. ݁+ ⊹ general physics 2 ; "exam 3"

last hurrah!!!

☆ Blackbody Radiation

• Blackbody radiation = electromagnetic radiation emitted by hot objects.

• Blackbody

  • Ideal object that absorbs all radiation falling on it.

  • Also a perfect emitter.

  • Its spectrum depends only on its temperature.

• Color vs Temperature

  • At about 1000 K: objects glow red.

    • Lower frequency, longer wavelength.

  • Above about 2000 K: objects glow yellow/white.

    • Higher frequency, shorter wavelength.

  • Big idea: as temperature increases, the frequency of emitted radiation increases.

☆ Planck Hypothesis & Photons

• Planck’s hypothesis

  • Energy in electromagnetic radiation is not continuous.

  • It comes in discrete chunks called quanta.

• Energy quantum / photon

  • Energy of radiation:

    • E = nhf

      • n = integer number of quanta

      • h = Planck’s constant

      • f = frequency

  • Single photon energy:

    • E = hf = hc/λ

• Photon

  • A photon is a packet of electromagnetic energy.

  • It is not a classical particle with rest mass.

  • Higher frequency = higher photon energy.

☆ Photoelectric Effect — Basic Idea

• Photoelectric effect

  • When light shines on a metal surface, electrons are emitted from the surface.

• Why it matters

  • It gives strong evidence that light energy is carried in photons.

  • Electrons are only emitted if each photon has enough energy to overcome the metal’s work function.

☆ Photoelectric Effect — Experimental Setup

• Apparatus

  • Two metal plates/electrodes in a vacuum.

  • A galvanometer measures current.

  • A battery creates a potential difference.

• Basic process

  • Light shines on the emitting plate.

  • Electrons are ejected.

  • Electrons travel to the opposite plate.

  • A current is detected.

☆ Ionization Energy

• Ionization energy = energy needed to remove an electron completely from the atom.

• Removing an electron means taking it to n = infinity.

• At n = infinity, energy is 0 eV.

• Ionization energy from level n:

  • Eion = 0 − En

☆ Hydrogen-Like Ions

• Hydrogen-like ions have only one electron.

• Examples:

  • H has Z = 1

  • He+ has Z = 2

  • Li2+ has Z = 3

  • Be3+ has Z = 4

• They use the Bohr formulas with Z² included.

• Higher Z means stronger attraction and larger energy gaps.

☆ Energy Transitions

• An electron moving from high n to low n emits a photon.

• An electron moving from low n to high n absorbs energy.

• Photon energy equals the energy difference:

  • Ephoton = Ei − Ef for emission

  • Ephoton = Ef − Ei for absorption

• Since photon energy is positive, use the magnitude of the energy difference.

☆ Emission vs Absorption

• Emission

  • Electron drops to a lower energy level.

  • Photon is released.

• Absorption

  • Electron jumps to a higher energy level.

  • Photon energy is absorbed.

• The photon energy must exactly match the energy gap.

☆ Rydberg Formula

• The Rydberg formula finds wavelengths of spectral lines.

• Formula for hydrogen-like atoms:

  • 1/λ = RZ²(1/nf² − 1/ni²)

• ni = initial level.

• nf = final level.

• R = 1.097 × 10⁷ m⁻¹

• For emission, ni > nf.

• For absorption, ni < nf, but use the energy gap.

☆ Spectral Series

• A spectral series is a group of transitions ending at the same final level.

• Series names for hydrogen:

  • Lyman series: transitions ending at n = 1.

  • Balmer series: transitions ending at n = 2.

  • Paschen series: transitions ending at n = 3.

• The final level determines the series.

☆ Lyman Series

• Lyman series = transitions ending at n = 1.

• These photons are high energy.

• They are usually in the ultraviolet region.

• Formula setup:

  • nf = 1

☆ Balmer Series

• Balmer series = transitions ending at n = 2.

• Some Balmer lines are visible.

• Formula setup:

  • nf = 2

• The shortest wavelength in the Balmer series happens when:

  • ni = infinity

☆ Paschen Series

• Paschen series = transitions ending at n = 3.

• These photons have lower energy than Lyman and Balmer transitions.

• They are usually infrared.

• Formula setup:

  • nf = 3

☆ Shortest Wavelength in a Series

• Shortest wavelength means highest energy photon.

• Highest energy transition in a series comes from:

  • ni = infinity

• Use Rydberg formula with:

  • 1/ni² = 0

• Higher energy = shorter wavelength.

☆ Longest Wavelength in a Set of Transitions

• Longest wavelength means lowest energy photon.

• Lowest energy photon comes from the smallest energy gap.

• For an electron starting at n = 4, the smallest drop is usually:

  • 4 → 3

• Smaller energy gap = lower frequency = longer wavelength.

