# Radioactive decay

> IB Physics SL · Unit 5: Nuclear and quantum physics
> Source: https://www.owlsprep.com/study/ib-physics-sl-u5-radioactive-decay/

This sub-topic covers the spontaneous random nature of radioactive decay, key relationships between decay constant, half-life and activity, and how to solve quantitative exponential decay problems for IB Physics SL exams.

**Prerequisites:** [Nuclear structure and isotopes](https://www.owlsprep.com/study/ib-physics-sl-u5-nuclear-structure/)

## Learning objectives

- Distinguish between the spontaneous and random nature of radioactive decay
- Define decay constant, activity and half-life
- Derive and use the relationship $T_{1/2} = \frac{\ln 2}{\lambda}$
- Solve quantitative problems involving exponential radioactive decay

## Nature of Radioactive Decay

**Radioactive decay** — A spontaneous, random process where an unstable nucleus emits ionizing radiation to form a more stable daughter nucleus.

*Example:* Carbon-14 decays via beta-minus emission to form stable nitrogen-14.

Radioactive decay has two core properties that are regularly tested in IB exams. Spontaneous means the process is not triggered or changed by external conditions: temperature, pressure, and chemical bonding have no effect on decay rate. Random means we can only predict the probability of decay for a group of nuclei, never when an individual nucleus will decay.

> **info**
>
> Unlike chemical reaction rates, which change with temperature, radioactive decay rate is completely constant regardless of external conditions. This is why radiocarbon dating works reliably across thousands of years.

**Worked example:** Explain why radioactive decay is described as both spontaneous and random.

1. First, explain the meaning of spontaneous in this context:
2. Spontaneous means decay occurs without any external trigger or cause. It is not affected by external environmental factors like temperature, pressure, or chemical state.
3. Next, explain the meaning of random:
4. Random means it is impossible to predict exactly when any single individual nucleus will decay. We can only state the probability of decay for a nucleus over a given time interval.

## Decay Constant and Activity

**Decay constant** — The probability that a single undecayed nucleus will decay per unit time.

*Notation:* $\lambda$

*Example:* A decay constant of $0.1 \text{ s}^{-1}$ means a 10% chance of decay per second for each nucleus.

**Activity** — The total rate of decay of a radioactive sample, equal to the number of decays per second.

*Notation:* $A$

*Example:* 1 becquerel (Bq) = 1 decay per second.

The fundamental relationship for radioactive decay connects activity, decay constant, and the number of undecayed nuclei:

$$A = \lambda N$$

This leads to the exponential decay law for undecayed nuclei and activity over time:

$$N = N_0 e^{-\lambda t} \quad \text{and} \quad A = A_0 e^{-\lambda t}$$

**Worked example:** A radioactive sample contains $2.0 \times 10^{15}$ undecayed nuclei, and has a decay constant of $3.4 \times 10^{-7} \text{ s}^{-1}$. Calculate the activity of the sample in Bq.

1. Recall the relationship between activity, decay constant and number of undecayed nuclei:
2. $$A = \lambda N$$
3. Substitute the given values into the equation:
4. $$A = (3.4 \times 10^{-7} \text{ s}^{-1}) \times (2.0 \times 10^{15}) = 6.8 \times 10^8 \text{ Bq$$

## Half-Life

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

*Notation:* $T_{1/2}$

**Derivation:** Derive the relationship between half-life and decay constant

*Starting from:* The exponential decay law: $N = N_0 e^{-\lambda t}$

1. By definition, after one half-life $t = T_{1/2}$, $N = \frac{N_0}{2}$. Substitute into the decay law:
2. $$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$$
3. Cancel $N_0$ from both sides:
4. $$\frac{1}{2} = e^{-\lambda T_{1/2}}$$
5. Take the natural logarithm of both sides:
6. $$\ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} \implies -\ln 2 = -\lambda T_{1/2}$$
7. Rearrange to get the final relationship:
8. $$T_{1/2} = \frac{\ln 2}{\lambda}$$

*Conclusion:* Half-life is inversely proportional to decay constant: isotopes with faster decay (larger $\lambda$) have shorter half-lives.

**Worked example:** The half-life of strontium-90 is 28 years. Calculate the decay constant of strontium-90 in $\text{year}^{-1}$.

