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IB Physics HL/Notes/E.3 Radioactive decay

IB Physics HLE.3 Radioactive decayNotes

Track Isotopes

Isotope questions are notation questions first. Compare Z to decide whether the element is the same, then compare A or A-Z to see whether the neutron number differs.

Isotopes are atoms of the same element with the same proton number Z.
They have different numbers of neutrons, so they have different nucleon numbers A.
Neutron number is A - Z.
Same Z means the same chemical element even if A changes.
Unstable isotopes are radionuclides that can undergo radioactive decay.

Sort each pair by isotope relationship.

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same isotope set
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different element
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not enough information
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Define isotope and use nuclear notation to compare two nuclides.

Using same mass number as the isotope condition.

Define isotope and use nuclear notation to compare two nuclides.

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Track Binding energy and mass defect

Mass defect is not missing matter; it is mass-equivalent energy. A bound nucleus has lower mass because energy was released when it formed. To break it apart, that same binding energy must be supplied.

The mass of a bound nucleus is less than the sum of the masses of its separated protons and neutrons.
The difference is the mass defect Δm.
Binding energy is E_b = Δmc².
Binding energy is released when the nucleus forms and must be supplied to separate it into individual nucleons.
In nuclear calculations, 1 u corresponds to about 931.5 MeV of energy.

Assemble the mass-defect to binding-energy calculation.

Formula
Target formula E_b = Δmc²
E_b
binding energy
J or MeV
Δm
mass defect
kg or u
c
speed of light
m s^-1
u
atomic mass unit
u
1Find mass of separated nucleons.Zmp + (A-Z)mn
2Subtract bound nucleus mass.Δm = mass separated nucleons - mass nucleus
3Convert mass defect to energy.E_b = Δmc²
4Use nuclear unit conversion if Δm is in u.1 u -> 931.5 MeV

Calculate binding energy from a mass defect.

Using the bound nucleus mass alone instead of the mass difference.

Calculate binding energy from a mass defect.

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Track Binding energy curve

The curve is a stability map. Energy is released when a nuclear reaction moves products upward on the binding-energy-per-nucleon curve.

Binding energy per nucleon is total binding energy divided by nucleon number A.
The graph peaks near iron-56, around the most stable nuclei.
Higher binding energy per nucleon means nucleons are more tightly bound.
Light nuclei can release energy by fusion because the product is closer to the peak.
Heavy nuclei can release energy by fission because the products are closer to the peak.
Nuclei near the peak do not release energy by ordinary fusion or fission toward greater stability.

Interpret energy release from the binding-energy curve.

Graph

Average binding energy per nucleon is plotted against nucleon number. The curve rises steeply for light nuclei, peaks near iron, then falls slowly for heavy nuclei.

1identify the peak near iron
2compare product and reactant positions
3link increased binding energy per nucleon to energy release

Use the binding-energy-per-nucleon graph to explain why fission and fusion can release energy.

Confusing total binding energy with average binding energy per nucleon.

Use the binding-energy-per-nucleon graph to explain why fission and fusion can release energy.

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Mass-energy equivalence

Mass-energy equivalence is the bridge from nuclear masses to released energy. The direction matters: if final mass is lower than initial mass, energy is released.

Einstein’s mass-energy equivalence is E = mc².
In nuclear processes, a decrease in mass corresponds to energy released.
A required energy input can increase the mass-energy of a system.
Because c² is very large, small mass changes correspond to large energy changes.
Mass differences may be handled in kg for joules or in atomic mass units for MeV.

Assemble E=mc² for nuclear mass changes.

Formula
Target formula E = Δmc²
E
energy equivalent of mass difference
J
Δm
mass difference
kg
c
speed of light
m s^-1
MeV
nuclear energy unit
MeV
1Find the mass difference.Δm = m_initial - m_final for released energy
2Use SI form if masses are in kg.E = Δmc²
3Convert to MeV if needed.1 MeV = 1.60 × 10^-13 J
4Interpret the sign physically.lower final mass -> energy released

Define mass-energy equivalence and apply it to a nuclear mass change.

Using E=mc² with the whole nucleus mass instead of the mass change.

Define mass-energy equivalence and apply it to a nuclear mass change.

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Track Strong nuclear force

The nucleus is a force-balance story. The electric force pushes protons apart, but the strong nuclear force binds nearby nucleons. Its short range helps explain why large nuclei need more neutrons.

