Radioactive decay (A-level only)

Welcome to the mathematics of radioactive decay! You already know what alpha, beta, and gamma radiation are, but now we need to predict how quickly a radioactive sample will disappear.

Here is what you'll learn:

Why radioactive decay is a truly random process, yet behaves predictably for large numbers of nuclei.
How to calculate the activity of a sample using the decay constant and Avogadro's constant.
How to use the exponential decay equations and natural logarithms.
How to analyse graphs (both decay curves and straight-line log graphs) to determine a substance's half-life.

The Random Nature of Decay

Radioactive decay is completely random. This means two things:

We cannot predict which specific nucleus will decay next.
We cannot predict exactly when a specific nucleus will decay.

However, each individual nucleus has a constant decay probability per unit time.

Analogy

Rolling Dice

Imagine you have $1000$ six-sided dice, and rolling a $6$ represents a nucleus "decaying". You can't point to a specific die and know it will roll a $6$ on the next throw. But because the probability of rolling a $6$ is exactly $16\frac{1}{6}$ for every die, you can confidently predict that about $167$ dice will "decay" on the first throw. As the number of dice (undecayed nuclei) drops, the number of $6$ s you roll per throw (the activity) will also drop.

Activity and the Decay Constant

To formalise this constant probability, we introduce a new term called the decay constant.

Definition

Decay Constant, $λ$

The decay constant ( $λ\lambda$ ) is the probability of an individual nucleus decaying per unit time interval. Its unit is $s−1\text{s}^{-1}$ (though it is sometimes given in $hr−1\text{hr}^{-1}$ or $yr−1\text{yr}^{-1}$ ).

If we know the probability of one nucleus decaying ( $λ\lambda$ ), and we know the total number of undecayed nuclei in the sample ( $N$ ), we can find the overall rate of decay. We call this rate the Activity, $A$ .

\lambda N

Activity is measured in Becquerels ( $Bq\text{Bq}$ ), where $\text{ Bq} = 1 \text{ decay per second}$ .

Often, exam questions won't just give you $N$ . Instead, you'll be given the mass of the radioactive sample. You must use molar mass and the Avogadro constant ( $mol−1N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$ ) to find $N$ first.

Example

Calculating Activity using Molar Mass

A sample contains $\text{ g}$ of pure Carbon-14. The decay constant of Carbon-14 is $3.84 \times 10^{-12} \text{ s}^{-1}$ . Calculate the activity of the sample.

Find the number of moles: Divide the mass by the molar mass ( $\text{ g mol}^{-1}$ ).

\text{Moles} = \frac{1.50}{14} = 0.1071 \text{ mol}

Find the total number of nuclei ( $N$ ): Multiply the moles by the Avogadro constant.

\times (6.02 \times 10^{23}) = 6.45 \times 10^{22} \text{ nuclei}

Calculate Activity ( $A$ ): Use the equation $\lambda N$ .

\times 10^{-12}) \times (6.45 \times 10^{22})

\times 10^{11} \text{ Bq}

The Exponential Decay Equations

Activity is the rate of change of the number of nuclei over time. Mathematically, we write this as:

\frac{\Delta N}{\Delta t} = -\lambda N

The minus sign is crucial: it shows that the number of undecayed nuclei $N$ is decreasing over time.

Because the rate of decay is directly proportional to the number of nuclei remaining, the decay follows an exponential curve. Using A-Level maths (integration), that rate-of-change equation turns into the exponential decay equation:

N_0 e^{-\lambda t}

Where:

$N$ is the number of undecayed nuclei remaining at time $t$ .
$N_0$ is the initial number of undecayed nuclei (at $t = 0$ ).
$e$ is Euler's number (the exponential function on your calculator).
$λ\lambda$ is the decay constant.
$t$ is the elapsed time.

Because Activity ( $A$ ) and Count Rate ( $C$ ) are both directly proportional to $N$ , they follow the exact same exponential pattern:

A_0 e^{-\lambda t}

Common Mistake

Watch your units!

The unit of time $t$ must always match the unit of the decay constant $λ\lambda$ . If $λ\lambda$ is in $years−1\text{years}^{-1}$ , $t$ must be in $years\text{years}$ . If $λ\lambda$ is in $s−1\text{s}^{-1}$ , $t$ must be in seconds.

Half-Life

Instead of the decay constant, we often describe how fast a substance decays using its half-life.

Definition

Half-life, $T_{1/2}$

The half-life ( $T_{1/2}$ ) is the time taken for half the number of radioactive nuclei in an initial sample to decay. (Or, equivalently, the time for the activity to halve).

