# Lesson 15: Half-life part 2

## Introduction

This lesson follows on from lesson 3 so it might be worthwhile quickly revising that before you start.

We’re going to see what 'half-life' means and why radioactivity changes with time. We’ll also see how carbon dating can be used to date ancient remains.

## 1 bequerel (Bq) means 1 count per second

First of all let’s look at half-life. Radioactivity is measured in becquerels. 1 becquerel is 1 count per second. 2 becquerels is 2 counts per second and so on. The unit of radioactivity is named after Henri Becquerel, who discovered it.

## Half-life: the time for the count rate to decrease by a half

Radioactivity decreases with time. A given isotope always takes the same amount of time for the count rate to decrease by a half.

For example, it might take 10 years for the count rate to drop from 80 Bq to 40 Bq; another 10 years to drop from 40 B to 20 Bq; another 10 years to drop from 20 Bq to 10 Bq and so on. In this case the half-life is 10 years.

## Different samples of the same isotope all have the same half-life

The half-life of a particular isotope is always the same. It doesn’t depend on the size of the sample and it doesn’t change with time.

If we had a bigger sample of the same isotope then the count would be higher, say 200 becquerels.

## Using half-life in simple calculations

What would its radioactivity be after 30 years?

Each half-life is 10 years. So we imagine going in forward one half-life at a time from ZERO years: 10 years, 20 years, 30 years, etc.

Then we halve the count for each half-life: 100 Bq after 10 years; 50 Bq after 20 years; 25 Bq after 30 years So we can see the radioactivity would be 25 becquerels afer 30 years.

We can use the same idea to find out how long it would take for a sample with radioactivity 120 Bq to drop to 30 Bq.

60 Bq after 10 years; 30 Bq after 20 years.

## Carbon dating can be used to find out when living things died

We can use radioactive decay to calculate the age of things. The best-known technique is called ‘radiocarbon dating’ or just 'carbon dating'.

Carbon dating can only be used to find the age of things that were once alive, like wood, leather, paper and bones.

If you have a wooden box, carbon dating can tell you when the tree to make it was cut down but not when the box was made.

Carbon dating can be used to date things up to about 60 000 years old. So how does it work?

## Trees are made from reorganised air

Many people think that plants grow by taking food from the soil through their roots but this is not true.

All green plants make their own food in their leaves. They make it from carbon dioxide in the air. This process is called photosynthesis. You need energy from the Sun and lots of water. Carbon dioxide is made into simple sugars and it is these that are the building blocks that make up wood, bark and leaves.

If we ignore water then over 90% of a green plant’s mass is just rearranged carbon dioxide.

Animals eat plants (or other animals that eat plants) so animals are also mostly rearranged carbon dioxide.

## Carbon-14: as rare as a grain of sand in a swimming pool of salt

A tiny fraction of carbon atoms are the radioactive isotope carbon-14. Carbon-14 is produced in the upper atmosphere by cosmic rays. It is a beta emitter with a half-life of about 5600 years.

As it's produced carbon-14 reacts with oxygen to form carbon dioxide and some of it is taken in by green plants and made into sugars along with the 'normal' carbon.

## Carbon-14 is constantly created and constantly decays

Carbon-14 is produced all the time but it also decays all the time back into nitrogen-14. An equilibrium is reached whereby about one in a trillion carbon atoms in the atmosphere is carbon-14.

Carbon-14 in living things decays all the time but is replaced by carbon-14 in food.

All new cells are made from food. In fact, all living things are rearranged food and NOTHING else. All food ultimately comes from green plants making sugars from carbon dioxide.

So all living things contain exactly the same proportion of carbon-14 compared to carbon-12: the proportion in the atmosphere. This is assumed to have stayed fairly constant.

A living adult human body contains about a billionth of a gram of carbon-14.

This means a human adult has a radioactivity of around 3000-4000 becquerels due to carbon-14. This is actually very small.

## When a living thing dies they stop eating so no new carbon-14

When a living thing dies the cells are no longer replaced so no new carbon enters it. The radioactivity of the carbon-14 begins to decrease. It halves about every 5600 years.

By measuring the radioactivity we can tell how long ago the living thing died.

Remember that the carbon-14 decays all the time whether the thing's alive or not. It's just that when it's living the carbon-14 is constantly replaced so the overall radioactivity stays constant.