Protons in a nucleus repel each other electrically.
The strong nuclear force is attractive between neighbouring nucleons at nuclear distances.
It is much stronger than the electric force at very short range.
It is short range, so it does not act significantly beyond nuclear dimensions.
Neutrons add strong-force attraction without adding electric repulsion.

Sort each nuclear-force statement.

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strong nuclear force
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electric force
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stability consequence
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Explain why the strong nuclear force is needed for nuclear stability.

Only saying nuclei are stable because protons and neutrons touch.

Explain why the strong nuclear force is needed for nuclear stability.

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Track Random radioactive decay

Random does not mean patternless for a large sample. It means individual nuclei cannot be timed, while the sample follows predictable half-life or exponential behaviour.

Radioactive decay is spontaneous: it is not triggered by ordinary external conditions.
For an individual unstable nucleus, the exact decay time cannot be predicted.
The probability of decay per unit time is constant for a given nuclide.
For a large sample, the overall decay pattern is statistically predictable.
A radioactive nucleus decays into a different nuclear state or nuclide that is more stable.

Sort claims about random decay.

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true for one nucleus
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true for a large sample
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Describe radioactive decay as a random and spontaneous process.

Saying random means half-life cannot be defined.

Describe radioactive decay as a random and spontaneous process.

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Track Decay nuclear changes

Decay equations are conservation puzzles. Write A and Z totals on both sides and choose the missing particle or daughter nucleus that balances both.

Alpha decay emits a helium-4 nucleus, so A decreases by 4 and Z decreases by 2.
Beta-minus decay converts a neutron into a proton, so A is unchanged and Z increases by 1.
Beta-plus decay converts a proton into a neutron, so A is unchanged and Z decreases by 1.
Gamma decay emits a high-energy photon from an excited nucleus, so A and Z are unchanged.
Decay equations must conserve total nucleon number A and proton number Z.

Sort each nuclear change by decay type.

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alpha
0
beta-minus
0
beta-plus
0
gamma
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State the changes in A and Z for alpha, beta-minus, beta-plus, and gamma decay.

Changing A during beta decay.

State the changes in A and Z for alpha, beta-minus, beta-plus, and gamma decay.

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Track Decay equations

Practice

Complete decay equations by balancing A first and Z second. If it is beta decay, include the correct neutrino partner when the syllabus context asks for it.

In any nuclear decay equation, total nucleon number A is conserved.
Total proton number Z is also conserved.
Alpha particles are written as helium-4 nuclei.
Beta-minus particles are electrons and beta-plus particles are positrons.
Beta-minus decay also emits an antineutrino; beta-plus decay emits a neutrino.
Gamma emission may be included as a photon with no change to A or Z.

Match each emitted particle to its equation role.

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Reasons
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Complete a nuclear decay equation.

Balancing mass number but not proton number.

Complete a nuclear decay equation.

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Track Neutrinos and antineutrinos

The beta particle alone is not the whole beta-decay story. Include the neutrino partner and know which decay uses which one.

In beta-minus decay, a neutron changes into a proton and emits an electron plus an antineutrino.
In beta-plus decay, a proton changes into a neutron and emits a positron plus a neutrino.
Neutrinos and antineutrinos carry away energy and momentum.
Their inclusion helps conservation laws hold in beta decay.
HL also uses the continuous beta spectrum as evidence that another particle shares the decay energy.

Sort beta-decay products and ideas.

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beta-minus
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beta-plus
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both beta decays
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Describe beta-minus and beta-plus decay, including neutrinos.

Omitting neutrino or antineutrino from beta decay.

Describe beta-minus and beta-plus decay, including neutrinos.

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Radiation penetration and ionization

Radiation risk depends on context. Alpha is dangerous inside the body despite low penetration. Gamma is hard to shield despite lower ionisation per interaction.

Alpha radiation is a helium nucleus; it is strongly ionising and weakly penetrating.
Alpha is stopped by paper, skin, or a few centimetres of air, but dangerous if inhaled or ingested.
Beta radiation has moderate ionising ability and penetration; it is stopped by thin aluminium.
Gamma radiation is a high-energy photon; it is weakly ionising but highly penetrating.
Gamma requires thick lead or concrete shielding to reduce intensity significantly.

Sort radiation properties by type.

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alpha
0
beta
0
gamma
0

Compare the penetration and ionising ability of alpha, beta, and gamma radiation.

