✨ The Photon Model of Light

Part 1 of 7 — Energy in Packets

In the early 1900s, physicists discovered that light behaves not just as a wave, but as a stream of discrete energy packets called photons. This revolutionary idea launched modern physics.

Photon Energy

A photon carries energy proportional to its frequency:

$E = hf$

where:

• $E$ = photon energy (in joules)
• $h = 6.63 \times 10^{-34}$ J·s (Planck's constant)
• $f$ = frequency of the light (in Hz)

Since $c = f\lambda$, we can also write:

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

Key Constants

Energy in Electron Volts

For atomic-scale problems, joules are inconveniently small. We use electron volts (eV):

$1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$

$E_{\text{eV}} = \frac{E_{\text{J}}}{1.60 \times 10^{-19}}$

A useful shortcut:

$E = \frac{hc}{\lambda} = \frac{1240 \text{ eV·nm}}{\lambda \text{ (nm)}}$

Photon Energy Quiz

The Photoelectric Effect

When light shines on a metal surface, it can eject electrons. This is the photoelectric effect.

The Energy Equation

$E_{\text{photon}} = \phi + KE_{\max}$

$hf = \phi + KE_{\max}$

where:

• $hf$ = energy of the incoming photon
• $\phi$ = work function — the minimum energy needed to free an electron from the metal surface
• $KE_{\max}$ = maximum kinetic energy of the ejected electron

Threshold Frequency

The threshold frequency $f_0$ is the minimum frequency needed to eject any electrons:

$\phi = hf_0 \quad \Rightarrow \quad f_0 = \frac{\phi}{h}$

If $f < f_0$: no electrons are ejected, regardless of intensity.

Stopping Voltage

The stopping voltage $V_s$ is the voltage needed to stop the fastest photoelectrons:

$eV_s = KE_{\max}$

$eV_s = hf - \phi$

Key Observations (AP Exam Favorites!)

Photoelectric Effect Concept Check

Photon & Photoelectric Effect Drill

Use $h = 6.63 \times 10^{-34}$ J·s, $c = 3.00 \times 10^{8}$ m/s, $1 \text{ eV} = 1.60 \times 10^{-19}$ J.

A sodium surface has work function $\phi = 2.28$ eV. Ultraviolet light of wavelength 250 nm shines on it.

1. Photon energy in joules ($\times 10^{-19}$ J, 3 significant figures)
2. Photon energy in eV (3 significant figures)
3. Maximum KE of ejected electrons in eV (3 significant figures)
4. Stopping voltage in V (3 significant figures)

Exit Quiz — Photon Model

🌊 Wave-Particle Duality

Part 2 of 7 — Matter Waves

Light acts as both a wave and a particle. In 1924, Louis de Broglie proposed that matter also has wave properties. Every moving particle has an associated wavelength — the de Broglie wavelength.

The de Broglie Wavelength

Any particle with momentum $p$ has an associated wavelength:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where:

• $\lambda$ = de Broglie wavelength (m)
• $h = 6.63 \times 10^{-34}$ J·s
• $p = mv$ = momentum of the particle
• $m$ = mass of the particle (kg)
• $v$ = speed of the particle (m/s)

Why Don't We See Waves for Everyday Objects?

For a 0.15 kg baseball at 40 m/s:

$\lambda = \frac{6.63 \times 10^{-34}}{(0.15)(40)} = 1.1 \times 10^{-34} \text{ m}$

This is $10^{19}$ times smaller than a proton — completely undetectable!

For an electron ($m = 9.11 \times 10^{-31}$ kg) at $10^6$ m/s:

$\lambda = \frac{6.63 \times 10^{-34}}{(9.11 \times 10^{-31})(10^6)} = 7.3 \times 10^{-10} \text{ m} = 0.73 \text{ nm}$

This is comparable to atomic spacings — electron waves are observable!

de Broglie Wavelength Quiz

Evidence for Matter Waves

Electron Diffraction

In 1927, Davisson and Germer fired electrons at a nickel crystal and observed a diffraction pattern — the same behavior expected of waves scattering off a periodic structure.