Decay curve showing half-life

We can derive a simple equation linking $T_{1/2}$ and $λ\lambda$ . After one half-life ( $t = T_{1/2}$ ), the number of nuclei drops to half of the original amount ( $\frac{N_0}{2}$ ). Let's substitute this into the exponential equation:

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

Cancel $N_0$ from both sides:

e^{-\lambda T_{1/2}}

Take the natural logarithm ( $ln⁡\ln$ ) of both sides. Remember that $ln(e^x) = x$ , and $ln⁡(0.5)=−ln⁡2\ln(0.5) = -\ln 2$ :

-\ln 2 = -\lambda T_{1/2}

Rearranging gives us the half-life equation:

T_{1/2} = \frac{\ln 2}{\lambda}

Example

Using exponential decay in Carbon Dating

Living wood has a Carbon-14 activity of $\text{ Bq}$ per gram of carbon. A piece of ancient wood found in an archaeological dig has an activity of $\text{ Bq}$ per gram. The half-life of Carbon-14 is $\text{ years}$ . Calculate the age of the ancient wood.

Find the decay constant ( $λ\lambda$ ): Using the half-life equation. Since $T_{1/2}$ is in years, $λ\lambda$ will be in $yr−1\text{yr}^{-1}$ .

\lambda = \frac{\ln 2}{T_{1/2}} = \frac{\ln 2}{5730} = 1.21 \times 10^{-4} \text{ yr}^{-1}

Set up the exponential equation: We know $A_0 = 0.25$ , $A = 0.15$ , and $λ=1.21×10−4\lambda = 1.21 \times 10^{-4}$ . We need to find $t$ .

A_0 e^{-\lambda t}

e^{-(1.21 \times 10^{-4}) t}

Rearrange and solve for $t$ : First, divide both sides by $0.25$ .

\frac{0.15}{0.25} = e^{-(1.21 \times 10^{-4}) t}

e^{-(1.21 \times 10^{-4}) t}

Use natural logs: Take $ln⁡\ln$ of both sides to remove the exponential.

\ln(0.60) = -1.21 \times 10^{-4} t

\times 10^{-4} t

\frac{-0.5108}{-1.21 \times 10^{-4}} = 4220 \text{ years}

Graphical Analysis using Log Graphs

Drawing a curve like the one above is useful, but curves are hard to analyse accurately. Physicists prefer straight lines. We can turn the exponential curve into a straight line by taking the natural log of the exponential equation:

A_0 e^{-\lambda t}

\ln A = \ln(A_0 e^{-\lambda t})

\ln A = \ln A_0 + \ln(e^{-\lambda t})

\ln A = -\lambda t + \ln A_0

Notice how this exactly matches the equation of a straight line, $y = m x + c$ :

$y$ -axis: $ln⁡A\ln A$
$x$ -axis: $t$
Gradient ( $m$ ): $−λ-\lambda$
$y$ -intercept ( $c$ ): $ln A_0$

Log graph of Activity vs Time

If you are given a table of Activity against Time in an exam, the standard practical technique is to add a column for $ln⁡A\ln A$ , plot $ln⁡A\ln A$ against $t$ , find the gradient of the line of best fit, and use it to find $λ\lambda$ . You can then easily calculate $T_{1/2}$ .

Common Mistake

Missing the negative sign

When you calculate the gradient from your log graph, you will get a negative number (because the line slopes downwards). Remember that the gradient is $−λ-\lambda$ . Since $λ\lambda$ is a probability, it must be positive. Don't forget to drop the minus sign before you calculate the half-life!

Applications in the Real World

You must be aware of how these concepts apply to real situations:

Radioactive Dating: As shown in the example above, measuring the ratio of Carbon-14 to stable Carbon-12 in organic material allows us to date once-living objects. Limits: Carbon dating is only useful for things up to about $\text{ years}$ old. Beyond that, the activity drops so low it becomes indistinguishable from background radiation. For older objects (like rocks), we use elements with much longer half-lives, such as Uranium-238 decaying to Lead-206.

Storage of Radioactive Waste: Nuclear reactors produce waste with a huge variety of half-lives.

Short half-lives: Highly radioactive and dangerous now (high activity because $\lambda N$ and $λ\lambda$ is large), but they decay to safe levels in a few years. They are stored in cooling ponds until safe.
Long half-lives: Not as intensely radioactive day-to-day, but they will remain dangerous for thousands of years. They must be sealed in glass (vitrification) and buried deep underground in geologically stable areas to prevent them ever entering the water supply.

