helium atom.

19. Work Backward from a Measured Line

A hydrogen line is measured near 434 nm and is known to be in the Balmer series. First convert wavelength to energy:

\[E_\gamma\approx\frac{1240}{434}=2.86\,\mathrm{eV}\]

Because \(n_f=2\), solve

\[2.86=13.6\left(\frac14-\frac{1}{n_i^2}\right)\quad\Rightarrow\quad n_i\approx5\]

The nearest allowed integer is \(n_i=5\), so the line is assigned to \(5\to2\). Integer quantum numbers are a powerful consistency check against measurement rounding.

20. Cascades Can Produce Several Photons

An atom in \(n=4\) need not fall directly to \(n=1\). It may follow \(4\to3\to2\to1\), producing three photons with different energies.

\[E_{4}-E_{1}=(E_{4}-E_{3})+(E_{3}-E_{2})+(E_{2}-E_{1})\]

The photon energies in a cascade sum to the direct energy gap. Different atoms in an ensemble can follow different allowed routes, so one excited population can produce multiple lines.

21. Bound–Bound Lines Versus Ionization

Bound–boundA photon must match an allowed discrete energy gap, producing a line.
Bound–continuumAt or above the ionization threshold, surplus photon energy can become electron kinetic energy.
\[K_{\mathrm{electron}}=E_\gamma-E_{\mathrm{ion}}\]

A continuum of photon energies can ionize once the threshold is crossed. That is why an absorption edge and continuum are physically different from isolated bound-state lines.

22. Why Real Lines Have Width or Split

  • Thermal Doppler broadening: atoms moving toward or away from the detector contribute slightly shifted wavelengths.
  • Collision broadening: frequent interactions perturb emitting states.
  • Finite lifetime: a state with a limited lifetime has an intrinsic energy uncertainty.
  • Instrument response: a spectrometer has finite resolving power.
  • Field splitting: magnetic or electric fields can separate sublevels and create nearby components.

Before claiming a new transition, compare observed separation with the instrument's resolution and other plausible broadening mechanisms.

23. Reading an Unknown Spectrum as Evidence

1 · LocateCalibrate wavelength and identify repeatable peaks or absorption minima.
2 · Compare patternUse several relative spacings, not one visually similar line, to test a species assignment.
3 · Test shiftA shared fractional displacement of the whole pattern may indicate relative motion rather than changed atomic gaps.
4 · State limitsBlend, noise, temperature, pressure, and resolution affect confidence.

A spectral “fingerprint” is strongest when a set of predicted lines and their spacing agrees. One isolated match can be coincidental.

24. Open Simulation Investigation

A. Build the spectrum
  1. Open PhET Models of the Hydrogen Atom.
  2. Select the spectra display, shine white light, and record the wavelengths that appear.
  3. For each recorded line, calculate \(1240/\lambda\) in eV and match it to a level gap.
  4. Predict which line has the greatest photon energy before checking the calculation.
B. Test a model
  1. Use monochromatic light and scan across one predicted transition.
  2. Record whether the atom responds below, at, and above the matching energy.
  3. Compare the Bohr, de Broglie, and Schrödinger model views.
  4. Write a claim supported by both an energy-level diagram and the simulated spectrum.

25. Safe Spectrum Data Activity

  1. Use a teacher-provided or open spectrum image with a labeled wavelength scale.
  2. Calibrate pixel position against at least two known wavelengths.
  3. Estimate peak wavelengths and propagate the reading uncertainty.
  4. Compare several peaks with a reference pattern and justify an identification.

Do not assemble or handle high-voltage discharge equipment without trained supervision. A public dataset or simulation can support the same energy-transition analysis safely.

26. Common Reasoning Traps

  • Subtracting energies in the wrong order: photon energy is always the positive magnitude of the gap.
  • Using \(n_i-n_f\): level energy varies as \(-1/n^2\), not linearly with \(n\).
  • Equating brightness with photon energy: position gives photon energy; intensity counts weighted events.
  • Assuming every gap is bright: selection rules and state populations matter.
  • Forgetting the final level in a series: Balmer lines all end at \(n=2\).
  • Calling the series limit the longest wavelength: it is the greatest gap and shortest wavelength for that series.

27. AP-Style Reasoning Checks

  1. An atom emits a 2.0 eV photon. What happens to the atom's energy?
  2. Which has shorter wavelength, hydrogen \(4\to2\) or \(3\to2\)? Explain without arithmetic.
  3. Why can emission and absorption lines occur at the same wavelength yet have different intensities?
  4. What final level defines the Lyman series?
  5. A He\(^+\) and H ion make the same \(3\to2\) transition. Compare ideal photon energies.
  6. Why do lines crowd near a series limit?
  7. Can a 1.0 eV photon be partly absorbed to drive a 1.5 eV bound-state transition?

Answers: it decreases by 2.0 eV; \(4\to2\), because its gap is larger; position comes from the shared gap while populations and probabilities differ; \(n_f=1\); He\(^+\) photon energy is four times H; high-\(n\) levels crowd near zero; not in the ideal one-photon bound–bound model.

Checkpoint · Topic 15.3

Explain how connecting spectral lines to energy transitions supports or limits this conclusion: Kmax=5-2=3 eV.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.