D3.11 Explain radioactive half-life, applications, and consequences

Ontario Grade 11 Physics

Energy and Society

SPH3U study topic D3.11

Atoms of some elements have unstable nuclei. An unstable nucleus can change spontaneously and emit radiation. We call this radioactive decay. We cannot predict exactly when one particular nucleus will decay, but we can describe how the number of undecayed nuclei in a large sample changes over time. This lesson focuses on half-life, how it is used, and why radioactive materials must be handled and managed carefully.

What you will learn

Explain what radioactive half-life means.
Use a half-life model to find the amount remaining or the time elapsed.
Describe applications of radioactive materials and consider their benefits and risks.

1. The sample and the half-life model

First define the system: it is the radioactive sample being considered. We track the undecayed radioactive nuclei in that sample. The quantity we count,

N

, is a number of nuclei, so it has no SI unit. We can also track a sample's mass or its activity. Activity is the number of nuclear decays per second, measured in becquerels; one becquerel means one decay per second.

A half-life is the time needed for half of the radioactive nuclei in a sample to decay. It is a property of a particular radioactive isotope. An isotope is a form of an element with a particular number of neutrons. After one half-life, half the original nuclei remain undecayed. After a second half-life, half of that remainder is left.

The sample's amount and activity are scalar quantities: they have size but no direction. The sample does not move in a positive or negative direction. For this lesson, time

t

is measured forward from the start of observation, and the remaining amount decreases as time increases. This is a sign convention for describing change, not a motion direction.

The half-life model assumes the sample follows its characteristic half-life over the time considered. If

T_{1/2}

is the half-life and

N_0

is the initial number of radioactive nuclei, then the remaining number after time

t

is found by counting how many half-lives have passed. The same fraction model applies to the mass of the radioactive isotope or its activity, provided the comparison is for the same sample and isotope.

Time units must match. If the half-life is given in days, use days for elapsed time; if it is in years, use years. The model describes the expected pattern for a large sample. It does not say that exactly half of a small handful of nuclei must decay during each half-life.

N=N_0\left(\frac{1}{2}\right)^{t/T_{1/2}}

Half-life is the time for half of the radioactive nuclei in a sample to decay.
After each half-life, half of the amount remaining is left.
Keep the units of elapsed time and half-life consistent.

2. Reading and using the pattern

The ratio

t/T_{1/2}

tells how many half-lives have passed. It is a count, so it has no unit. If that count is a whole number, repeated halving is an easy way to find the remainder. For a fraction of a half-life, the same model gives a fractional exponent.

A simple amount-versus-time sketch would start at

N_0

when

t=0

, fall to

N_0/2

at

t=T_{1/2}

, then to

N_0/4

at

t=2T_{1/2}

and

N_0/8

at

t=3T_{1/2}

. The curve slopes downward but does not reach zero in a fixed number of half-lives. A real sample has a finite number of nuclei, so the model is most useful for describing the overall pattern.

Sometimes the question gives the remaining fraction and asks for elapsed time. Identify how many halvings match that fraction, then multiply by the half-life. For example, one quarter remaining means two half-lives have passed. If a fraction does not correspond to a whole number of halvings, use the model to find the elapsed time.

A useful check is that a positive elapsed time should give a remaining amount less than the initial amount. A longer elapsed time should not give a larger remainder. Keep the answer's units attached to time quantities, and state clearly whether a result refers to nuclei, mass, or activity.

t=nT_{1/2}

The amount remaining falls by a factor of two each half-life.
A time ratio such as $t/T_{1/2}$ has no unit.
For forward time, the remaining amount cannot exceed the initial amount.

3. Applications and consequences

Half-life helps people choose and use radioactive materials. In medicine, a radioactive tracer can be used to follow a process in the body or help produce an image. A tracer is a radioactive substance used to provide information. Its radiation can be detected from outside the body or measured with suitable equipment. A relatively short half-life can be useful because the activity decreases over time, but the substance must last long enough for the intended procedure.

