Radioactive decay | PhysicsRank

01 / Radioactive decay, half-life and mean life

Radioactive decay is unpredictable for one nucleus but predictable for a population

For a specified radioactive species in the constant-rate model, each surviving nucleus has probability approximately λΔt of decaying during a sufficiently short interval Δt. The decay constant λ has units of inverse time.

The probability is conditional on the nucleus still being present. It is not a rule that every nucleus waits for one fixed lifetime and then decays. Two otherwise identical nuclei can decay at very different times.

For a large population with N surviving parents, the expected loss rate is proportional to N: dN/dt = −λN. The minus sign describes a decreasing parent population. Actual small samples fluctuate around the statistical expectation.

The model assumes the decay constant stays fixed and there is no additional production or removal of parents. Nuclear decay constants are often stable under ordinary conditions, but the model should not be confused with a universal statement that every decay mode is completely unaffected by its environment.

\frac{dN}{dt}=-\lambda N,\qquad P(\text{decay in }\Delta t\mid\text{survival})\approx\lambda\Delta t

The probability approximation requires λΔt much smaller than one. For finite intervals, use the exponential survival law.

Explain this equation

Read it in words

The expected number of undecayed nuclei decreases at a rate proportional to how many remain. Over a sufficiently short interval, each surviving nucleus has decay probability approximately λ times that interval.

Symbols and units

$N$: Number of undecayed parent nuclei; the smooth law describes a large population or its expectation.Dimensionless count.
$\lambda$: Decay constant: probability rate for each surviving nucleus.s⁻¹, or the reciprocal of the chosen time unit.
$t$: Elapsed time.Seconds (s), consistent with λ.
$\Delta t$: A short positive time interval.Same time unit as t.
$P$: Conditional probability that a surviving nucleus decays in that interval.Dimensionless number between 0 and 1.

Signs and operations

The derivative is negative because parent nuclei disappear from this population. The vertical bar means “given that the nucleus has survived so far”. ≈ means an approximation; λΔt is not a valid large-time probability.

When it applies

Assume a constant decay constant and no new parent nuclei being added. The short-interval estimate needs λΔt much less than 1. Individual decays remain random even when the population law is smooth.

Make it concrete

For λ = 0.020 s⁻¹ and Δt = 0.10 s, the short-interval probability is about 0.002, or 0.2%. For 100 s the linear expression would give 2, which cannot be a probability; use 1 − exp(−λΔt) instead.

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Equation (LaTeX): \frac{dN}{dt}=-\lambda N,\qquad P(\text{decay in }\Delta t\mid\text{survival})\approx\lambda\Delta t
Read it as: The expected number of undecayed nuclei decreases at a rate proportional to how many remain. Over a sufficiently short interval, each surviving nucleus has decay probability approximately λ times that interval.
Symbols:
N: Number of undecayed parent nuclei; the smooth law describes a large population or its expectation.; Dimensionless count.
\lambda: Decay constant: probability rate for each surviving nucleus.; s⁻¹, or the reciprocal of the chosen time unit.
t: Elapsed time.; Seconds (s), consistent with λ.
\Delta t: A short positive time interval.; Same time unit as t.
P: Conditional probability that a surviving nucleus decays in that interval.; Dimensionless number between 0 and 1.
Signs and operations: The derivative is negative because parent nuclei disappear from this population. The vertical bar means “given that the nucleus has survived so far”. ≈ means an approximation; λΔt is not a valid large-time probability.
Conditions: Assume a constant decay constant and no new parent nuclei being added. The short-interval estimate needs λΔt much less than 1. Individual decays remain random even when the population law is smooth.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

A nuclide has a half-life of six hours. Does each individual nucleus decay exactly six hours after preparation?

Yes
No; six hours describes the population survival probability
Only if the sample is large

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Topic: Radioactive decay, half-life and mean life
Focus: Radioactive decay is unpredictable for one nucleus but predictable for a population
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
\frac{dN}{dt}=-\lambda N,\qquad P(\text{decay in }\Delta t\mid\text{survival})\approx\lambda\Delta t
Meaning: The probability approximation requires λΔt much smaller than one. For finite intervals, use the exponential survival law.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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02 / Radioactive decay, half-life and mean life

Equal time intervals remove the same fraction, not the same number

Separate variables in dN/dt = −λN and integrate from N0 at time zero. The result is ln(N/N0) = −λt, or N = N0e^(−λt). The surviving fraction depends only on λt, a dimensionless product.