☆ Maximum Number of Photons from Level n

• If an electron starts at level n, the maximum possible number of different photon energies is:

  • n(n − 1)/2

• Example: from n = 4:

  • 4(3)/2 = 6 possible photons

• This counts all possible downward transitions.

☆ Bohr Radius

• Allowed orbit radius for hydrogen-like atoms:

  • rn = a₀(n²/Z)

• a₀ = 5.29 × 10⁻¹¹ m

• For hydrogen, Z = 1, so:

  • rn = a₀n²

• Radius increases with n².

• Higher energy levels are farther from the nucleus.

☆ Electron Speed in Bohr Orbit

• Electron speed can be found using angular momentum quantization:

  • mvr = nh/(2π)

• Solve for speed:

  • v = nh/(2πmr)

• Higher Z generally means stronger attraction and faster electron speeds for the same n.

☆ X-Ray Production

• X-rays can be produced when fast electrons hit a metal target and slow down.

• Lost kinetic energy becomes photon energy.

• If all kinetic energy becomes one photon:

  • qV = hc/λ

• Minimum accelerating voltage:

  • V = hc/(qλ)

☆ Characteristic X-Rays

• Characteristic X-rays happen when inner-shell electrons are removed and outer electrons fall into the vacancy.

• The emitted photon has energy equal to the energy difference between atomic levels.

• Each element produces characteristic X-ray wavelengths.

☆ X-Ray Tube Voltage

• An electron accelerated through voltage V gains kinetic energy:

  • KE = qV

• If this energy becomes an X-ray photon:

  • qV = hc/λ

• Higher voltage produces higher energy X-rays and shorter wavelengths.

☆ Nucleus Basics

• The nucleus contains protons and neutrons.

• Protons have positive charge.

• Neutrons have no charge.

• The nucleus is tiny, dense, and contains most of the atom’s mass.

• Electrons occupy the space outside the nucleus.

☆ Atomic Number, Z

• Atomic number, Z = number of protons.

• Determines the element.

• Example:

  • Carbon has Z = 6.

  • Uranium has Z = 92.

• Changing Z changes the element.

☆ Mass Number, A

• Mass number, A = protons + neutrons.

• Formula:

  • A = Z + N

• N = number of neutrons.

• To find neutrons:

  • N = A − Z

☆ Isotopes

• Isotopes are atoms of the same element with different numbers of neutrons.

• Same Z, different A.

• Example:

  • Carbon-12 and carbon-14 both have 6 protons.

  • Carbon-14 has more neutrons.

☆ Nuclide Notation

• Nuclear notation shows mass number and atomic number:

  • A over Z X

• X = element symbol.

• A = mass number.

• Z = atomic number.

• Example: carbon-14 has:

  • A = 14

  • Z = 6

☆ Strong Nuclear Force

• The strong nuclear force holds protons and neutrons together in the nucleus.

• It is very strong but acts only over very short distances.

• It overcomes repulsion between protons.

• Without it, the nucleus would fly apart.

☆ Mass Defect

• Mass defect = missing mass when nucleons bind together.

• A nucleus has less mass than the total mass of its separated protons and neutrons.

• Formula:

  • Δm = mass of separated nucleons − mass of nucleus

• Missing mass becomes binding energy.

☆ Binding Energy

• Binding energy = energy needed to separate a nucleus into individual protons and neutrons.

• It is also the energy released when the nucleus forms.

• Formula:

  • BE = Δmc²

• Using atomic mass units:

  • BE = Δm(931.5 MeV/u)

☆ Binding Energy per Nucleon

• Binding energy per nucleon measures nuclear stability.

• Formula:

  • BE per nucleon = total binding energy / A

• Higher binding energy per nucleon usually means a more stable nucleus.

• Iron/nickel region has very high stability.

☆ Atomic Mass vs Nuclear Mass

• Atomic mass includes electrons.

• Nuclear mass does not include electrons.

• To find nuclear mass:

  • nuclear mass = atomic mass − electron masses

• Electron mass in u:

  • me = 0.000548 u

☆ Using Atomic Masses in Nuclear Reactions

• Atomic masses often work directly if electrons cancel on both sides.

• For beta-minus decay, atomic electron masses usually cancel.

• For beta-plus decay, be careful because positron/electron counting matters.

• If unsure, use nuclear masses.

☆ Radioactivity

• Radioactivity = spontaneous decay of unstable nuclei.

• Unstable nuclei emit particles or energy to become more stable.

• Decay is random for individual nuclei but predictable statistically for large samples.

☆ Alpha Decay

• Alpha decay emits an alpha particle.

• Alpha particle = helium nucleus:

  • 4 over 2 He

• Changes:

  • A decreases by 4

  • Z decreases by 2

• General form:

  • Parent → daughter + alpha particle

☆ Beta-Minus Decay

• Beta-minus decay emits an electron.

• A neutron turns into a proton.