## Preparing a sample for carbon dating

Say we want to find the age of an old dead tree. We don’t just stick a Geiger counter in front of it and hope for the best.

Animals and plants have similar amounts of radioactive isotopes, particularly potassium-40, another beta emitter. We need to make sure that we’re only measuring the beta radiation from the carbon-14. Even though carbon-14 causes around half of the internal radioactivity of living things, it’s only around 0.25 Bq per gram.

A common way to isolate the carbon is to carefully burn a piece of the wood and use the carbon dioxide given off. The carbon dioxide is separated out from the other gases. It is mostly carbon-12 with tiny amounts of the radioactive carbon-14.

## Calculating the age by comparing the count with the atmosphere

We measure the radioactivity of the carbon dioxide in a special chamber to shield it from background radiation. We can then compare it with the radioactivity of the same amount of carbon dioxide from the atmosphere.

The oldest samples have the lowest radioactivity.

The radioactivity halves with each half-life. This means we can calculate the age of a sample. For example, a sample with a count of only 25% of atmospheric carbon dioxide must be two half-lives old:

100% - 50% takes 1 half-life

50% - 25% takes a second half-life

If the half-life is 5600 years then the sample must be 5600 x 2 = 11 200 years old.

## Counting carbon-14 atoms directly: another way of calculating age

You can use a much smaller sample of the material you want to test if you count the carbon-14 atoms directly rather than having to wait for them to decay. Even this kind of carbon dating can only be used to date things that were once alive and died less than about 60 000 years ago.

Other radio-dating techniques are used to date ancient rocks.

## Displaying radioactive decay on a graph

We can plot a graph of radioactivity against time for our sample that had a half-life of 10 years.

We can use our graph to show that it always takes 10 years for the radioactivity to drop by a half regardless of where you are on the graph.

## The decay of protactinium experiment

A common school experiment is to find the half-life of an isotope called protactinium-234m.

It's useful because it has a half-life of the order of a minute and a pure sample can be prepared simply by shaking a bottle of liquid.

## There's nothing special about half-life

It's not that radioactive isotopes 'have' a half-life.

They get less radioactive in a way that's called an exponential. Exponential decay means that equal periods of time give equal *proportional* changes in radioactivity.

So you can pick any period of time, say 1 minute, and measure how much the radioactivity drops to in that minute. Say it drops to 63% of its initial value. After another minute it will drop to 63% of this value. Another minute, another 63% drop.

Now 63% isn't a very obvious number so we pick a time where the drop is something simple, like 50%, or a half. We could equally well choose the one-third life or the four-fifths life.

## Why radioactivity decreases with time: fewer and fewer nuclei left *to* decay

Let’s try and explain why radioactivity decreases with time.

When a nucleus decays two things happen

- The nucleus changes to become more stable
- Some nuclear radiation is given off, e.g. a single alpha or a single beta particle

Note that radioactive decay *never* means a nucleus just disappears. Remember the nucleus is just part of an atom, except that, when we talk about radioactivity, we tend to ignore the electrons that take up most of the volume. The atoms can't just vanish into nothingness and neither can its nucleus. The nucleus simply changes.

Let's think about a sample of a beta emitter. The sample consists of billions of atoms. The nucleus of each atom is unstable. Each nucleus will emit a single beta particle and then become stable.

So every time a nucleus changes it gives out a beta particle. One nucleus, one particle.

But as time passes there are fewer undecayed nuclei left TO decay. You can say this about the undecayed nuclei: ‘The fewer you have, the slower you lose them.’

Every time you ‘lose’ an undecayed nucleus a beta particle is given off. So the fewer undecayed nuclei you have, the slower you lose them, and the lower the radioactivity.

## The decay of a given nucleus is completely random

Now let’s try and explain why radioactive decay has a CONSTANT ‘half-life’.

Only curves called ‘exponentials’ have a constant half-life. There are lots of curves that look like exponentials but they don't have constant half-lives.

Half-life is constant because every nucleus has a constant chance of decay each second. We may simply talk about ‘chance of decay’ as shorthand for ‘chance of decay of every nucleus each second’.

But the decay of a given nucleus is completely random. You can’t predict at all when it will decay.