In equal time intervals, the survival factor is the same. The number lost in later intervals is smaller because fewer parents remain. A straight-line decrease of N with time is therefore not the radioactive-decay law.

The probability that a particular parent survives to time t is e^(−λt). The probability that it has decayed by then is 1 − e^(−λt). These probabilities sum to one for a single irreversible parent decay.

A semilog plot of corrected parent number or activity has constant slope −λ. A linear plot of the number itself curves downward and approaches zero asymptotically in the expectation model.

N(t)=N_0e^{-\lambda t},\qquad P(T>t)=e^{-\lambda t},\qquad P(T\le t)=1-e^{-\lambda t}

The exponential gives expected population or survival probability. It need not be an integer when applied to a small initial sample.

WORKED EXAMPLE

Distinguish the fraction remaining from the fraction decayed

A species has decay constant 0.20 h⁻¹. Find the expected fraction remaining after 5.0 h and the fraction that has decayed.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Equal time intervals remove the same fraction, not the same number
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
N(t)=N_0e^{-\lambda t},\qquad P(T>t)=e^{-\lambda t},\qquad P(T\le t)=1-e^{-\lambda t}
Meaning: The exponential gives expected population or survival probability. It need not be an integer when applied to a small initial sample.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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03 / Radioactive decay, half-life and mean life

Half-life converts exponential decay into repeated halving

Set N/N0 = 1/2 in the decay law. Taking logarithms gives T1/2 = ln 2/λ. After t, the surviving fraction can therefore be written 2^(−t/T1/2).

After one half-life, half remains; after two, one quarter; after three, one eighth. The material lost during successive half-life intervals is N0/2, N0/4, N0/8 and so on.

The initial amount changes the absolute population and activity but does not change the half-life in this model. Splitting a sample in half halves its activity at that instant; it does not halve the nuclide’s half-life.

For a noninteger number of half-lives, keep the exponential power or logarithm. Linear interpolation between halves is generally incorrect because the population changes exponentially.

T_{1/2}=\frac{\ln2}{\lambda},\qquad\frac N{N_0}=2^{-t/T_{1/2}},\qquad t=\frac{\ln(N_0/N)}{\lambda}

Use the same time unit for t and T1/2. Convert λ to s⁻¹ when an activity in Bq is required.

WORKED EXAMPLE

Find both survivors and decays after several half-lives

An initially pure parent sample contains 8.0 × 10⁶ nuclei and has half-life 3.0 h. Find the expected parents remaining after 9.0 h and the number that have decayed.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Half-life converts exponential decay into repeated halving
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
T_{1/2}=\frac{\ln2}{\lambda},\qquad\frac N{N_0}=2^{-t/T_{1/2}},\qquad t=\frac{\ln(N_0/N)}{\lambda}
Meaning: Use the same time unit for t and T1/2. Convert λ to s⁻¹ when an activity in Bq is required.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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04 / Radioactive decay, half-life and mean life

Mean life and half-life describe different features of the distribution

The probability density for a decay time is p(t) = λe^(−λt) for t ≥ 0. The mean lifetime is the expectation ∫t p(t)dt, which evaluates to τ = 1/λ.

Equivalently, integrating the survival probability gives τ = ∫₀∞e^(−λt)dt. This provides an intuitive interpretation: expected lifetime is the area under the survival-probability curve.

The half-life is the median decay time because half the nuclei have decayed by then. The mean life is longer: τ = T1/2/ln 2 ≈ 1.443 T1/2. The long tail of late decays pulls the mean above the median.

At t = τ the surviving fraction is 1/e, about 36.8%, not one half. There is no finite time at which the continuous expectation abruptly becomes exactly zero, though a finite sample can of course eventually lose its final parent nucleus.

\tau=\int_0^\infty t\lambda e^{-\lambda t}\,dt=\frac1\lambda,\qquad T_{1/2}=\tau\ln2

τ is the mean of the exponential decay-time distribution; T1/2 is its median.

