# Radioactive decay law

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u27-radioactive-decay-law/

This module covers the exponential law governing spontaneous radioactive decay, explains the relationships between decay constant, half-life and activity, and teaches you to solve common CIE exam problems on this core topic.

**Prerequisites:** [Introduction to radioactivity](https://www.owlsprep.com/study/cie-9702-u27-introduction-to-radioactivity/); Exponential functions and logarithms

## Learning objectives

- State and explain the radioactive decay law for spontaneous nuclear decay
- Relate decay constant, activity and half-life using standard formulas
- Calculate decay quantities from given data and graphical plots
- Solve common exam problems involving radioactive decay

## Foundations of the Radioactive Decay Law

**Radioactive decay law** — A statistical law that describes the rate of decay of a large sample of radioactive nuclei, stating that the rate of decay is proportional to the number of undecayed nuclei remaining at that time.

*Example:* If half the sample remains after one half-life, the rate of decay will be half the original rate.

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

**Decay constant ($\lambda$)** — The constant of proportionality in the decay law, equal to the probability that any single undecayed nucleus will decay per unit time. Units are always inverse time (e.g. $s^{-1}$).

**Worked example:** The decay constant of tritium is $1.5 \times 10^{-3} \text{ year}^{-1}$. Write the equation for the number of undecayed nuclei $N(t)$ if the initial number of nuclei is $5.0 \times 10^{12}$.

1. Integrate the differential decay law starting from $N=N_0$ at $t=0$:
2. $$\int_{N_0}^{N} \frac{dN}{N} = -\lambda \int_0^t dt$$
3. Evaluate the integrals and rearrange:
4. $$\ln\left(\frac{N}{N_0}\right) = -\lambda t \implies N(t) = N_0 e^{-\lambda t}$$
5. Substitute the given values of $N_0$ and $\lambda$:
6. $$N(t) = (5.0 \times 10^{12}) e^{-(1.5 \times 10^{-3})t}$$

> **Exam tip:** Always confirm the exponent is negative for decay: the number of undecayed nuclei always decreases over time.

## Activity and the Half-Life Relationship

**Activity ($A$)** — The total rate of decay of a radioactive sample, measured in becquerels (Bq), where 1 Bq = 1 decay per second.

*Notation:* A = \left|\frac{dN}{dt}\right|

*Example:* A 100 Bq source has 100 decays every second on average.

Since $A = |dN/dt| = \lambda N$, activity also follows the exponential decay law: $A(t) = A_0 e^{-\lambda t}$, where $A_0 = \lambda N_0$ is the initial activity.

**Half-life ($t_{1/2}$)** — The time taken for half of the original undecayed nuclei to decay, or for the activity of a sample to decrease to half its initial value.

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

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

1. By definition, at $t = t_{1/2}$, $N = \frac{N_0}{2}$
2. Substitute into the decay law: $\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$
3. Cancel $N_0$ and take natural logarithms of both sides: $\ln\left(\frac{1}{2}\right) = -\lambda t_{1/2}$
4. Simplify: $-\ln 2 = -\lambda t_{1/2}$

*Conclusion:* The relationship is: $$t_{1/2} = \frac{\ln 2}{\lambda}$$

**Worked example:** Cobalt-60 has a half-life of 5.27 years. Calculate its decay constant in $s^{-1}$.

1. Convert half-life from years to seconds:
2. $$5.27 \text{ years} = 5.27 \times 365 \times 24 \times 3600 = 1.662 \times 10^8 \text{ s}$$
3. Rearrange the half-life formula to solve for $\lambda$:
4. $$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{1.662 \times 10^8} \approx 4.2 \times 10^{-9} \text{ s}^{-1}$$

> **Exam tip:** Always convert half-life to match the required time units for the answer before calculating $\lambda$. This is a common marking point.

## Solving Common Exam Decay Problems

**Comparing methods**

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

- **Exponential ($e$) form** — Uses $N = N_0 e^{-\lambda t}$ for any value of $t$
  - Pros: General form, required when calculating decay constant
  - Cons: More calculation steps, higher chance of logarithm errors

- **Half-life power form** — Uses $N = N_0 \left(\frac{1}{2}\right)^{n}$, where $n = t/t_{1/2}$ is the number of half-lives
  - Pros: Much faster for problems with integer numbers of half-lives
  - Cons: Not useful if you need to find the decay constant

**Worked example:** A radioactive sample has an initial activity of 4800 Bq. After 24 hours, the activity is 300 Bq. Calculate the half-life of the sample.

