# Radioactive Decay

## Radioactive Decay Revision

**Radioactive Decay**

We have looked at the properties of **alpha**,** beta **and** gamma** decay. Each of these may be emitted as part of **radioactive decay**.

**What is Radioactive Decay?**

**Radioactive decay** is the random and spontaneous decay of the **nucleus** of an atom to become more **stable**, resulting in the emission of **alpha**,** beta **or** gamma radiation**.

When looking closely at a **radioactive source**, it is impossible to predict which nucleus will decay and when it will decay. This is because the process is **completely random**. It is also important to remember that no changes in the conditions surrounding a source will affect its **rate of decay**.

**Activity**

The **activity (A) **of a radioactive source is the average number of nuclei that **decay per unit of time**. Notice two things from this definition:

**Average**– as radioactive decay is random, the number of nuclei that decay per unit of time varies. We can however state the average number of decaying nuclei per unit of time.- The definition states
**per unit of time**. This unit of time can vary depending upon which isotope is being observed and how quickly they decay.

A radioactive source that is described as highly radioactive has a high activity. This is measured in **Becquerels **\text{(Bq)} where \bold{1} \text{ Bq}= \bold{1}** decaying nuclei per second**.

The equation used to calculate activity is as follows:

A=\dfrac{\Delta N}{\Delta t}

- A= the
**activity**in becquerels \text{(Bq)} - \Delta N= the
**number of decayed nuclei** - \Delta t= the
**change in time**in seconds \text{(s)}

**Example:** Over a period of one hour, a GM counter detects 25,000 counts of radiation. What is the activity of the source? The background radiation has been measured at 1 count per second.

**[3 marks]**

Substitute the given values into the activity equation:

\begin{aligned} \bold{A} &= \bold{\dfrac{\Delta N}{\Delta t}} \\ &= \dfrac{\textcolor{f43364}{25, 000}}{\textcolor{10a6f3}{60 \times 60}} \\ &= \bold{6.9} \text{ Bq} \end{aligned}

Subtract the background radiation:

6.9 - \textcolor{00d865}{1} = \bold{5.9} \textbf{ Bq}

We can also use the **decay constant **(\lambda) to describe how radioactive a source is. The **decay constant **is the **probability** that a nuclei will decay per second. Decay constant is related to activity using the following equation:

A= \lambda N

- A= the
**activity**in becquerels \text{(Bq)} - \lambda = the
**decay constant**per second \text{(s}^{-1}\text{)} - N= the
**number of nuclei**in a sample

**Example**: A source contains 6 \times 10^{24} undecayed nuclei. If the decay constant is 2.1 \times 10^{-6} \text{ s}^{-1}, calculate the activity of the source.

**[2 marks]**

Substitute the values into the equation linking activity and decay constant:

\begin{aligned} \bold{A} &= \bold{\lambda N} \\ &= \textcolor{f95d27}{2.1 \times 10^{-6}} \times \textcolor{ffad05}{6 \times 10^{24}} \\ &= \bold{1.3 \times 10^{19}} \textbf{ Bq} \end{aligned}

**Exponential Decay**

It’s common to see **radioactive decay graphs** like the one below. This graph shows how the number of undecayed nuclei \text{(N)} varies with time \text{(t)}.

This type of **decay curve **is an **exponential decay** curve. The number of nuclei begins to decay from N0 (initial number of nuclei) rapidly but the **rate of decay slows over time** until almost reaching zero.

The steeper slope, represented above in black, shows a decay with a **higher decay constant **compared to the shallower red line.

**Radioactive Decay Equations**

Because of the nature of **exponential decay**, it can be difficult to predict how many nuclei would be remaining in a sample after a certain amount of time. A calculation can be completed to calculate this number:

N=N_0e^{-\lambda t}

- N= the
**number of undecayed nuclei** - N_0= the
**initial number of undecayed nuclei** - \lambda = the
**decay constant**per second \text{(s}^{-1}\text{)} - t= the
**time**in seconds \text{(s)}

As the **activity of a source** and the **count rate **also **decays exponentially**, the equation may also be written as:

A=A_0 e^{- \lambda t}

- A= the
**activity**at a given time (t) in becquerels \text{(Bq)} - A_0 the
**initial activity**in becquerels \text{(Bq)} - \lambda= the
**decay constant**per second \text{(s}^{-1}\text{)} - t= the
**time**in seconds \text{(s)}