WORKED EXAMPLE

Translate a half-life into a mean life

A nuclide has half-life 6.0 h. Find its mean life and the surviving fraction at that mean life.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Mean life and half-life describe different features of the distribution
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
\tau=\int_0^\infty t\lambda e^{-\lambda t}\,dt=\frac1\lambda,\qquad T_{1/2}=\tau\ln2
Meaning: τ is the mean of the exponential decay-time distribution; T1/2 is its median.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Radioactive decay, half-life and mean life

A surviving nucleus does not acquire an expiry date

In the constant-λ model, the chance of surviving an additional time s given survival to age t is P(T > t + s)/P(T > t). The exponential factors cancel to give e^(−λs), independent of the elapsed age t.

This is the memoryless property. A parent nucleus that has survived several half-lives has the same conditional probability of surviving the next half-life as a newly selected surviving parent: one half.

The expected remaining lifetime of a survivor is still τ under this model. That does not imply the whole original sample has reset; the number of survivors has already decreased, so its total activity is smaller.

The memoryless property is specific to the exponential model. Ageing objects such as worn components do not automatically have the same survival law, and a mixture of nuclides need not look memoryless when their identities are ignored.

WORKED EXAMPLE

Condition on the nuclei that are still present

A species has half-life 4.0 h. A particular nucleus has already survived 12 h. What is its conditional probability of surviving another 4.0 h in the constant-rate model?

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: A surviving nucleus does not acquire an expiry date
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Radioactive decay, half-life and mean life

Activity counts decays per second, not nuclei remaining

Activity is the positive expected decay rate A = −dN/dt = λN. Its SI unit is the becquerel, where 1 Bq means one decay per second. The population N is a count, while activity is a rate.

For a single species with fixed λ, activity falls with the same exponential factor as its parent population: A(t) = A0e^(−λt). A ratio of activities can therefore determine elapsed time when both readings refer to the same species and detection conditions.

A detector records only some emitted radiation. A simple measured count-rate model is R = εbA + Rbackground, where ε is detection efficiency and b is the mean number or branching yield of the counted emissions per parent decay. Dead time or changing geometry can require further correction.

Activity is not absorbed dose and does not by itself specify a biological effect. In these calculations it is simply a nuclear decay rate. Keep it separate from energy per decay, detected counts and energy deposited in a material.

A=\lambda N,\qquad A(t)=A_0e^{-\lambda t},\qquad1\ \mathrm{Bq}=1\ \mathrm{s^{-1}}\text{ decay}

Convert λ to inverse seconds for Bq. Detection efficiency, emission yield and background are separate from the intrinsic activity.

WORKED EXAMPLE

Convert the half-life before reporting becquerels

A sample has 1.2 × 10¹² parent nuclei and half-life 12 h. Estimate its activity in Bq.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Activity counts decays per second, not nuclei remaining
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
A=\lambda N,\qquad A(t)=A_0e^{-\lambda t},\qquad1\ \mathrm{Bq}=1\ \mathrm{s^{-1}}\text{ decay}
Meaning: Convert λ to inverse seconds for Bq. Detection efficiency, emission yield and background are separate from the intrinsic activity.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Radioactive decay, half-life and mean life

Subtract background and account for a finite counting interval

If the same detector has constant efficiency and background, the corrected count rate R − Rbackground is proportional to parent activity. Ratios of those corrected rates can be used without knowing the efficiency, provided the emission yield also stays the same.

A background that is not subtracted makes a decaying signal appear to flatten at late times. Taking a logarithm of the raw rate can then give a false slope and an incorrect half-life.

Over a finite interval from t1 to t2, the expected number of parent decays is N(t1) − N(t2), or the integral of activity over that interval. Approximating counts by A(t1)(t2 − t1) is valid only when activity changes little during the interval.

Observed counts also fluctuate. For independent ideal counting events, a Poisson model gives standard deviation approximately √C for a count C, before additional background and instrumental uncertainty are combined. The deterministic exponential describes an expectation, not a noise-free measurement.