The electrons' de Broglie wavelength matched the wavelength predicted by the diffraction pattern, confirming de Broglie's hypothesis.

Double-Slit Experiment with Particles

When electrons (or even larger particles like neutrons and molecules) pass through a double slit:

• Many particles: An interference pattern builds up on the detector
• One particle at a time: Each particle lands at a single point, but after many particles, the interference pattern still emerges
• Observation: If you detect which slit the particle goes through, the interference pattern disappears

This demonstrates that each particle interferes with itself — it passes through both slits as a wave!

Compton Scattering

When X-ray photons collide with electrons, the scattered photon has a longer wavelength (lower energy). The wavelength shift depends on the scattering angle:

$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$

This proved that photons carry momentum: $p = h/\lambda = E/c$.

Wave-Particle Duality Concepts

de Broglie Wavelength Calculation Drill

Use $h = 6.63 \times 10^{-34}$ J·s, $m_e = 9.11 \times 10^{-31}$ kg, $1 \text{ eV} = 1.60 \times 10^{-19}$ J.

An electron is accelerated from rest through a potential difference of 100 V.

1. Kinetic energy gained by the electron ($\times 10^{-17}$ J, 3 significant figures)
2. Speed of the electron ($\times 10^{6}$ m/s, 3 significant figures)
3. de Broglie wavelength ($\times 10^{-10}$ m, 3 significant figures)

Exit Quiz — Wave-Particle Duality

⚛️ Atomic Models & Energy Levels

Part 3 of 7 — The Bohr Model

How do atoms emit and absorb light? The Bohr model of hydrogen explains discrete spectral lines by quantizing electron orbits into specific energy levels.

The Bohr Model of Hydrogen

Niels Bohr (1913) proposed that:

1. Electrons orbit the nucleus only in specific allowed orbits (energy levels)
2. Each orbit has a quantized energy — electrons cannot have energies between levels
3. Electrons can jump between levels by absorbing or emitting a photon

Energy Levels of Hydrogen

$E_n = -\frac{13.6}{n^2} \text{ eV}$

where $n = 1, 2, 3, \ldots$ is the principal quantum number.

Key Features

• Negative energies mean the electron is bound to the atom
• The ground state ($n = 1$) is the lowest energy: $-13.6$ eV
• Ionization energy = energy needed to remove the electron from ground state = 13.6 eV
• As $n \to \infty$, levels get closer together and approach 0 eV

Photon Emission and Absorption

Emission

When an electron drops from a higher level $n_i$ to a lower level $n_f$:

$\Delta E = E_{n_i} - E_{n_f} = hf_{\text{photon}}$

The atom emits a photon with energy equal to the energy difference between levels.

Absorption

When a photon with exactly the right energy hits the atom, the electron jumps up:

$E_{\text{photon}} = E_{n_f} - E_{n_i} = hf$

The photon must have exactly the energy of a transition — partial absorption doesn't happen.

Calculating Photon Wavelength

For a transition between levels $n_i$ and $n_f$:

$\Delta E = 13.6 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \text{ eV}$

$\lambda = \frac{hc}{\Delta E} = \frac{1240 \text{ eV·nm}}{\Delta E \text{ (eV)}}$

Spectral Lines vs. Continuous Spectra

Line Spectra (Discrete)

• Emission spectrum: Bright colored lines on a dark background
  • Hot, low-density gas emits only specific wavelengths
  • Each element has a unique "fingerprint" of spectral lines
• Absorption spectrum: Dark lines on a continuous rainbow background
  • Cool gas absorbs specific wavelengths from white light passing through

Continuous Spectrum

• Hot, dense objects (solids, liquids, dense gases) emit all wavelengths
• Produces a smooth rainbow with no gaps

Why Line Spectra?

Because energy levels are quantized, only specific energy differences exist → only specific photon energies (and wavelengths) are emitted or absorbed.