Confusing penetrating power with ionising ability.

Compare the penetration and ionising ability of alpha, beta, and gamma radiation.

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Track Activity, count rate and half-life

Activity belongs to the sample; count rate belongs to the detector. Half-life can be read from corrected count rate because count rate is proportional to activity under fixed geometry.

Activity is the number of nuclear decays per second and is measured in becquerels, Bq.
Count rate is the number of counts recorded per second by a detector.
Count rate may be less than activity because detectors do not catch every emission.
Half-life is the time for the number of undecayed nuclei in a sample to halve.
Because activity is proportional to the number of undecayed nuclei, activity also halves every half-life.
Measured count rate should be corrected for background radiation when determining half-life.

Match each decay measurement term to its meaning.

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Define activity, count rate, and half-life.

Equating raw detector count rate directly with activity.

Define activity, count rate, and half-life.

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Track Half-life changes

Half-life is repeated halving, not repeated subtraction. On a graph, choose two points where activity halves and read the time difference.

After one half-life, half the undecayed nuclei remain.
After n whole half-lives, the remaining fraction is (1/2)^n.
Activity and corrected count rate halve with the same half-life as the number of undecayed nuclei.
A decay curve is exponential, not linear.
For times that are not whole-number multiples of half-life, use exponential decay at HL.

Use the decay curve to apply half-life.

Graph

A corrected count-rate curve starts at 800 counts per second and halves every 5 minutes.

1read the time for a halving
2count whole half-life intervals
3apply repeated halving

Determine half-life from a decay curve and predict activity after whole half-lives.

Treating decay as a straight-line decrease.

Determine half-life from a decay curve and predict activity after whole half-lives.

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Background radiation

Background correction is not optional in count-rate experiments. If the raw count has a constant background added, it will not halve correctly even when the source activity does.

Background radiation comes from sources such as cosmic rays, rocks, radon, food, and medical or industrial sources.
A detector records background counts even when no experimental source is present.
Measure background count rate without the source under the same conditions.
Corrected count rate = measured count rate - background count rate.
Half-life calculations should use corrected count rate, not raw count rate.

Sort background radiation claims.

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background source
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correction step
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incorrect step
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Explain how background radiation affects count-rate measurements.

Using raw count rate directly for half-life.

Explain how background radiation affects count-rate measurements.

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Evidence for strong force

This HL card asks for evidence, not just the name of the force. Stable nuclei and binding energy show a strong attractive interaction at short distances.

Protons repel each other electrically, so a nucleus of multiple protons would not be stable without another attractive force.
Many nuclei are stable, showing an attractive force acts between nucleons at nuclear distances.
Large binding energies show that substantial energy is required to separate nucleons.
The force must be short range because its effects are confined to nuclear scales.
The need for extra neutrons in larger stable nuclei is consistent with short-range attraction plus long-range proton repulsion.

Match each strong-force evidence cue to the inference.

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Reasons
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Describe evidence for the strong nuclear force.

Only defining the strong force without evidence.

Describe evidence for the strong nuclear force.

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Neutron-proton ratio

The neutron-proton ratio explains why stability is not just about total size. Too many neutrons or too many protons moves a nucleus away from the stability band, and beta decay can move it back.

Light stable nuclei often have neutron number close to proton number.
Heavier stable nuclei need more neutrons than protons.
Extra neutrons add strong-force attraction without adding electric repulsion.
Neutron-rich nuclei tend to undergo beta-minus decay, converting a neutron to a proton.
Proton-rich nuclei may undergo beta-plus decay, converting a proton to a neutron.
The stability band is a qualitative guide to likely decay direction.

Sort stability-band statements.

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stable trend
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neutron-rich
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Explain the role of neutron-to-proton ratio in nuclear stability.

Saying all stable nuclei have equal proton and neutron numbers.

Explain the role of neutron-to-proton ratio in nuclear stability.

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Track Binding energy above A≈60

This card zooms in on the heavy side of the binding-energy curve. Heavy nuclei can release energy by moving products upward toward the peak.

The binding-energy-per-nucleon curve peaks near A around 56 to 60.
For nuclei much heavier than this region, average binding energy per nucleon is lower.
If a heavy nucleus splits into medium-mass nuclei closer to the peak, the products are more tightly bound per nucleon.
The increase in total binding energy corresponds to energy released.
This is the E.3 graph basis for fission energy, used again in E.4.