WORKED EXAMPLE

A fixed background changes the apparent decay ratio

A detector records total rates of 200 counts/s and 80 counts/s, separated by 12 min. The constant background is 40 counts/s, and all other detection factors are unchanged for a single nuclide. Find its half-life.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Subtract background and account for a finite counting interval
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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08 / Radioactive decay, half-life and mean life

A stable daughter count can record how many parents decayed

Suppose each parent decay produces one stable daughter, no daughters are initially present, and the system remains closed to parent and daughter transfer. Then Ndaughter = N0 − Nparent.

The daughter-to-parent ratio is therefore Nd/Np = e^(λt) − 1. Solving gives t = ln(1 + Nd/Np)/λ. This is a simple dating model based on the accumulated decay product.

Initial daughter material changes the equation, as does loss or gain of either species. A measured ratio alone does not establish an age unless those assumptions or corrections are justified.

If the daughter is itself radioactive, its population is affected by production and its own decay. The stable-daughter subtraction rule cannot be reused without including that extra loss term.

WORKED EXAMPLE

Recover an age from a closed stable-daughter model

A closed sample initially contained only a parent species. Each decay produces one stable daughter. The present daughter-to-parent count ratio is 7, and parent half-life is 5.0 h. Find the elapsed time.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: A stable daughter count can record how many parents decayed
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

09 / Radioactive decay, half-life and mean life

Several species do not usually make one exponential

For independent radioactive species, activities add: Atotal = ΣλiNi. If their decay constants differ, the total is a sum of exponentials and generally has no single constant half-life.

A short-lived component can dominate early measurements while a long-lived component dominates later. A semilog plot of total activity then bends rather than following one straight line.

Equal initial activity is different from equal initial parent population. Since A = λN, a shorter-lived species needs fewer nuclei to give the same initial activity. Specify which quantity is equal before comparing components.

In a parent–daughter chain, production of the daughter makes the equations coupled: dNd/dt = λpNp − λdNd. A daughter’s activity may initially rise even while the parent population falls, so a falling-only assumption can fail.

A mixture’s surviving activity does not follow one repeated halving

Inspect a plotted point: Time (h) = 0

2 h species: 0.5
8 h species: 0.5
Total: 1

Two independent species begin with equal activity. Their half-lives are 2 h and 8 h. Each plotted value is divided by the combined initial activity, so the component curves start at 0.5 and their sum starts at 1. Lines connect the plotted samples.

WORKED EXAMPLE

Add activities after applying each species’ own decay law

Two independent species each start with activity 100 Bq. Their half-lives are 2 h and 8 h. Find the combined activity after 8 h.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: Several species do not usually make one exponential
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

10 / Radioactive decay, half-life and mean life

A constant source can balance radioactive loss

If new parents are produced at constant rate P while existing ones decay, the population equation becomes dN/dt = P − λN. The production term changes the solution; pure exponential decay no longer applies during irradiation or feeding.

At steady state, production balances decay: Neq = P/λ and equilibrium activity equals P. Starting from zero population, N(t) = (P/λ)(1 − e^(−λt)).

Once production stops, the population present at that moment becomes the initial condition for ordinary exponential decay. A multi-stage problem therefore needs a growth stage and a decay stage joined with a continuous population.

The same balance structure appears in charging circuits and thermal relaxation. Recognizing the source-minus-loss form helps derive the result and check the initial slope and long-time limit.

WORKED EXAMPLE

Find the population during constant production

A hypothetical species is produced at 20 nuclei/s and has λ = 0.10 s⁻¹. It starts with zero population. Find the equilibrium expected population and expected population after 10 s.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Radioactive decay, half-life and mean life
Focus: A constant source can balance radioactive loss
Learning goals:
- Derive exponential survival from a constant decay probability rate.
- Distinguish fraction remaining from fraction decayed.
- Convert among decay constant, half-life and mean life.
- Explain conditional survival and the memoryless property.
- Calculate activity with correct inverse-time units.
- Correct background and integrate counts over finite intervals.
- Infer a model age only with stated daughter and closure assumptions.
- Handle mixtures and production without inventing one half-life.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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BRING THE IDEAS TOGETHER

What should stay with you

The exponential law describes survival probability and expected populations.
Equal time intervals remove equal fractions, not equal numbers.
Half-life is ln 2/λ; mean life is 1/λ and is longer.
A survivor’s conditional future probability is unchanged in the constant-rate model.
Activity is λN in decays per second; detector counts require efficiency and background accounting.
Stable-daughter dating needs initial-composition and closed-system assumptions.
Mixtures of decay constants generally do not have one exponential law.
With production, solve source minus loss and join stages with continuous population.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A sample has half-life 4 h. What fraction remains after 12 h?