C=C_0 e^{-\lambda t}

- C= the
**count rate**at a given time (t) (\text{s}^{-1}\text{, min}^{-1}\text{ or hour}^{-1} \text{ etc}) - C_0= the
**initial count rate**(\text{s}^{-1}\text{, min}^{-1}\text{ or hour}^{-1} \text{ etc}) - \lambda = the
**decay constant**per second \text{(s}^{-1}\text{)} - t=
**time**in seconds \text{(s)}

**Example:** A source contains 6 \times 10^{24}. If the decay constant of 2.1 \times 10^{-6} \text{ s}^{-1}, calculate the activity of the source after 1 year.

**[4 marks]**

Convert 1 year to seconds:

t= \textcolor{ffad05}{1 \text{ year}} = 1 \times 365 \times 24 \times 60 \times 60 = \textcolor{f95d27}{\bold{31 \, 536 \, 000} \textbf{ s} }

Calculate the initial activity:

\begin{aligned} A &= \lambda N \\ A_0 &= \textcolor{00bfa8}{2.1 \times 10^{-6}} \times \textcolor{d11149}{6 \times 10^{24}}) \\ &= \textcolor{aa57ff}{\bold{1.26 \times 10^{19}}} \textbf{ Bq} \end{aligned}

Substitute values into the equation for decay in activity:

\begin{aligned} \bold{A} &= \bold{A_0 e^{-\lambda t}} \\ &= \textcolor{aa57ff}{1.26 \times 10^{19}} \times e^{-(\textcolor{00bfa8}{2.1 \times 10^{-6}} \times \textcolor{f95d27}{31 \, 536 \, 000}}) \\ &= \bold{2.25 \times 10^{-10}} \textbf{ Bq} \end{aligned}

**Half-life**

The **half-life **of a sample is the time taken for the number of **radioactive nuclei** to halve. Therefore, it could also be defined as the **time taken for the activity or count rate to halve**. The half-life of a sample can be calculated using the equation:

t_{\dfrac{1}{2}}= \dfrac{ln(2)}{\lambda}

- t_{\dfrac{1}{2}}= the
**half-life**in seconds, minutes, hours, days or years - \lambda= the
**decay constant**per second \text{(s}^{-1}\text{)}

**Example:** A source of radiation has a decay constant of 0.04 \text{ s}^{-1}. What is its half life?

Substitute into the equation for half-life:

\begin{aligned} \bold{t_{\dfrac{1}{2}}} &= \bold{\dfrac{ln(2)}{\lambda}} \\ &= \dfrac{ln(2)}{\textcolor{00d865}{0.04}} \\ &= \bold{17.3} \textbf{ s} \end{aligned}

Alternatively, half-life can be calculated graphically. By taking two values for **activity** (one exactly half of the other) and their corresponding **times**, the half-life can be found.

**Example:** Use the graph to calculate the half-life of this radioactive source.

**[2 marks]**

Choose two values for activity from the graph, one half the othet:

A_0=290 \text{ Bq and } A_1=145 \text{ Bq}

Read off the corresponding times: = 0 s and 9.5 s

t_0= 0 \text{ s and }t_1=9.5 \text{ s}

Therefore:

\text{half-life}= \bold{9.5} \textbf{ s}

Knowledge of half-life can have some practical applications. For example, **carbon**-14 is a radioactive isotope with a half-life of 5400 years. By measuring the amount of carbon-14 in an artefact, the artefact can be dated. This is known as **carbon dating**.

## Radioactive Decay Example Questions

**Question 1:** Over a period of one day, a GM counter detects 6\times 10^6 counts of radiation. What is the activity of the source?

**[2 marks]**

1 \text{ day} = 24 \times 60 \times 60 = 86400 \text{ s} \\ \begin{aligned} \bold{A} &= \bold{\dfrac{\Delta N}{\Delta t}} \\ &= \dfrac{6 \times 10^6}{86400} \\ &= \bold{69.4} \textbf{ Bq}\end{aligned}

**Question 2:** State what is meant by the half-life of a radioactive sample?

**[1 mark]**

The **time taken for the number of nuclei/activity/count rate to decrease by half**.

**Question 3:** A radioactive source has a half-life of 20 \text{ s}. Calculate its decay constant.

**[2 marks]**

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