Radioactive decay can also help estimate the age of some materials. The method compares the amount of a radioactive isotope remaining with an amount or pattern used for comparison. It is useful only when the isotope and material suit the question and the assumptions of the method are reasonable. Different isotopes have different half-lives, so the choice affects what time spans can be studied.

Radioactive materials can also be used to irradiate objects, such as in some sterilization processes. Irradiation means exposing an object to radiation. It can reduce harmful microorganisms, but it requires controlled equipment and procedures. The use of radiation does not mean that the treated object itself necessarily becomes radioactive.

The benefits come with consequences. Radiation can damage living cells, and the level of risk depends on the radiation exposure. Radioactive materials therefore need appropriate handling, shielding where suitable, access controls, and monitoring. Radioactive waste must be managed so that people and the environment are not exposed to harmful levels. Materials with long half-lives can remain radioactive for a long time; short half-life does not by itself make a material safe while its activity is still significant.

A half-life alone does not tell the whole safety story. The amount of material, its activity, the kind of radiation, the route of exposure, and the time near the source all matter. Follow trained guidance and safety rules rather than judging safety from a half-life calculation alone.

Applications include medical tracers, age estimation, and controlled irradiation.
Radiation can be useful, but it can also damage living tissue.
Safe use and waste management depend on more than half-life alone.

Amount remaining after whole half-lives

Half-lives passed	Fraction remaining	Percent remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%

Worked example

Finding the remaining amount

A sample begins with

80.0\ \mathrm{mg}

of a radioactive isotope. Its half-life is

6.00\ \mathrm{h}

. Find the radioactive isotope mass remaining after

18.0\ \mathrm{h}

.

Set up the sample
The system is the radioactive isotope in the sample. Time is measured forward from the start, so the remaining mass decreases. The initial mass is $80.0\ \mathrm{mg}$ , the half-life is $6.00\ \mathrm{h}$ , and the elapsed time is $18.0\ \mathrm{h}$ . The unknown is the remaining mass.
Count the half-lives
The elapsed time and half-life are both in hours. Their ratio gives the number of half-lives that have passed.
$n=\frac{18.0\ \mathrm{h}}{6.00\ \mathrm{h}}=3.00$
Apply repeated halving
After three half-lives, the mass has been halved three times. Substitute the initial mass and the half-life count.
$m=80.0\ \mathrm{mg}\left(\frac{1}{2}\right)^3=10.0\ \mathrm{mg}$

Answer: The remaining radioactive isotope mass is

10.0\ \mathrm{mg}

.

Check: The units remain milligrams. Three half-lives leave one eighth of the initial mass, and

10.0\ \mathrm{mg}

is one eighth of

80.0\ \mathrm{mg}

. It is positive and less than the starting mass, as expected.

Worked example

Finding the elapsed time

A radioactive sample has a half-life of

4.50\ \mathrm{d}

. Its measured activity is one quarter of its initial activity. How much time has elapsed?

Define the quantity
The system is the radioactive sample. Activity is the number of decays per second, measured in becquerels, and it is a scalar. The fraction remaining is one quarter. The unknown is elapsed time, measured in days.
Relate the fraction to halvings
One half-life leaves one half. A second half-life leaves half of that, which is one quarter of the initial activity. Therefore, two half-lives have elapsed.
$\left(\frac{1}{2}\right)^2=\frac{1}{4}$
Calculate the time
Multiply the number of half-lives by the half-life. The answer is reported to three significant figures, matching the given half-life.
$t=2(4.50\ \mathrm{d})=9.00\ \mathrm{d}$

Answer: The elapsed time is

9.00\ \mathrm{d}

.

Check: Days are the correct time unit. The time is positive, and two half-lives produce one quarter of the initial activity. The result is consistent with the stated remaining fraction.

Worked example

Comparing two samples

Two samples each begin with

64.0\ \mathrm{g}

of radioactive isotope. Sample A has a half-life of

2.00\ \mathrm{h}

, and Sample B has a half-life of

8.00\ \mathrm{h}

. Find the remaining isotope mass in each sample after

8.00\ \mathrm{h}

.

