Decay is random for one nucleus but predictable for a whole sample

Definition

Half-life

The time taken for the number of unstable nuclei in a sample to halve, or for the count-rate (or activity) to fall to half its starting value.

You cannot predict when any single nucleus will decay, because decay is random.

A real sample contains a huge number of nuclei, so on average its activity falls in a predictable way.

This predictable pattern is described by the half-life.

Half-life: the time for the count-rate to halve

The half-life is the time it takes for the number of unstable nuclei, or the count-rate, to fall to a half.

After each half-life the count-rate halves again, so after

n

half-lives the fraction remaining is

(12)n\left(\tfrac{1}{2}\right)^{n}

This gives: remaining count-rate

=

initial count-rate

×(12)n\times \left(\tfrac{1}{2}\right)^{n}

A graph showing exponential decay over four half-lives. The y-axis shows the amount starting at 1.0 and halving to 0.5, 0.25, 0.125, and 0.0625 at each successive half-life on the x-axis.

Example

A source has a count-rate of $640$ counts/s and a half-life of $6$ days. Find the count-rate after $18$ days.

$days18\ \text{days}$ is $18 \div 6 = 3$ half-lives, so halve the count-rate three times:

\rightarrow 320 \rightarrow 160 \rightarrow 80\ \text{counts/s}

After $18$ days the count-rate is $80$ counts/s.

Common Mistake

Half-life is not the time for the sample to decay completely, nor half of that time.
It is the time for the count-rate to fall by half, and the same length of time halves it again.

Self review

What is meant by the half-life of a radioactive isotope?
Why can the decay of a whole sample be predicted even though single nuclei are random?
What fraction of the original nuclei remains after 3 half-lives?
A source has a count-rate of $800$ counts/s and a half-life of $2$ hours; what is the count-rate after $6$ hours?

Net decline compares the radiation left with what you started with

Definition

Net decline

The overall fall in a radioactive emission after a number of half-lives, usually written as a ratio comparing the final amount with the initial amount.

After each half-life the count-rate falls to a half, so after

n

half-lives the fraction remaining is

(12)n=12n\left(\tfrac{1}{2}\right)^{n} = \dfrac{1}{2^{n}}

The net decline is written as a ratio comparing the initial and final count-rates.

After

1

2

3

and

4

half-lives the fraction left is

12\tfrac{1}{2}

14\tfrac{1}{4}

18\tfrac{1}{8}

and

116\tfrac{1}{16}

$A decay curve graph illustrating the net decline of a substance over four half-lives. The amount remaining decreases from 1.0 to 0.5, 0.25, 0.125, and 0.0625, corresponding to the fractions 1/2, 1/4, 1/8, and 1/16.$

Example

A source falls through $4$ half-lives. Give the net decline as a ratio.

After $4$ half-lives the fraction remaining is $124=116\dfrac{1}{2^{4}} = \dfrac{1}{16}$ .

So final : initial $= 1 : 16$ , which is the same as an initial : final ratio of $16 : 1$ .

Note

This ratio calculation is only needed at Higher tier.

Common Mistake

Read whether the question wants the amount remaining compared with the start ( $1 : 16$ ) or the decline written as start compared with end ( $16 : 1$ ).
State which way round your ratio goes so its meaning is clear.

Self review

What fraction of a source remains after 5 half-lives?
Write the net decline after 5 half-lives as an initial : final ratio.
What fraction remains after 3 half-lives?
How do you work out the fraction remaining after $n$ half-lives?

4.2.3a Half-lives and the random nature of radioactive decay

Decay is random for one nucleus but predictable for a whole sample

Definition

Half-life

The time taken for the number of unstable nuclei in a sample to halve, or for the count-rate (or activity) to fall to half its starting value.

You cannot predict when any single nucleus will decay, because decay is random.
A real sample contains a huge number of nuclei, so on average its activity falls in a predictable way.
This predictable pattern is described by the half-life.

Half-life: the time for the count-rate to halve

The half-life is the time it takes for the number of unstable nuclei, or the count-rate, to fall to a half.
After each half-life the count-rate halves again, so after $n$ half-lives the fraction remaining is $(12)n\left(\tfrac{1}{2}\right)^{n}$ .
This gives: remaining count-rate $=$ initial count-rate $×(12)n\times \left(\tfrac{1}{2}\right)^{n}$ .

