# Radioactive decay

> Physics · CIE A-Level AS
> Source: https://www.owlsprep.com/study/cie-9702-u11-radioactive-decay/

This module covers the core properties of radioactive decay, including the random and spontaneous nature of decay, balancing nuclear equations, the exponential decay law, and calculations involving decay constant and half-life.

**Prerequisites:** [Atomic and nuclear structure (proton, nucleon number)](https://www.owlsprep.com/study/cie-9702-u10-nuclear-structure/)

## Learning objectives

- Distinguish between spontaneous and random nature of radioactive decay
- Write balanced nuclear decay equations for alpha, beta and gamma decay
- Apply the exponential decay law to calculate activity, number of nuclei and decay constant
- Relate half-life to decay constant and solve related calculation problems

## Nature of Radioactive Decay

**Radioactive decay** — A spontaneous, random process where an unstable parent nucleus emits radiation to become more stable, and is unaffected by external physical conditions such as temperature or pressure.

It is critical to distinguish between the two key descriptors of radioactive decay that are regularly tested in CIE exams: spontaneous and random.

**Worked example:** Distinguish between the terms 'spontaneous' and 'random' when describing radioactive decay.

1. Define spontaneous decay:
2. Spontaneous means the decay process is not triggered by any external factor (e.g. temperature, pressure, chemical bonding) and cannot be controlled by external conditions.
3. Define random decay:
4. Random means that it is impossible to predict exactly when any single unstable nucleus will decay, only the probability of decay over a given time period.

> **tip**
>
> This question is extremely common in both multiple choice and structured questions – always define both terms separately when asked to distinguish.

## Nuclear Decay Equations

**Balanced nuclear decay equation** — An equation representing the transformation of an unstable parent nucleus, where total proton (atomic) number and total nucleon (mass) number are conserved on both sides of the reaction.

For CIE AS Level, you need to know three common decay modes:
1. Alpha ($\alpha$) decay: emits a $^4_2\alpha$ particle, proton number decreases by 2, mass number decreases by 4
2. Beta-minus ($\beta^-$) decay: a neutron decays to a proton, emits an electron and electron antineutrino, proton number increases by 1, mass number unchanged
3. Gamma ($\gamma$) decay: emits a high-energy photon, no change to proton or mass number

**Worked example:** Write the balanced nuclear equation for the alpha decay of uranium-238 ($^{238}_{92}\text{U}$).

1. Conserve mass number: total mass on left = 238, so $238 = A + 4 \rightarrow A = 234$
2. Conserve proton number: total proton number on left = 92, so $92 = Z + 2 \rightarrow Z = 90$
3. The element with proton number 90 is thorium (Th), so the full balanced equation is:
4. $$^{238}_{92}\text{U} \rightarrow ^{234}_{90}\text{Th} + ^{4}_{2}\alpha$$

## Exponential Decay Law

**Activity** — The rate of decay of a radioactive sample, equal to the number of decays per second. Activity is proportional to the number of undecayed nuclei remaining in the sample, and is measured in becquerels (Bq), where $1 \text{ Bq} = 1 \text{ decay per second}$.

*Notation:* A

The decay constant $\lambda$ is the probability of an individual nucleus decaying per unit time. The rate of decay is given by the differential equation:

$$\frac{dN}{dt} = -\lambda N$$

Integrating this gives the exponential decay law for the number of undecayed nuclei $N$ at time $t$, where $N_0$ is the initial number of undecayed nuclei:

$$N = N_0 e^{-\lambda t}$$

Since activity $A \propto N$, we also have:

$$A = A_0 e^{-\lambda t}$$

**Worked example:** A radioactive sample has an initial activity of 800 Bq. The decay constant is $0.02 \text{ s}^{-1}$. Calculate the activity after 100 seconds.

1. Write down the decay law for activity:
2. $$A = A_0 e^{-\lambda t}$$
3. Substitute the given values: $A_0 = 800 \text{ Bq}$, $\lambda = 0.02 \text{ s}^{-1}$, $t = 100 \text{ s}$
4. Calculate $\lambda t = 0.02 \times 100 = 2$
5. Compute the final activity:
6. $$A = 800 e^{-2} \approx 800 \times 0.135 = 108 \text{ Bq}$$

## Half-Life

**Half-life** — The time taken for half of the original number of undecayed nuclei in a sample to decay, or equivalently the time taken for the activity of a sample to decrease to half of its initial value.

