Radioactive Decay Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radioactive Decay.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Radioactive decay is the spontaneous change of an unstable atomic nucleus into a more stable one, often releasing particles or electromagnetic radiation in the process.

Some nuclei are unstable and naturally break down over time.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Radioactive Decay asks whether the system is nuclear, quantum, or relativistic before using an everyday model.

Common stuck point: Students often know a formula related to radioactive decay but skip the recognition step: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough?

Worked Examples

Example 1

hard

A sample contains two isotopes equally at

t = 0

. Isotope A has

T = 1\,\text{hr}

, isotope B has

T = 4\,\text{hr}

. At

t = 4\,\text{hr}

, what fraction of the surviving sample is isotope B?

Answer

\approx 88.9\%

First step

After

4\,\text{hr}

, A:

(1/2)^4 = 1/16

remains.

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Example 2

hard

A radioactive source has activity

A_0 = 8000\,\text{Bq}

initially and

1000\,\text{Bq}

after

9\,\text{hours}

. Find

\lambda

and

T_{1/2}

Example 3

challenge

Radon-222 (

T_{1/2} = 3.82\,\text{days}

) is in equilibrium with its parent. After the parent is removed, find the time for the radon activity to drop to

5\%

of its initial value.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A sample has half-life

5 \text{ years}

. What fraction remains after

5 \text{ years}

Example 2

easy

A sample has half-life

3 \text{ days}

. What fraction remains after

9 \text{ days}

Example 3

easy

Starting with

800

atoms and half-life

T

, how many remain after

2

half-lives?

Example 4

easy

Is radioactive decay a linear or exponential decrease over time?

Example 5

easy

1000 \text{ g}

sample has half-life

10 \text{ hours}

. How much remains after

10 \text{ hours}

Example 6

easy

Does the half-life depend on how much of the substance you start with?

Example 7

easy

A sample drops to

1/4

of its original amount. How many half-lives passed?

Example 8

easy

After many half-lives, does the sample ever reach exactly zero atoms by this model?

Example 9

medium

A sample has half-life

4 \text{ hours}

. What fraction remains after

12 \text{ hours}

Example 10

medium

640 \text{ g}

of an isotope (half-life

2 \text{ days}

) decays for

8 \text{ days}

. Find the mass remaining.

Example 11

medium

A sample decays from

1000

125

atoms. Given half-life

6 \text{ hours}

, find the elapsed time.

Example 12

medium

Carbon-14 has half-life

5730 \text{ years}

. A fossil has

1/4

of its original C-14. Find its age.

Example 13

medium

200 \text{ g}

sample (half-life

1 \text{ hour}

) is left for

3 \text{ hours}

. How much has decayed (not remaining)?

Example 14

medium

Two isotopes have half-lives

2 \text{ days}

and

6 \text{ days}

. After

6 \text{ days}

, which has more of its original fraction remaining?

Example 15

medium

A radioactive source has half-life

20 \text{ minutes}

. What fraction remains after

1 \text{ hour}

Example 16

medium

960 ext{ g}

sample (half-life

5 ext{ years}

) decays for

15 ext{ years}

. Find the mass remaining.

Example 17

medium

A sample falls to

1/16

of its original amount. Its half-life is

4 ext{ days}

. Find the elapsed time.

Example 18

challenge

A sample has half-life

T

. After

2.5

half-lives, find the fraction remaining (use

(1/2)^{2.5}

Example 19

challenge

A sample of

N_0

decays with half-life

T

. Express its decay constant

\lambda

and find the fraction remaining after time

T

using

N = N_0 e^{-\lambda t}

Example 20

challenge

Two samples start equal. Sample A (half-life

1 \text{ h}

) and B (half-life

3 \text{ h}

). After

3 \text{ hours}

, find the ratio of remaining amounts A:B.

Example 21

easy

Starting with

1600

atoms with half-life

T

, how many remain after

3T

Example 22

easy

200\,\text{g}

sample with half-life

4\,\text{hours}

is left for

8\,\text{hours}

. How much remains?

Example 23

easy

A sample of

10000

atoms (half-life

1\,\text{day}

) is observed for

4\,\text{days}

. How many atoms remain?

Example 24

easy

After exactly one half-life, what fraction of the original sample has decayed?

Example 25

medium

A sample has half-life

6\,\text{minutes}

. What fraction remains after

24\,\text{minutes}

Example 26

medium

A sample drops from

1024

atoms to

32

atoms in

25\,\text{minutes}

. Find the half-life.

Example 27

medium

Carbon-14 (

T_{1/2} = 5730\,\text{yr}

). A fossil has

1/8

of its original C-14 left. Find its age.

Example 28

medium

A sample has

\lambda = 0.231\,\text{hr}^{-1}

. Find its half-life.

Example 29

medium

An isotope with half-life

T = 10\,\text{years}

has decay constant

\lambda =

Example 30

medium

4000\,\text{g}

sample (half-life

5\,\text{years}

) decays for

20\,\text{years}

. Find the mass remaining.

Example 31

medium

Two isotopes have half-lives

T

and

3T

. After time

3T

, what is the ratio of remaining fractions (short:long)?

Example 32

medium

An isotope has activity

A_0

initially and

A_0/8

after

30\,\text{minutes}

. Find its half-life.

Example 33

medium

A sample has

N_0 = 10^6

atoms and

\lambda = 0.1\,\text{day}^{-1}

. How many remain after

10\,\text{days}

Example 34

medium

What is the mean lifetime

\tau

of an isotope with

T_{1/2} = 14\,\text{days}

Example 35

medium

Compute the activity

A = \lambda N

N = 10^{20}

atoms with

\lambda = 10^{-10}\,\text{s}^{-1}

Example 36

hard

Iodine-131 has

T_{1/2} = 8.0\,\text{days}

. A medical dose has initial activity

200\,\text{MBq}

. Find the activity after

24\,\text{days}

Example 37

hard

An ancient organic sample contains

10\%

of its original carbon-14. Use

T_{1/2} = 5730\,\text{yr}

to estimate its age.

Example 38

hard

How long until

99\%

of a sample has decayed if its half-life is

50\,\text{years}

Example 39

hard

After

1.5

half-lives, what fraction of a radioactive sample remains? Use

(1/2)^{1.5}

Example 40

challenge

An isotope decays into a stable daughter. After

4

half-lives, what is the ratio of daughter atoms to surviving parent atoms?

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Background Knowledge

These ideas may be useful before you work through the harder examples.

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