A graph showing exponential decay over four half-lives. The y-axis shows the amount starting at 1.0 and halving to 0.5, 0.25, 0.125, and 0.0625 at each successive half-life on the x-axis.

Example

A source has a count-rate of $640$ counts/s and a half-life of $6$ days. Find the count-rate after $18$ days.

$days18\ \text{days}$ is $18 \div 6 = 3$ half-lives, so halve the count-rate three times:

\rightarrow 320 \rightarrow 160 \rightarrow 80\ \text{counts/s}

After $18$ days the count-rate is $80$ counts/s.

Common Mistake

Half-life is not the time for the sample to decay completely, nor half of that time.
It is the time for the count-rate to fall by half, and the same length of time halves it again.

Self review

What is meant by the half-life of a radioactive isotope?
Why can the decay of a whole sample be predicted even though single nuclei are random?
What fraction of the original nuclei remains after 3 half-lives?
A source has a count-rate of $800$ counts/s and a half-life of $2$ hours; what is the count-rate after $6$ hours?

4.2.3b Net decline after a number of half-lives

Net decline compares the radiation left with what you started with

Definition

Net decline

The overall fall in a radioactive emission after a number of half-lives, usually written as a ratio comparing the final amount with the initial amount.

After each half-life the count-rate falls to a half, so after $n$ half-lives the fraction remaining is $(12)n=12n\left(\tfrac{1}{2}\right)^{n} = \dfrac{1}{2^{n}}$ .
The net decline is written as a ratio comparing the initial and final count-rates.
After $1$ , $2$ , $3$ and $4$ half-lives the fraction left is $12\tfrac{1}{2}$ , $14\tfrac{1}{4}$ , $18\tfrac{1}{8}$ and $116\tfrac{1}{16}$ .

Example

A source falls through $4$ half-lives. Give the net decline as a ratio.

After $4$ half-lives the fraction remaining is $124=116\dfrac{1}{2^{4}} = \dfrac{1}{16}$ .

So final : initial $= 1 : 16$ , which is the same as an initial : final ratio of $16 : 1$ .

Note

This ratio calculation is only needed at Higher tier.

Common Mistake

Read whether the question wants the amount remaining compared with the start ( $1 : 16$ ) or the decline written as start compared with end ( $16 : 1$ ).
State which way round your ratio goes so its meaning is clear.

Self review

What fraction of a source remains after 5 half-lives?
Write the net decline after 5 half-lives as an initial : final ratio.
What fraction remains after 3 half-lives?
How do you work out the fraction remaining after $n$ half-lives?

4.2.3 Half-lives and the random nature of radioactive decay

4.2.3a Half-lives and the random nature of radioactive decay

Decay is random for one nucleus but predictable for a whole sample

Half-life: the time for the count-rate to halve

4.2.3b Net decline after a number of half-lives

Net decline compares the radiation left with what you started with

Review 4.2.3 Half-lives and the random nature of radioactive decay by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

4.2.3 Half-lives and the random nature of radioactive decay

4.2.3a Half-lives and the random nature of radioactive decay

Decay is random for one nucleus but predictable for a whole sample

Half-life: the time for the count-rate to halve

4.2.3b Net decline after a number of half-lives

Net decline compares the radiation left with what you started with

Review 4.2.3 Half-lives and the random nature of radioactive decay by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

4.2.3 Half-lives and the random nature of radioactive decay

4.2.3a Half-lives and the random nature of radioactive decay

Decay is random for one nucleus but predictable for a whole sample

Half-life: the time for the count-rate to halve

4.2.3b Net decline after a number of half-lives

Net decline compares the radiation left with what you started with

Review 4.2.3 Half-lives and the random nature of radioactive decay by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

4.2.3 Half-lives and the random nature of radioactive decay Revision Guide

4.2.3 Half-lives and the random nature of radioactive decay

4.2.3a Half-lives and the random nature of radioactive decay

Decay is random for one nucleus but predictable for a whole sample

Half-life: the time for the count-rate to halve

4.2.3b Net decline after a number of half-lives

Net decline compares the radiation left with what you started with

Review 4.2.3 Half-lives and the random nature of radioactive decay by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

4.2.3 Half-lives and the random nature of radioactive decay Revision Guide