Interpret the heavy side of the binding-energy curve.

Graph

A heavy nucleus lies on the slowly falling right-hand side of the average binding energy per nucleon graph. Its fission products lie closer to the iron-region peak.

1locate heavy nucleus and products on the curve
2compare binding energy per nucleon
3connect increase in binding energy to released energy

Use the binding-energy-per-nucleon graph to explain why a heavy nucleus can release energy by fission.

Saying fission releases energy simply because a nucleus splits.

Use the binding-energy-per-nucleon graph to explain why a heavy nucleus can release energy by fission.

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Explain Discrete nuclear energy levels

This is the nuclear analogue of atomic line spectra, but the transitions are nuclear rather than electronic. Discrete emitted energies point to discrete nuclear levels.

Nuclei can exist in discrete energy states.
Gamma emission occurs when an excited nucleus transitions to a lower nuclear energy level.
Discrete gamma photon energies show that nuclear energy levels are discrete.
Alpha particles can also be emitted with discrete energies, supporting discrete nuclear energy levels.
For a gamma photon, transition energy can be related to frequency or wavelength using E = hf = hc/λ.

Connect nuclear spectra to level differences.

Formula
Target formula ΔE = hf = hc/λ
ΔE
nuclear energy-level difference
J or eV
h
Planck constant
J s
f
gamma photon frequency
Hz
c
speed of light
m s^-1
λ
gamma photon wavelength
m
1Identify the nuclear transition energy.ΔE = E_high - E_low
2Relate transition energy to photon frequency.ΔE = hf
3Relate it to wavelength.ΔE = hc/λ
4Interpret discrete emitted energies.discrete spectra -> discrete nuclear energy levels

Explain how alpha and gamma spectra provide evidence for discrete nuclear energy levels.

Calling the energy levels atomic instead of nuclear.

Explain how alpha and gamma spectra provide evidence for discrete nuclear energy levels.

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Track Beta spectrum and neutrino

The beta spectrum is an evidence argument. Continuous beta energies are not a measurement mistake; they show the decay energy is shared among more than two products.

Beta particles are emitted with a continuous range of kinetic energies.
If beta decay produced only a daughter nucleus and beta particle, energy and momentum conservation would imply a fixed beta energy for each transition.
The continuous spectrum shows that another particle shares the available energy.
The neutrino or antineutrino carries away variable energy and momentum.
This explained the beta spectrum while preserving conservation laws.

Sort beta-spectrum and neutrino claims.

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beta spectrum evidence
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neutrino role
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incorrect claim
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Explain why the continuous beta spectrum is evidence for the neutrino.

Saying beta decay violates conservation of energy.

Explain why the continuous beta spectrum is evidence for the neutrino.

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Track Radioactive decay law

This is the HL decay equation for arbitrary times. Whole half-life steps are a shortcut; the exponential law is the full model.

The number of undecayed nuclei in a sample follows N = N0e^(-λt).
N0 is the initial number of undecayed nuclei.
λ is the decay constant, with unit s^-1.
The exponential model applies to large samples statistically.
It allows calculations at any time, not only after whole numbers of half-lives.
The curve approaches zero asymptotically rather than reaching zero after a fixed time.

Assemble the exponential decay law.

Formula
Target formula N = N0e^(-λt)
N
number of undecayed nuclei at time t
N0
initial number of undecayed nuclei
λ
decay constant
s^-1
t
elapsed time
s
1Identify the initial number of undecayed nuclei.N0
2Use the decay constant for the nuclide.λ
3Insert elapsed time with consistent units.t
4Apply exponential decay.N = N0e^(-λt)

Use the radioactive decay law to calculate the number of undecayed nuclei after an arbitrary time.

Using repeated halving when the time is not a whole number of half-lives.

Use the radioactive decay law to calculate the number of undecayed nuclei after an arbitrary time.

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Track Decay constant meaning

λ is not just a fitting constant. It is a probability rate for one undecayed nucleus. The small-time approximation is useful, but only when λΔt is small.

The decay constant λ is the probability per unit time that an undecayed nucleus will decay.
Its unit is s^-1 if time is measured in seconds.
A larger λ means a greater chance of decay per unit time and a shorter half-life.
For a sufficiently small time interval Δt, the probability of decay is approximately λΔt.
This approximation works only when λΔt is much less than 1.

Sort statements about decay constant.