Each element has different energy levels → different spectral lines → spectral "fingerprints" allow identification of elements in stars!

Atomic Model Concept Quiz

Energy Level Calculation Drill

Use $E_n = -13.6/n^2$ eV and $\lambda = 1240/\Delta E$ (nm).

A hydrogen atom transitions from the $n = 5$ level to the $n = 2$ level.

1. Energy of $n = 5$ level (in eV, 3 significant figures)
2. Energy of $n = 2$ level (in eV, 3 significant figures)
3. Energy of emitted photon (in eV, 3 significant figures)
4. Wavelength of emitted photon (in nm, round to nearest whole number)

Exit Quiz — Bohr Model

🌈 Atomic Transitions & Spectral Series

Part 4 of 7 — Reading the Light

Every element produces a unique set of spectral lines. By understanding energy level diagrams and the spectral series of hydrogen, you can predict and calculate the wavelengths of emitted or absorbed light.

Energy Level Diagrams

An energy level diagram shows allowed electron energies as horizontal lines, with transitions as arrows:

Emission (arrow pointing DOWN)

• Electron drops to a lower level
• Photon is emitted with energy $\Delta E = E_{\text{upper}} - E_{\text{lower}}$
• Longer arrow → higher energy photon → shorter wavelength

Absorption (arrow pointing UP)

• Electron jumps to a higher level
• Photon is absorbed with energy exactly matching $\Delta E$
• The incoming photon disappears — its energy goes into the electron

Reading the Diagram

$E_{\infty} = 0 \text{ eV (ionized)} \quad \cdots$ $E_4 = -0.85 \text{ eV} \quad \rule{3cm}{0.5pt}$ $E_3 = -1.51 \text{ eV} \quad \rule{3cm}{0.5pt}$ $E_2 = -3.40 \text{ eV} \quad \rule{3cm}{0.5pt}$ $E_1 = -13.6 \text{ eV} \quad \rule{3cm}{0.5pt}$

A transition from $n = 3 \to n = 1$ releases $\Delta E = 13.6 - 1.51 = 12.09$ eV — an ultraviolet photon.

Hydrogen Spectral Series

Transitions are grouped by their final (lower) level:

Lyman Series (to $n = 1$) — Ultraviolet

All transitions ending at the ground state:

• $2 \to 1$: $\Delta E = 10.2$ eV, $\lambda = 122$ nm
• $3 \to 1$: $\Delta E = 12.1$ eV, $\lambda = 103$ nm
• $4 \to 1$: $\Delta E = 12.75$ eV, $\lambda = 97.3$ nm

Balmer Series (to $n = 2$) — Visible Light!

The only series in the visible range:

• $3 \to 2$: $\Delta E = 1.89$ eV, $\lambda = 656$ nm (red, H-alpha)
• $4 \to 2$: $\Delta E = 2.55$ eV, $\lambda = 486$ nm (blue-green)
• $5 \to 2$: $\Delta E = 2.86$ eV, $\lambda = 434$ nm (violet)
• $6 \to 2$: $\Delta E = 3.02$ eV, $\lambda = 410$ nm (violet)

Paschen Series (to $n = 3$) — Infrared

• $4 \to 3$: $\Delta E = 0.66$ eV, $\lambda = 1875$ nm
• $5 \to 3$: $\Delta E = 0.97$ eV, $\lambda = 1282$ nm

AP Exam Tip

The Balmer series is the most commonly tested because it falls in the visible spectrum. Remember: Balmer → n = 2 → Visible.

Emission vs. Absorption Quiz

Spectral Line Calculation Drill

Use $E_n = -13.6/n^2$ eV and $\lambda = 1240/\Delta E$ (nm).

1. Energy of the photon emitted in the $n = 3 \to n = 2$ transition (in eV, 2 decimal places)
2. Wavelength of that photon (in nm, round to nearest whole number)
3. Energy of the photon emitted in the $n = 4 \to n = 1$ transition (in eV, 2 decimal places)
4. Wavelength of that photon (in nm, 1 decimal place)
5. Is the $4 \to 1$ photon UV, visible, or IR? (type: UV)

Advanced Transition Drill

A hydrogen atom in the $n = 4$ state can transition to several lower levels.