Exam technique

In the exam

Check your units before you calculate: If time $t$ is given in hours and half-life $T_{1/2}$ is in seconds, you must convert them so they match before you calculate $λ\lambda$ or put them into the exponential equation.
Be ready to find $λ\lambda$ from a graph: AQA loves to give you a graph of $ln⁡N\ln N$ against $t$ . Pick two widely spaced points on the line of best fit, calculate the gradient ( $\frac{\Delta y}{\Delta x}$ ), and state $λ=−m\lambda = -m$ .
Know your Avogadro trick: If a question gives you mass instead of $N$ , immediately think: $Mass→Moles→Nuclei\text{Mass} \to \text{Moles} \to \text{Nuclei}$ . Use the nucleon number as the molar mass (e.g., Uranium-235 has a molar mass of $\text{ g mol}^{-1}$ ).

Self review

Check yourself

What does it mean to say that radioactive decay is a "random" process?
Can you write down the relationship linking activity, decay constant, and number of undecayed nuclei?
If you plot a graph of $Rate)\ln(\text{Count Rate})$ against time, what do the gradient and $y$ -intercept represent?
Why is it difficult to carbon-date an object that is hundreds of millions of years old?

Welcome to the mathematics of radioactive decay! You already know what alpha, beta, and gamma radiation are, but now we need to predict how quickly a radioactive sample will disappear.

Here is what you'll learn:

Why radioactive decay is a truly random process, yet behaves predictably for large numbers of nuclei.
How to calculate the activity of a sample using the decay constant and Avogadro's constant.
How to use the exponential decay equations and natural logarithms.
How to analyse graphs (both decay curves and straight-line log graphs) to determine a substance's half-life.

The Random Nature of Decay

Radioactive decay is completely random. This means two things:

We cannot predict which specific nucleus will decay next.
We cannot predict exactly when a specific nucleus will decay.

However, each individual nucleus has a constant decay probability per unit time.

Analogy

Rolling Dice

Activity and the Decay Constant

To formalise this constant probability, we introduce a new term called the decay constant.

Definition

Decay Constant, $λ$

\lambda N

Activity is measured in Becquerels ( $Bq\text{Bq}$ ), where $\text{ Bq} = 1 \text{ decay per second}$ .

Example

Calculating Activity using Molar Mass

A sample contains $\text{ g}$ of pure Carbon-14. The decay constant of Carbon-14 is $3.84 \times 10^{-12} \text{ s}^{-1}$ . Calculate the activity of the sample.

Find the number of moles: Divide the mass by the molar mass ( $\text{ g mol}^{-1}$ ).

\text{Moles} = \frac{1.50}{14} = 0.1071 \text{ mol}

Find the total number of nuclei ( $N$ ): Multiply the moles by the Avogadro constant.

\times (6.02 \times 10^{23}) = 6.45 \times 10^{22} \text{ nuclei}

Calculate Activity ( $A$ ): Use the equation $\lambda N$ .

\times 10^{-12}) \times (6.45 \times 10^{22})

\times 10^{11} \text{ Bq}

The Exponential Decay Equations

Activity is the rate of change of the number of nuclei over time. Mathematically, we write this as:

\frac{\Delta N}{\Delta t} = -\lambda N

The minus sign is crucial: it shows that the number of undecayed nuclei $N$ is decreasing over time.

N_0 e^{-\lambda t}

Where:

$N$ is the number of undecayed nuclei remaining at time $t$ .
$N_0$ is the initial number of undecayed nuclei (at $t = 0$ ).
$e$ is Euler's number (the exponential function on your calculator).
$λ\lambda$ is the decay constant.
$t$ is the elapsed time.

Because Activity ( $A$ ) and Count Rate ( $C$ ) are both directly proportional to $N$ , they follow the exact same exponential pattern:

A_0 e^{-\lambda t}

Common Mistake

Watch your units!

Half-Life

Instead of the decay constant, we often describe how fast a substance decays using its half-life.

Definition

Half-life, $T_{1/2}$

The half-life ( $T_{1/2}$ ) is the time taken for half the number of radioactive nuclei in an initial sample to decay. (Or, equivalently, the time for the activity to halve).

Decay curve showing half-life

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

Cancel $N_0$ from both sides:

e^{-\lambda T_{1/2}}

Take the natural logarithm ( $ln⁡\ln$ ) of both sides. Remember that $ln(e^x) = x$ , and $ln⁡(0.5)=−ln⁡2\ln(0.5) = -\ln 2$ :

-\ln 2 = -\lambda T_{1/2}

Rearranging gives us the half-life equation:

T_{1/2} = \frac{\ln 2}{\lambda}

Example

Using exponential decay in Carbon Dating

Find the decay constant ( $λ\lambda$ ): Using the half-life equation. Since $T_{1/2}$ is in years, $λ\lambda$ will be in $yr−1\text{yr}^{-1}$ .