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decay constant meaning
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small-time approximation
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Define decay constant and state the condition for using λt as an approximate decay probability.

Treating λt as exact for large times.

Define decay constant and state the condition for using λt as an approximate decay probability.

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Activity equation

Practice

Activity is not a separate decay law. It is λ times the number of undecayed nuclei, so it decays exponentially with the same λ and same half-life.

Activity A is the decay rate of a radioactive sample.
For a sample with N undecayed nuclei, A = λN.
Initial activity is A0 = λN0.
Activity follows A = A0e^(-λt).
Activity and N have the same half-life because they are proportional.
A large sample of a short-lived isotope can have high activity because both N and λ matter.

Assemble activity equations.

Formula
Target formula A = λN
A
activity at time t
Bq
λ
decay constant
s^-1
N
number of undecayed nuclei
A0
initial activity
Bq
t
elapsed time
s
1Activity is decay rate.A = λN
2Initial activity follows from initial nuclei.A0 = λN0
3Substitute exponential N.A = A0e^(-λt)
4State proportional consequence.A and N share the same half-life

Calculate activity from decay constant and number of undecayed nuclei.

Using count rate as A without correction or detector context.

Calculate activity from decay constant and number of undecayed nuclei.

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Track Half-life and decay constant

The half-life relation comes from setting N/N0 = 1/2 in N=N0e^-λt. In practice, it is the conversion key between graph data and HL equations.

Half-life T1/2 is the time for N or activity to fall to half its initial value.
For exponential decay, T1/2 = ln2/λ.
Equivalently, λ = ln2/T1/2.
A larger decay constant means a shorter half-life.
Time units must be consistent: if T1/2 is in seconds, λ is in s^-1.
This relation allows conversion between half-life data and exponential-decay equations.

Assemble the half-life decay-constant relation.

Formula
Target formula T1/2 = ln2/λ
T1/2
half-life
s
λ
decay constant
s^-1
ln2
natural log of 2
N
number of undecayed nuclei
1Start from exponential decay.N = N0e^(-λt)
2At one half-life, N=N0/2.1/2 = e^(-λT1/2)
3Solve for half-life.T1/2 = ln2/λ
4Rearrange if half-life is known.λ = ln2/T1/2

Convert between half-life and decay constant.

Forgetting the ln2 factor or mixing minutes and seconds.

Convert between half-life and decay constant.

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Retrieve the Core E.3 Radioactive decay Model

Review

This retrieval card covers the SL spine of E.3: identify the nuclide, calculate nuclear energy, choose the decay type, and interpret measurements correctly.

Isotopes have the same proton number but different neutron numbers and mass numbers.
Mass defect gives binding energy through E=Δmc²; 1 u corresponds to about 931.5 MeV.
Binding energy per nucleon peaks near iron; moving toward the peak releases energy.
The strong nuclear force binds nucleons at short range despite proton-proton repulsion.
Alpha, beta-minus, beta-plus, and gamma decays have specific A and Z changes, and beta decays include neutrino partners.
Activity is decays per second in Bq; half-life is the time for N or activity to halve; background count must be subtracted from raw count rate.

Match each core E.3 cue to the correct model statement.

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Reasons
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Summarize the core E.3 radioactive-decay model.

Listing radiation names without nuclear changes or measurement definitions.

Summarize the core E.3 radioactive-decay model.

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Retrieve the HL E.3 Radioactive decay Model

Review

HL E.3 adds evidence and arbitrary-time calculation. The evidence cards ask why a model was needed; the equation cards ask for consistent units and exponential reasoning.

Evidence for the strong force includes stable nuclei despite proton repulsion and large binding energies.
The neutron-to-proton ratio controls stability; heavy stable nuclei need more neutrons than protons.
Discrete alpha and gamma spectra provide evidence for discrete nuclear energy levels.
The continuous beta spectrum is evidence for the neutrino or antineutrino sharing energy and momentum.
The decay constant λ is probability per unit time; for sufficiently small Δt, decay probability is approximately λΔt.
Exponential decay uses N=N0e^(-λt), A=λN, A=A0e^(-λt), and T1/2=ln2/λ.

Match each HL E.3 cue to its model or equation.

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Reasons
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Summarize the HL E.3 radioactive-decay model.

Using equations without explaining evidence for neutrinos or nuclear levels.

Summarize the HL E.3 radioactive-decay model.

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