1. How many distinct spectral lines can be emitted by a collection of hydrogen atoms all starting in $n = 4$? (Use $N(N-1)/2$ for $N$ levels.)
2. Which transition produces the shortest wavelength photon? (Write as: 4to1)
3. Wavelength of the $4 \to 3$ transition (in nm, round to nearest whole number)

Exit Quiz — Atomic Transitions

☢️ Nuclear Physics Fundamentals

Part 5 of 7 — Inside the Nucleus

Atoms have a tiny, dense nucleus containing protons and neutrons held together by the strong nuclear force. Understanding nuclear structure is essential for radioactivity, nuclear energy, and AP Physics 2.

Nuclear Notation

A nucleus is described by:

$^{A}_{Z}X$

where:

• $X$ = element symbol
• $Z$ = atomic number = number of protons (defines the element)
• $A$ = mass number = total number of nucleons (protons + neutrons)
• Number of neutrons: $N = A - Z$

Examples

Isotopes

Isotopes are atoms of the same element (same $Z$) with different numbers of neutrons (different $A$):

• $^{12}_{6}\text{C}$ (6 neutrons) and $^{14}_{6}\text{C}$ (8 neutrons) are both carbon
• Same chemical properties, different nuclear properties
• Some isotopes are stable, others are radioactive

Nuclear Forces

The Problem

Protons are all positively charged → they repel each other via the electromagnetic (Coulomb) force. So why doesn't the nucleus fly apart?

The Strong Nuclear Force

The strong nuclear force holds nucleons together:

Key Points for AP

• The strong force is attractive and acts between all nucleon pairs
• It is short-range — only acts between neighboring nucleons
• In large nuclei, distant protons repel but the strong force cannot reach across the entire nucleus → large nuclei tend to be unstable
• Neutrons help: they contribute to the strong force without adding electromagnetic repulsion

Binding Energy & Mass-Energy Equivalence

Mass Defect

The mass of a nucleus is less than the sum of its individual protons and neutrons:

$\Delta m = (Zm_p + Nm_n) - m_{\text{nucleus}}$

This "missing mass" is the mass defect.

Where Did the Mass Go?

Einstein's mass-energy equivalence:

$E = mc^2$

The mass defect was converted to binding energy — the energy holding the nucleus together:

$E_b = \Delta m \cdot c^2$

Binding Energy per Nucleon

$\frac{E_b}{A} = \text{binding energy per nucleon}$

This tells us how tightly bound each nucleon is:

• Iron-56 ($^{56}\text{Fe}$) has the highest binding energy per nucleon (~8.8 MeV/nucleon) — the most stable nucleus
• Lighter nuclei: can fuse to move toward iron → releases energy
• Heavier nuclei: can fission to move toward iron → releases energy

Useful Conversion

$1 \text{ u} = 931.5 \text{ MeV/c}^2$

where 1 u = 1 atomic mass unit = $1.66 \times 10^{-27}$ kg.

Nuclear Physics Concept Quiz

Binding Energy Drill

The mass of $^{4}_{2}\text{He}$ is 4.0026 u. Use $m_p = 1.0073$ u, $m_n = 1.0087$ u, and 1 u = 931.5 MeV/$c^2$.

1. Total mass of 2 free protons + 2 free neutrons (in u, 3 significant figures)
2. Mass defect $\Delta m$ (in u, 3 significant figures)
3. Binding energy of $^{4}_{2}\text{He}$ (in MeV, 3 significant figures)
4. Binding energy per nucleon (in MeV, 3 significant figures)

Exit Quiz — Nuclear Physics

☢️ Radioactive Decay

Part 6 of 7 — Nuclear Transformations

Unstable nuclei spontaneously transform by emitting particles and energy. Each type of radioactive decay follows strict conservation laws — conserving charge, mass number, and lepton number.