\lambda = \frac{\ln 2}{T_{1/2}} = \frac{\ln 2}{5730} = 1.21 \times 10^{-4} \text{ yr}^{-1}

Set up the exponential equation: We know $A_0 = 0.25$ , $A = 0.15$ , and $λ=1.21×10−4\lambda = 1.21 \times 10^{-4}$ . We need to find $t$ .

A_0 e^{-\lambda t}

e^{-(1.21 \times 10^{-4}) t}

Rearrange and solve for $t$ : First, divide both sides by $0.25$ .

\frac{0.15}{0.25} = e^{-(1.21 \times 10^{-4}) t}

e^{-(1.21 \times 10^{-4}) t}

Use natural logs: Take $ln⁡\ln$ of both sides to remove the exponential.

\ln(0.60) = -1.21 \times 10^{-4} t

\times 10^{-4} t

\frac{-0.5108}{-1.21 \times 10^{-4}} = 4220 \text{ years}

Graphical Analysis using Log Graphs

A_0 e^{-\lambda t}

\ln A = \ln(A_0 e^{-\lambda t})

\ln A = \ln A_0 + \ln(e^{-\lambda t})

\ln A = -\lambda t + \ln A_0

Notice how this exactly matches the equation of a straight line, $y = m x + c$ :

$y$ -axis: $ln⁡A\ln A$
$x$ -axis: $t$
Gradient ( $m$ ): $−λ-\lambda$
$y$ -intercept ( $c$ ): $ln A_0$

Log graph of Activity vs Time

Common Mistake

Missing the negative sign

Applications in the Real World

You must be aware of how these concepts apply to real situations:

Storage of Radioactive Waste: Nuclear reactors produce waste with a huge variety of half-lives.

Short half-lives: Highly radioactive and dangerous now (high activity because $\lambda N$ and $λ\lambda$ is large), but they decay to safe levels in a few years. They are stored in cooling ponds until safe.
Long half-lives: Not as intensely radioactive day-to-day, but they will remain dangerous for thousands of years. They must be sealed in glass (vitrification) and buried deep underground in geologically stable areas to prevent them ever entering the water supply.

Exam technique

In the exam

Check your units before you calculate: If time $t$ is given in hours and half-life $T_{1/2}$ is in seconds, you must convert them so they match before you calculate $λ\lambda$ or put them into the exponential equation.
Be ready to find $λ\lambda$ from a graph: AQA loves to give you a graph of $ln⁡N\ln N$ against $t$ . Pick two widely spaced points on the line of best fit, calculate the gradient ( $\frac{\Delta y}{\Delta x}$ ), and state $λ=−m\lambda = -m$ .
Know your Avogadro trick: If a question gives you mass instead of $N$ , immediately think: $Mass→Moles→Nuclei\text{Mass} \to \text{Moles} \to \text{Nuclei}$ . Use the nucleon number as the molar mass (e.g., Uranium-235 has a molar mass of $\text{ g mol}^{-1}$ ).

Self review

Check yourself

What does it mean to say that radioactive decay is a "random" process?
Can you write down the relationship linking activity, decay constant, and number of undecayed nuclei?
If you plot a graph of $Rate)\ln(\text{Count Rate})$ against time, what do the gradient and $y$ -intercept represent?
Why is it difficult to carbon-date an object that is hundreds of millions of years old?

The Random Nature of Decay

Activity and the Decay Constant

The Exponential Decay Equations

Half-Life

Graphical Analysis using Log Graphs

Applications in the Real World

Review Radioactive decay (A-level only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Radioactive decay (A-level only)

The Random Nature of Decay

Activity and the Decay Constant

The Exponential Decay Equations

Half-Life

Graphical Analysis using Log Graphs

Applications in the Real World

Review Radioactive decay (A-level only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Radioactive decay (A-level only)

The Random Nature of Decay

Activity and the Decay Constant

The Exponential Decay Equations

Half-Life

Graphical Analysis using Log Graphs

Applications in the Real World

Review Radioactive decay (A-level only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Radioactive decay (A-level only) Revision Guide

Radioactive decay (A-level only)

The Random Nature of Decay

Activity and the Decay Constant

The Exponential Decay Equations

Half-Life

Graphical Analysis using Log Graphs

Applications in the Real World

Review Radioactive decay (A-level only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Radioactive decay (A-level only) Revision Guide