1. Rearrange the half-life relationship to solve for decay constant:
2. $$\lambda = \frac{\ln 2}{T_{1/2}}$$
3. Substitute the given value for half-life:
4. $$\lambda = \frac{0.693}{28} \approx 0.025 \text{ year}^{-1}$$

**Check your understanding**

Test your understanding of the half-life relationship:

1. What is the half-life of a sample with decay constant $0.1 \text{ s}^{-1}$?

   - 0.069 s
   - 6.9 s
   - 1.44 s
   - 14.4 s

   *Why:* Correct! $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{0.1} = 6.9 \text{ s}$. If you got 0.069, you inverted the relationship between half-life and decay constant.

## Solving Decay Problems

**Comparing methods**

There are two common methods to solve decay problems, each suited to different scenarios:

- **Counting half-lives** — Calculate the number of half-lives $n = \frac{\text{total time}}{T_{1/2}}$, then $A = \frac{A_0}{2^n}$ or $N = \frac{N_0}{2^n}$
  - Pros: Fast and simple; Low chance of calculation error
  - Cons: Only works for whole numbers of half-lives

- **Exponential decay equation** — Calculate $\lambda = \frac{\ln 2}{T_{1/2}}$, then substitute into $N = N_0 e^{-\lambda t}$ or $A = A_0 e^{-\lambda t}$
  - Pros: Works for any time, including non-whole half-lives
  - Cons: More steps, higher chance of unit or calculation error

**Worked example:** A sample of iodine-131 has an initial activity of 800 Bq. The half-life of iodine-131 is 8 days. Calculate the activity after 32 days.

1. Method 1 (counting half-lives): Calculate number of half-lives:
2. $$n = \frac{32 \text{ days}}{8 \text{ days per half-life}} = 4$$
3. Calculate final activity:
4. $$A = \frac{A_0}{2^n} = \frac{800}{2^4} = \frac{800}{16} = 50 \text{ Bq}$$
5. Method 2 (exponential decay equation, for confirmation):
6. $$\lambda = \frac{\ln 2}{8} \approx 0.0866 \text{ day}^{-1}$$
7. $$A = 800 e^{-(0.0866)(32)} = 800 e^{-2.77} \approx 50 \text{ Bq}$$

## Common pitfalls

- **Wrong:** Defining spontaneous decay as decay that happens quickly.
  - Why it fails: Spontaneous does not refer to the speed of decay, it refers to the process not being triggered by external factors.
  - Correct: Define spontaneous decay as decay that occurs without external cause, and is unaffected by external conditions like temperature or pressure.
- **Wrong:** Using base 10 exponents instead of base $e$ in the decay equation.
  - Why it fails: IB Physics always uses the natural exponential form of the decay law, so mixing up bases will give wrong results.
  - Correct: Always use $N = N_0 e^{-\lambda t}$ and remember $T_{1/2} = \frac{\ln 2}{\lambda}$.
- **Wrong:** Mixing up units for decay constant and time $t$.
  - Why it fails: Decay constant must have the same time unit as the time input to the decay equation, or results will be orders of magnitude wrong.
  - Correct: Always convert decay constant to match the time unit of $t$ before substituting into the decay equation.
- **Wrong:** Assuming half-life is half the time it takes for all nuclei to decay.
  - Why it fails: Half-life is a constant property of an isotope, it does not depend on the original number of nuclei.
  - Correct: Remember half-life is always the time for half the current number of undecayed nuclei to decay.

## Cheatsheet

| Quantity | Symbol | Key Relationship | Units |
| --- | --- | --- | --- |
| Undecayed nuclei | $N$ | $N = N_0 e^{-\lambda t}$ | Unitless (count) |
| Activity | $A$ | $A = \lambda N = A_0 e^{-\lambda t}$ | Becquerel (Bq) |
| Decay constant | $\lambda$ | $\lambda = \frac{\ln 2}{T_{1/2}}$ | Time$^{-1}$ |
| Half-life | $T_{1/2}$ | $T_{1/2} = \frac{\ln 2}{\lambda}$ | Time |
| After $n$ half-lives | $A_n$ | $A_n = \frac{A_0}{2^n}$ | Bq |

## What's next

Mastering radioactive decay is critical for understanding real-world applications of nuclear physics, from radiocarbon dating to medical radiation dosing. This sub-topic also forms the foundation for the rest of the nuclear physics unit, including binding energy, nuclear reactions, fission and fusion. Exponential decay concepts often appear in combination with other topics, so practicing decay problems now will help you earn easy marks on both Paper 1 and Paper 2 exams.

- [Nuclear reactions](https://www.owlsprep.com/study/ib-physics-sl-u5-nuclear-reactions/)
- [Nuclear fission and fusion](https://www.owlsprep.com/study/ib-physics-sl-u5-nuclear-fission-and-fusion/)

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