Types of Radioactive Decay

Alpha Decay ($\alpha$)

The nucleus emits an alpha particle: $^{4}_{2}\text{He}$ (2 protons + 2 neutrons).

$^{A}_{Z}X \to ^{A-4}_{Z-2}Y + ^{4}_{2}\text{He}$

• $Z$ decreases by 2, $A$ decreases by 4
• Common in heavy nuclei (e.g., uranium, radium)
• Example: $^{238}_{92}\text{U} \to ^{234}_{90}\text{Th} + ^{4}_{2}\text{He}$

Beta-Minus Decay ($\beta^{-}$)

A neutron converts to a proton, emitting an electron and an antineutrino:

$n \to p + e^{-} + \bar{\nu}_e$

$^{A}_{Z}X \to ^{A}_{Z+1}Y + ^{\;\;0}_{-1}e + \bar{\nu}_e$

• $Z$ increases by 1, $A$ stays the same
• Occurs in neutron-rich nuclei
• Example: $^{14}_{6}\text{C} \to ^{14}_{7}\text{N} + ^{\;\;0}_{-1}e + \bar{\nu}_e$

Beta-Plus Decay ($\beta^{+}$)

A proton converts to a neutron, emitting a positron and a neutrino:

$p \to n + e^{+} + \nu_e$

$^{A}_{Z}X \to ^{A}_{Z-1}Y + ^{\;0}_{+1}e + \nu_e$

• $Z$ decreases by 1, $A$ stays the same
• Occurs in proton-rich nuclei
• Example: $^{11}_{6}\text{C} \to ^{11}_{5}\text{B} + ^{\;0}_{+1}e + \nu_e$

Gamma Decay ($\gamma$)

An excited nucleus emits a high-energy photon:

$^{A}_{Z}X^{*} \to ^{A}_{Z}X + \gamma$

• Neither $Z$ nor $A$ changes — just energy is released
• Often follows alpha or beta decay (daughter nucleus is in an excited state)

Conservation Laws in Nuclear Decay

Every nuclear reaction must conserve:

1. Conservation of Mass Number ($A$)

$\sum A_{\text{reactants}} = \sum A_{\text{products}}$

Total number of nucleons is conserved.

2. Conservation of Charge ($Z$)

$\sum Z_{\text{reactants}} = \sum Z_{\text{products}}$

Total charge (atomic number) is conserved.

3. Conservation of Lepton Number

• Electron ($e^-$) and neutrino ($\nu_e$): lepton number = $+1$
• Positron ($e^+$) and antineutrino ($\bar{\nu}_e$): lepton number = $-1$

In beta decay, lepton number is conserved (starts at 0, products sum to 0):

• $\beta^-$: $e^-$ (+1) and $\bar{\nu}_e$ (−1) → total = 0 ✓
• $\beta^+$: $e^+$ (−1) and $\nu_e$ (+1) → total = 0 ✓

4. Conservation of Energy and Momentum

The total mass-energy and momentum of the system are conserved. The kinetic energy of the products comes from the mass defect.

Radioactive Decay Quiz

Half-Life

The half-life ($t_{1/2}$) is the time for half of a radioactive sample to decay:

$N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

where:

• $N$ = number of remaining undecayed nuclei
• $N_0$ = initial number of nuclei
• $t$ = elapsed time
• $t_{1/2}$ = half-life

After Each Half-Life

Activity

The activity (decay rate) also halves every half-life:

$A = A_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

Activity is measured in becquerels (Bq) = decays per second.

Half-Life Calculation Drill

A radioactive isotope has a half-life of 8.0 days. You start with $6.4 \times 10^{20}$ atoms.

1. Number of atoms remaining after 24 days ($\times 10^{19}$)
2. Number of half-lives that have elapsed after 24 days
3. Fraction of the original sample that has decayed after 24 days (as a decimal)
4. If the initial activity is 2400 Bq, activity after 32 days (in Bq)

Round all answers to 3 significant figures.