## Nuclei never grow old

The next point is slightly more subtle. A radioactive nucleus doesn’t have a particular ‘life time’ like living things do. A ninety year-old person is more likely to die this year than a sixteen year-old.

But a nucleus doesn’t grow old like this.

At the start of every second it has exactly the same chance of decay. It doesn’t matter whether it’s already survived a trillion seconds or just one. But if we have no idea at all exactly when a particular nucleus will decay how can we know how the radioactivity of a sample of trillions of nuclei will change with time?

## You know how many nuclei will decay, you just don't know which ones

Imagine a large number of nuclei. Each nucleus has a 25% chance of decay each second, say. At the end of 1 second about 25% will have decayed but we have no idea which ones.

The bigger the number we started with the closer we'll get to 25%.

At the end of each second 25% *of the number remaining* have decayed. The PROPORTION depends only on the chance of decay. It doesn’t depend on how many have already decayed.

And that’s why there is a half-life (and a third-life, two fifths-life, etc…). It’s because every nucleus of the same isotope has the same chance of decay each second. And this chance never changes.

## High chance of decay means short half-life

But different isotopes have different chances of decay. In other words different isotopes have different half-lives. If you measured half-life with enough precision you could say that every half-life is unique.

If you have three nuclei, each from different isotopes, then one will have the highest chance of decay and one will have the lowest. But you have no idea which one will decay first. It only makes sense to talk about likelihood when you have lots of nuclei for each isotope.

## Radioactivity is proportional to the number of undecayed nuclei

Imagine you have a sample of millions of nuclei of a beta emitter.

Over time the undecayed nuclei decay. So the number of undecayed nuclei decreases. The rate of beta emission (i.e. the radioactivity) also decreases. This makes sense because it’s only the *undecayed* nuclei that CAN emit a beta particle.

If you have fewer of these undecayed nuclei you’re bound to see fewer betas.

In fact the radioactivity is directly proportional to the number of undecayed nuclei. If you halve the number of undecayed nuclei, you halve the radioactivity.

## High chance of decay gives steep curve

We can plot a graph of number of undecayed nuclei against time. This has an identical shape to the graph of radioactivity against time. This makes perfect sense because the fewer undecayed nuclei there are left, the fewer there are left to decay and give out a beta particle.

If the chance of decay is high, the nuclei decay quickly, the graph is steep and the half-life is short.

The graph drops steeply because at the end of each second there are far fewer undecayed nuclei than there were. If there are fat fewer undecayed nuclei then the radioactivity must also be much lower.

If the chance of decay is low, the nuclei decay slowly, the graph is shallow and the half-life is long.

## Half-life: the time for the number of undecayed nuclei to decrease by a half

Every half-life the number of undecayed nuclei decreases by a half.

So if the half-life is 10 seconds and there are 64 million undecayed nuclei, at the end of ten seconds there would 32 million, another ten seconds 16 million and so on.

## Comparing the undecayed nuclei graph with the radioactivity graph

The graph of undecayed nuclei against time and the graph of radioactivity against time have a similar shape. That’s why they can both be used to calculate half-life.

The two graphs are deeply linked. The more undecayed nuclei there are, the more will decay, giving off a beta (say) each second.

If you change the chance of decay (by choosing different isotopes) the number-of-nuclei curve always starts at the same place but the shape changes. This makes sense: you always start with 200 nuclei (say) no matter how fast they decay.

But the radioactivity (‘A’ for activity) curve starts in a different place. The greater the chance of decay the greater the initial activity. This also makes sense. If the nuclei are more likely to decay you’d expect more beta particles to be given off in the first second.

## A window on one big decay curve per isotope

If you change the number of nuclei you start with (by changing the size of your source) then what you’re effectively doing is taking different snap shots of one big decay curve.

Imagine you start with a ‘large’ sample of 80 undecayed nuclei. At some point you’ll have 60 left. The curve after 60 is exactly the same as the curve of a different sample that STARTED with 60.

After 30 it’s exactly the same as a different sample that started with 30. It’s the same for any other start number. But it’s all the same curve so the half-life is the same regardless of how big a sample you start with.

The curves may not look as if they joined up if you plotted them because you tend to choose a scale that uses all your graph paper.

In the next lesson we’ll extend these ideas so we can work out what happens when we’re not dealing with a whole number of half-lives.

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