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SELF CHECK 2

A nuclide has λ = 0.25 s⁻¹. What is its mean life?

Answer in s.

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SELF CHECK 3

A sample has 5000 parent nuclei and λ = 0.020 s⁻¹. What is its expected activity?

Answer in Bq.

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SELF CHECK 4

Under the initially pure closed stable-daughter model, daughter-to-parent ratio is 3. How many parent half-lives have elapsed?

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SELF CHECK 5

A species is produced at 30 nuclei/s and has λ = 0.15 s⁻¹. What equilibrium expected population follows?

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SELF CHECK 6

At one mean life, approximately what fraction remains?

One half
1/e, about 36.8%
Zero

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SELF CHECK 7

A sample of one nuclide is divided into two equal portions. What happens immediately to each portion?

Activity halves and half-life stays the same
Both activity and half-life halve
Activity stays the same and half-life doubles

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SELF CHECK 8

Why should a constant detector background be subtracted before fitting ln(count rate) against time?

It otherwise makes the late-time signal look artificially flat
It changes the nuclear decay constant physically
It guarantees there will be no statistical noise

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

Infer time from a remaining fractionIntermediate · Free sample · Hints and guided reasoning Two different decay timescalesIntermediate · Hints and guided reasoning After three half-lives, what fraction remains?Beginner · Hints and guided reasoning How many decays occur each second?Intermediate · Hints and guided reasoning Equal populations, unequal activitiesIntermediate · Hints and guided reasoning Removing material and waitingIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

If a nucleus has survived several half-lives, is it “due” to decay sooner in the constant-rate model?

Compare your reasoning

No. Conditional on survival, the model gives the same future decay probability over a fixed interval. Its elapsed lifetime does not make the next interval more likely.

Revisit the explanation →

A few days later: change the situation

Your detector count rate halves after subtracting background. What else must remain unchanged before interpreting this as one half-life of a single isotope?

Compare your reasoning

Detection efficiency, geometry and relevant attenuation must remain stable, and the signal must be described by one isotope without changing production or daughter contributions. A measured rate is an observation filtered through the apparatus.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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What you will learn to do

Bring back one idea

Radioactive decay is unpredictable for one nucleus but predictable for a population

Read it in words

Symbols and units

Signs and operations

When it applies

Make it concrete

A nuclide has a half-life of six hours. Does each individual nucleus decay exactly six hours after preparation?

Equal time intervals remove the same fraction, not the same number

Distinguish the fraction remaining from the fraction decayed

Half-life converts exponential decay into repeated halving

Find both survivors and decays after several half-lives

Mean life and half-life describe different features of the distribution

Translate a half-life into a mean life

A surviving nucleus does not acquire an expiry date

Condition on the nuclei that are still present

Activity counts decays per second, not nuclei remaining

Convert the half-life before reporting becquerels

Subtract background and account for a finite counting interval

A fixed background changes the apparent decay ratio

A stable daughter count can record how many parents decayed

Recover an age from a closed stable-daughter model

Several species do not usually make one exponential

A mixture’s surviving activity does not follow one repeated halving

Add activities after applying each species’ own decay law

A constant source can balance radioactive loss

Find the population during constant production

What should stay with you

Try it without the examples

A sample has half-life 4 h. What fraction remains after 12 h?

A nuclide has λ = 0.25 s⁻¹. What is its mean life?

A sample has 5000 parent nuclei and λ = 0.020 s⁻¹. What is its expected activity?

Under the initially pure closed stable-daughter model, daughter-to-parent ratio is 3. How many parent half-lives have elapsed?

A species is produced at 30 nuclei/s and has λ = 0.15 s⁻¹. What equilibrium expected population follows?

At one mean life, approximately what fraction remains?

A sample of one nuclide is divided into two equal portions. What happens immediately to each portion?

Why should a constant detector background be subtracted before fitting ln(count rate) against time?

Take the ideas into a problem

What can you explain later?

Next study session

A few days later: change the situation