Exit Quiz — Radioactive Decay

🧪 Synthesis & AP Review — Modern Physics

Part 7 of 7 — Fission, Fusion & Final Mastery

This final part covers nuclear reactions (fission and fusion), common AP exam mistakes, and a comprehensive mastery quiz spanning all of modern physics: photons, wave-particle duality, atomic models, and nuclear physics.

Nuclear Fission

Fission = a heavy nucleus splits into two lighter nuclei (plus neutrons and energy).

Example: Uranium-235

$^{235}_{92}\text{U} + ^{1}_{0}n \to ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3\,^{1}_{0}n + \text{energy}$

Key Features

• Triggered by absorbing a slow (thermal) neutron
• Releases ~200 MeV per fission event
• Products move toward iron on the binding energy curve → more tightly bound → energy released
• Released neutrons can trigger more fissions → chain reaction

Conservation Check

• Mass number: $235 + 1 = 141 + 92 + 3(1) = 236$ ✓
• Charge: $92 + 0 = 56 + 36 + 0 = 92$ ✓

Applications

• Nuclear reactors: controlled chain reaction, use control rods to absorb excess neutrons
• Nuclear weapons: uncontrolled chain reaction

Nuclear Fusion

Fusion = two light nuclei combine to form a heavier nucleus (plus energy).

Example: Hydrogen Fusion (in stars)

$^{2}_{1}\text{H} + ^{3}_{1}\text{H} \to ^{4}_{2}\text{He} + ^{1}_{0}n + 17.6 \text{ MeV}$

Key Features

• Products move toward iron on the binding energy curve → energy released
• Requires extremely high temperatures (~$10^7$ K) to overcome Coulomb repulsion between positive nuclei
• Powers the Sun and all main-sequence stars
• Releases more energy per nucleon than fission
• No long-lived radioactive waste (cleaner than fission)

Fission vs. Fusion Summary

Fission & Fusion Quiz

Common AP Mistakes to Avoid

Photoelectric Effect

• ❌ "Brighter light → faster electrons" → ✅ Brighter light → MORE electrons (same max KE)
• ❌ "Any light can eject electrons if bright enough" → ✅ Must be above threshold frequency
• ❌ "KE of photoelectrons depends on intensity" → ✅ KE depends only on frequency

Energy Levels

• ❌ "The electron orbits at any radius" → ✅ Only quantized orbits ($n = 1, 2, 3, \ldots$)
• ❌ "A 10.0 eV photon will excite hydrogen from $n = 1$ to $n = 2$" → ✅ Needs EXACTLY 10.2 eV
• ❌ "Higher $n$ = higher energy = more negative" → ✅ Higher $n$ = less negative = higher energy

Nuclear Physics

• ❌ "Beta decay changes $A$" → ✅ Beta decay keeps $A$ constant (changes $Z$ by ±1)
• ❌ "Gamma decay changes the element" → ✅ Gamma only releases energy ($Z$ and $A$ unchanged)
• ❌ "Fission and fusion both work with any nucleus" → ✅ Fission works for heavy nuclei, fusion for light nuclei

Half-Life

• ❌ "After 2 half-lives, all atoms have decayed" → ✅ After 2 half-lives, 1/4 remain
• ❌ "Half-life depends on how much sample you have" → ✅ Half-life is a fixed property of the isotope

AP FRQ-Style Problem

Light of wavelength 200 nm strikes a metal surface with work function $\phi = 4.20$ eV. Use $h = 6.63 \times 10^{-34}$ J·s, $c = 3.00 \times 10^{8}$ m/s, $1 \text{ eV} = 1.60 \times 10^{-19}$ J.

1. Photon energy in eV (3 significant figures)
2. Maximum KE of ejected electrons in eV (3 significant figures)
3. Stopping voltage in V (3 significant figures)
4. de Broglie wavelength of the fastest ejected electron ($\times 10^{-10}$ m, 3 significant figures). Use $m_e = 9.11 \times 10^{-31}$ kg.

Final Mastery Quiz — All Modern Physics