*Notation:* t_{1/2}

We can derive the relationship between half-life and decay constant by setting $N = \frac{N_0}{2}$ at $t = t_{1/2}$:

$$\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$$

Cancel $N_0$ and take natural logarithms of both sides, leading to the core relation:

$$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$$

**Worked example:** A radioactive isotope has a half-life of 5.0 years. A pure sample has an initial mass of 100 g. Calculate (a) the decay constant, (b) the mass remaining after 15 years.

1. Part (a): Use the relation between $t_{1/2}$ and $\lambda$:
2. $$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{5.0} = 0.14 \text{ yr}^{-1} \ (2 \text{ s.f.})$$
3. Part (b): Calculate the number of half-lives elapsed:
4. $n = \frac{\text{total time}}{t_{1/2}} = \frac{15}{5} = 3$ half-lives
5. Fraction of mass remaining after $n$ half-lives = $(\frac{1}{2})^n$
6. $$\text{Mass remaining} = 100 \times (\frac{1}{2})^3 = 100 \times \frac{1}{8} = 12.5 \text{ g}$$

## Common pitfalls

- **Wrong:** Confusing spontaneous and random decay, stating 'spontaneous means you cannot predict when an individual nucleus will decay'
  - Why it fails: This mixes the definitions of the two terms, which is a common question prompt that requires separate definitions for full marks
  - Correct: Spontaneous = decay is not triggered by external factors; Random = cannot predict the decay time of an individual nucleus
- **Wrong:** Forgetting to add an electron antineutrino to beta-minus decay equations
  - Why it fails: CIE exam markers require all products to be included for full marks, even though mass and proton number are conserved without it
  - Correct: Always add $\bar{\nu}_e$ (electron antineutrino) to the product side of any $\beta^-$ decay equation
- **Wrong:** Misremembering the relation between half-life and decay constant, using $\lambda = \frac{t_{1/2}}{\ln 2}$
  - Why it fails: Longer half-life means a smaller decay constant, so this inverse relationship gives incorrect values
  - Correct: Recall $\lambda = \frac{\ln 2}{t_{1/2}}$, which gives smaller $\lambda$ for larger $t_{1/2}$, matching the physical meaning
- **Wrong:** Claiming 50 g of a 100 g sample with half-life 5 years remains after 10 years
  - Why it fails: This counts only one half-life instead of two half-lives for 10 years
  - Correct: After 10 years = 2 half-lives, fraction remaining = $(1/2)^2 = 1/4$, so 25 g remains
- **Wrong:** Mixing units, e.g. using half-life in minutes with decay constant calculated in per second
  - Why it fails: Unit mismatch leads to incorrect numerical results, even if the method is correct
  - Correct: Always match units of $t_{1/2}$, $\lambda$ and time $t$ before starting any calculation

## Cheatsheet

| Quantity/Property | Symbol | Formula/Rule | Unit |
| --- | --- | --- | --- |
| Activity | $A$ | $A = A_0 e^{-\lambda t} = \lambda N$ | Bq |
| Decay constant | $\lambda$ | $\lambda = \ln 2 / t_{1/2}$ | time$^{-1}$ |
| Half-life | $t_{1/2}$ | $t_{1/2} = \ln 2 / \lambda$ | time |
| Undecayed nuclei | $N$ | $N = N_0 e^{-\lambda t}$ | nuclei |
| Alpha decay change | $\Delta Z, \Delta A$ | $\Delta Z = -2, \Delta A = -4$ | - |
| Beta⁻ decay change | $\Delta Z, \Delta A$ | $\Delta Z = +1, \Delta A = 0$ | - |

## What's next

Now that you have mastered the fundamentals of radioactive decay, you can build on this knowledge to explore more advanced topics in A-Level nuclear physics. Radioactive decay is the foundation for understanding radioactive dating, which uses known half-lives of isotopes to estimate the age of ancient materials, as well as nuclear energy generation and radiation safety. You will also encounter more complex decay chains in further topics, where multiple sequential decays occur before a stable nucleus is formed, requiring you to apply the exponential decay law to multiple steps. Mastery of this sub-topic is also essential for any questions on binding energy and nuclear fission/fusion that rely on understanding nuclear transformations.

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