1. Use the power form for activity: $A = A_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$
2. Substitute the given values:
3. $$300 = 4800 \left(\frac{1}{2}\right)^{24/t_{1/2}}$$
4. Simplify the left-hand side: $\frac{300}{4800} = \frac{1}{16} = \left(\frac{1}{2}\right)^4$
5. Equate the exponents: $\frac{24}{t_{1/2}} = 4$
6. Solve for half-life: $t_{1/2} = 6 \text{ hours}$

**Check your understanding**

Test your understanding

1. What is the activity after 3 half-lives, if the initial activity is 1000 Bq?

   - 125 Bq
   - 250 Bq
   - 333 Bq
   - 500 Bq

   *Why:* After $n$ half-lives, activity is $A = A_0 (1/2)^n$. For $n=3$, $(1/2)^3 = 1/8$, so $1000/8 = 125$ Bq.

## Graphical Analysis of Decay

A plot of $\ln N$ (or $\ln A$) against time $t$ gives a straight line for exponential decay. Rearranging the decay law gives: $\ln N = \ln N_0 - \lambda t$, which matches the straight line equation $y = c + mx$.

> **info**
>
> The gradient of the $\ln N$ vs $t$ line is equal to $-\lambda$, so decay constant is the magnitude (positive value) of the gradient.

**Worked example:** Values of $\ln A$ for a decay experiment are given below. Calculate the decay constant from the data. $t = 0 \text{ s}, \ln A = 6.9$; $t = 200 \text{ s}, \ln A = 5.5$

1. Calculate the gradient of the line:
2. $$\text{gradient} = \frac{\Delta \ln A}{\Delta t} = \frac{5.5 - 6.9}{200 - 0} = \frac{-1.4}{200} = -0.007 \text{ s}^{-1}$$
3. Decay constant is the absolute value of the gradient:
4. $$\lambda = |-0.007| = 0.007 \text{ s}^{-1}$$

## Common pitfalls

- **Wrong:** Forgetting to convert units of half-life to match the answer units, e.g. leaving half-life in days when asked for $\lambda$ in $s^{-1}$
  - Why it fails: Exam questions often specify different units for the answer than the given input value
  - Correct: Always check the required units for your final answer and convert all quantities to match before starting calculation
- **Wrong:** Using the gradient of $\ln N$ vs $t$ directly (negative value) as decay constant
  - Why it fails: The gradient is negative because $N$ decreases over time, but decay constant is a positive probability
  - Correct: Decay constant $\lambda$ is equal to the absolute value (magnitude) of the gradient of $\ln N$ against $t$
- **Wrong:** Writing the decay law with a positive exponent: $N = N_0 e^{\lambda t}$
  - Why it fails: Mixing up exponential decay and exponential growth
  - Correct: Remember the number of undecayed nuclei decreases over time, so the exponent must always be negative: $N = N_0 e^{-\lambda t}$
- **Wrong:** Applying the decay law to predict the exact decay time of a single nucleus
  - Why it fails: The decay law is a statistical law that only describes the average behavior of large samples
  - Correct: The decay law can only predict average behavior for large samples, it cannot predict when a single nucleus will decay
- **Wrong:** Confusing number of undecayed nuclei $N$ with activity $A$
  - Why it fails: Both quantities decay exponentially, but they have different units and meanings
  - Correct: Remember $A = \lambda N$: activity is rate of decay, with units of Bq, while $N$ is just a count of undecayed nuclei

## Cheatsheet

| Quantity | Formula | Units |
| --- | --- | --- |
| Number of undecayed nuclei | $N(t) = N_0 e^{-\lambda t}$ | Count (dimensionless) |
| Activity | $A(t) = \lambda N(t) = A_0 e^{-\lambda t}$ | Bq (decays per second) |
| Half-life / decay constant relation | $\lambda = \frac{\ln 2}{t_{1/2}}$ | $\lambda$: inverse time, $t_{1/2}$: time |
| Gradient of $\ln N$ vs $t$ | $\text{gradient} = -\lambda$ | Inverse time |
| Quantity after $n$ half-lives | $X = X_0 \left(\frac{1}{2}\right)^n$ | Same as $X_0$ |

## What's next

The radioactive decay law is a core foundation for all further topics in nuclear physics for CIE A-Level. You will apply this law to practical problems like radiocarbon dating, radioactive tracer calculations in medicine, and safety analysis for radioactive sources. It is also used to understand decay series, nuclear reaction rates and the practical measurement of half-lives. Next, you will build on this knowledge to learn about the different modes of radioactive decay, how to balance nuclear equations, and the properties of different types of nuclear radiation.

- [Half-life](https://www.owlsprep.com/study/cie-9702-u27-half-life/)
- [Nuclear fission and fusion](https://www.owlsprep.com/study/cie-9702-u27-nuclear-fission-and-fusion/)
- [Radiation safety](https://www.owlsprep.com/study/cie-9702-u27-radiation-safety/)

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