• Changes:

  • A stays the same

  • Z increases by 1

• General form:

  • neutron → proton + electron + antineutrino

• In nuclear notation:

  • Parent → daughter + beta-minus particle

☆ Beta-Plus Decay

• Beta-plus decay emits a positron.

• A proton turns into a neutron.

• Changes:

  • A stays the same

  • Z decreases by 1

• General form:

  • proton → neutron + positron + neutrino

☆ Gamma Decay

• Gamma decay emits high-energy electromagnetic radiation.

• It changes the energy state of the nucleus.

• Changes:

  • A does not change

  • Z does not change

• Gamma rays often occur after alpha or beta decay.

☆ Balancing Nuclear Equations

• Nuclear equations must conserve:

  • Mass number, A

  • Atomic number, Z

  • Charge

• Add A values on both sides.

• Add Z values on both sides.

• Both totals must match.

☆ Half-Life

• Half-life = time required for half of the radioactive nuclei to decay.

• After 1 half-life: 1/2 remains.

• After 2 half-lives: 1/4 remains.

• After 3 half-lives: 1/8 remains.

• Half-life is constant for a given isotope.

☆ Decay Constant

• Decay constant, λ measures probability of decay per unit time.

• Formula:

  • λ = 0.693/t1/2

• Larger λ means faster decay.

• Shorter half-life means larger λ.

☆ Radioactive Decay Law

• Number of nuclei remaining:

  • N = N₀e^(−λt)

• Activity also decays the same way:

  • A = A₀e^(−λt)

• If using half-lives:

  • N = N₀(1/2)^(t/t1/2)

  • A = A₀(1/2)^(t/t1/2)

☆ Activity

• Activity = number of decays per second.

• Formula:

  • A = λN

• SI unit:

  • Becquerel, Bq

• 1 Bq = 1 decay/s

☆ Curie

• Curie, Ci is another unit of activity.

• Conversion:

  • 1 Ci = 3.70 × 10¹⁰ Bq

• To convert Bq to Ci:

  • Ci = Bq / (3.70 × 10¹⁰)

☆ Carbon Dating

• Carbon dating uses carbon-14 decay to estimate the age of once-living material.

• Living organisms maintain a roughly constant carbon-14 ratio.

• After death, carbon-14 decays and is not replaced.

• Use activity decay:

  • A = A₀e^(−λt)

• Solve age:

  • t = −ln(A/A₀)/λ

☆ Number of Decays Between Two Times

• Activity changes with time:

  • A = A₀e^(−λt)

• Number of decays between t1 and t2 can be found from:

  • ΔN = (A1 − A2)/λ

• This works because A = λN.

☆ Energy Released in Nuclear Reactions

• Nuclear reaction energy comes from mass difference.

• Formula:

  • Δm = mass reactants − mass products

  • Q = Δm(931.5 MeV/u)

• If Δm > 0, energy is released.

• If Δm < 0, energy must be supplied.

☆ Q-Value

• Q-value = energy released or absorbed in a nuclear reaction.

• Formula:

  • Q = (mass reactants − mass products)c²

• Positive Q means exothermic/released energy.

• Negative Q means endothermic/energy required.

☆ Fission

• Fission = splitting a heavy nucleus into smaller nuclei.

• Usually releases energy because products have greater binding energy per nucleon.

• Often also releases neutrons.

• Neutrons can cause a chain reaction.

☆ Chain Reaction

• A chain reaction happens when neutrons from one fission event cause more fission events.

• If controlled, it can power a nuclear reactor.

• If uncontrolled, it can lead to explosive release of energy.

☆ Fusion

• Fusion = combining light nuclei to form a heavier nucleus.

• Fusion releases energy for light elements because products are more tightly bound.

• Fusion powers stars.

• Requires very high temperature/pressure because nuclei repel each other electrically.

☆ Alpha Particle Energy

• In alpha decay, released energy becomes kinetic energy of products.

• If recoil is ignored, alpha particle gets approximately all released energy.

• Use:

  • Q = KE = 1/2mv²

• Solve speed:

  • v = √(2KE/m)

☆ Beta Decay Energy with Atomic Masses

• For beta-minus decay, using atomic masses:

  • Q = [Mparent − Mdaughter](931.5 MeV/u)

• Electron masses cancel automatically.

• For beta-plus decay, electron/positron mass issues must be handled carefully.

• Safer method: use nuclear masses if the problem gives electron mass.

☆ Nuclear Stability

• Stability depends on the neutron-to-proton ratio.

• Light nuclei are stable near N = Z.

• Heavy nuclei need more neutrons than protons.

• Very heavy nuclei are often unstable and radioactive.

☆ Why Heavy Nuclei Decay

• Heavy nuclei have many protons.

• Proton-proton repulsion is large.

• Extra neutrons help, but beyond a point the nucleus becomes unstable.

• Decay helps move the nucleus toward a more stable