Define both systems
Each sample is a separate system. Time is measured forward. Mass is a scalar, so no direction is needed. Both initial masses are $64.0\ \mathrm{g}$ ; the elapsed time is $8.00\ \mathrm{h}$ .
Count Sample A half-lives
Sample A's elapsed time is four times its half-life. Halve its mass four times.
$m_A=64.0\ \mathrm{g}\left(\frac{1}{2}\right)^{8.00\ \mathrm{h}/2.00\ \mathrm{h}}=4.00\ \mathrm{g}$
Count Sample B half-lives
Sample B has completed one half-life in the same elapsed time. Only one halving has occurred.
$m_B=64.0\ \mathrm{g}\left(\frac{1}{2}\right)^{8.00\ \mathrm{h}/8.00\ \mathrm{h}}=32.0\ \mathrm{g}$

Answer: Sample A has

4.00\ \mathrm{g}

remaining, while Sample B has

32.0\ \mathrm{g}

remaining.

Check: Both answers have units of grams and are less than their initial masses. Sample A has the shorter half-life, so more of it decays in the same

8.00\ \mathrm{h}

; its smaller remainder is reasonable.

Common mistakes and how to avoid them

Assuming that one half-life means all radioactive nuclei have decayed.

Correction: One half-life means half of the radioactive nuclei have decayed; the other half remain undecayed.

Halving the original amount only once, even when several half-lives have passed.

Correction: Halve the amount remaining after each half-life, or use the half-life model.

Assuming a shorter half-life always means a sample is safe sooner.

Correction: A short half-life means activity decreases more quickly, but safety also depends on the amount, radiation, and exposure conditions.

Treating a half-life calculation as an exact prediction for every individual nucleus.

Correction: The half-life model describes the expected pattern for a sample. It does not predict the decay time of a particular nucleus.

Lesson summary

Radioactive half-life is the time for half of a sample's radioactive nuclei to decay.
After each half-life, half of the previous remainder is left.
Keep elapsed-time units consistent with the half-life units.
Radioactive materials have applications, but their use and waste require careful safety practices.

Check your understanding

Question 1

A sample has a half-life of

5.0\ \mathrm{d}

. What fraction remains after

15.0\ \mathrm{d}

?

One half
One quarter
One eighth
Three quarters

Show answer and explanation

One eighth

The elapsed time is three half-lives. Repeated halving leaves one eighth.

Question 2

A sample's activity falls to one half of its initial value. What has happened, according to the half-life model?

One half-life has passed.
Two half-lives have passed.
All radioactive nuclei have decayed.
The sample's half-life has doubled.

Show answer and explanation

One half-life has passed.

After one half-life, one half of the initial amount or activity remains.

Question 3

Which statement about radioactive materials is most accurate?

A long half-life means a sample has no radiation.
A short half-life makes every sample immediately safe.
Radiation has applications, and exposure and waste still need careful management.
Half-life tells exactly when each nucleus will decay.

Show answer and explanation

Radiation has applications, and exposure and waste still need careful management.

Radioactive materials can be useful, but half-life alone does not establish safety. Exposure and waste must be managed.

Key terms

Radioactive decay: A spontaneous change in an unstable nucleus that emits radiation.
Isotope: A form of an element with a particular number of neutrons.
Half-life: The time needed for half of the radioactive nuclei in a sample to decay.
Activity: The number of nuclear decays per second in a sample.
Becquerel: The SI unit of activity; one becquerel is one decay per second.
Tracer: A radioactive substance used to provide information about a process or location.
Irradiation: Exposure of an object or material to radiation.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation D3.11. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

D3.11 · Explain radioactive half-life, applications, and consequences

Ontario Grade 11 Physics

SPH3U study topic D3.11

What you will learn

1. The sample and the half-life model

2. Reading and using the pattern

3. Applications and consequences

Amount remaining after whole half-lives

Finding the remaining amount

Finding the elapsed time

Comparing two samples

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH3U

About this lesson and its review