Half-Life Examples in Chemistry

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Half-Life.

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

Concept Recap

Half-life is the time required for half of the radioactive nuclei in a sample to decay.

Radioactive samples do not lose the same amount each time; they lose the same fraction each 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: Half-Life starts by identifying the starting amount, the half-life of the isotope, and the elapsed time, then counting how many half-lives have passed (n = elapsed time divided by the half-life) to find how much remains.

Common stuck point: Students often know a formula related to half-life but skip the recognition step: Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion? That leads to a correct-looking substitution attached to the wrong chemical model.

Sense of Study hint: Ask: Am I using particle counts, nuclear charge, mass number, electron arrangement, or isotope notation to describe an atom or ion?

Worked Examples

Example 1

medium

An ancient artifact has

\tfrac{1}{16}

of its original C-14. With C-14 half-life of

5730

years, how old is it?

Answer

22920\ \text{years}

First step

(1/2)^n = 1/16 \Rightarrow n = 4

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Example 2

medium

An isotope has

t_{1/2} = 12

years. After

36

years, what percent of original remains?

Example 3

medium

Strontium-90 has half-life

29

years. A contaminated site has

20\text{ Bq/g}

activity initially. What is the activity after

87

years?

Example 4

hard

A medical isotope must drop below

1\%

of injected activity before discharge. If half-life is

6

hours, how long until below

1\%

Example 5

hard

Two isotopes A (

t_{1/2}=2

hr) and B (

t_{1/2}=6

hr) start with equal masses. After

6

hours, what is mass ratio

A:B

Example 6

hard

A rock contains equal moles of K-40 (parent) and Ar-40 (daughter), no Ar-40 originally. K-40 half-life is

1.25 \times 10^9

years. Age?

Example 7

challenge

An isotope decays via

N = N_0 e^{-\lambda t}

with

\lambda = 0.0231\text{ /year}

. Approximate its half-life.

Practice Problems

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

Example 1

easy

A radioactive sample has a half-life of 4 days. What fraction remains after one half-life?

Example 2

easy

Starting with 80 g of an isotope, how much remains after one half-life?

Example 3

easy

How many half-lives have passed when a sample is reduced to

\frac{1}{8}

of its original amount?

Example 4

easy

An isotope has a half-life of 10 years. How many years for 2 half-lives to pass?

Example 5

easy

Starting with 200 g, how much remains after 2 half-lives?

Example 6

easy

Will a radioactive sample ever reach exactly zero atoms after a finite number of half-lives, in theory?

Example 7

easy

Fraction remaining after 4 half-lives is what?

Example 8

easy

If 100 g decays to 25 g, how many half-lives passed?

Example 9

medium

An isotope has a half-life of 5 years. Starting from 100 g, how much remains after 15 years?

Example 10

medium

A sample's half-life is 2 hours. After 6 hours, what fraction of the original remains?

Example 11

medium

Carbon-14 has a half-life of about 5730 years. A fossil has

\frac14

of its original C-14. How old is it?

Example 12

medium

An isotope decays from 64 g to 4 g. If its half-life is 3 minutes, how much time has passed?

Example 13

medium

A 500 g sample drops to 62.5 g. How many half-lives passed?

Example 14

medium

A student says 'after one half-life half the sample is gone, so after two half-lives it is all gone.' Correct the error.

Example 15

medium

How much of a 1 g sample remains after 5 half-lives, as a fraction?

Example 16

medium

A sample of 320 g has a half-life of 8 days. How much remains after 24 days?

Example 17

medium

An isotope's half-life is 6 hours. What percent of the original remains after 18 hours?

Example 18

challenge

Two isotopes start with equal masses. Isotope A has half-life 1 hour, isotope B has half-life 2 hours. After 4 hours, what is the ratio of remaining mass A to mass B?

Example 19

challenge

A sample is 6.25% of its original amount. How many half-lives have passed?

Example 20

challenge

An isotope decays to 12.5% of its original amount in 30 years. What is its half-life?

Example 21

easy

An isotope has half-life

7

years. How long until 1 half-life elapses?

Example 22

easy

Starting with

160\text{ g}

of an isotope, how much remains after

1

half-life?

Example 23

easy

An isotope's half-life is

3

minutes. How long until

3

half-lives pass?

Example 24

easy

200\text{ g}

decays to

25\text{ g}

, how many half-lives passed?

Example 25

easy

An isotope with half-life

20

days starts at

400\text{ g}

. How much remains after

40

days?

Example 26

medium

Iodine-131 has half-life

8

days. Starting from

640\text{ mg}

, how much remains after

24

days?

Example 27

medium

96\text{ g}

sample of Tc-99m (half-life

6

hours) is administered. How much remains after

18

hours?

Example 28

medium

A sample loses

87.5\%

of its activity. How many half-lives have passed?

Example 29

medium

An isotope's mass drops from

800\text{ g}

25\text{ g}

30

days. What is its half-life?

Example 30

medium

Two isotopes start with equal masses. Isotope A: half-life

4

hours. Isotope B: half-life

8

hours. After

8

hours, which has more remaining?

Example 31

medium

An isotope has half-life

25

minutes. Starting with

1024\text{ atoms}

, how many remain after

100

minutes?

Example 32

medium

An isotope has half-life

50

years. After

200

years, what fraction of the original sample is left?

Example 33

medium

Uranium-238 has a half-life of

4.5 \times 10^9

years. After

9 \times 10^9

years, what fraction remains?

Example 34

hard

An isotope's activity drops from

480\text{ Bq}

30\text{ Bq}

20\text{ hr}

. Find the half-life.

Example 35

hard

640\text{ g}

of Cs-137 (

t_{1/2}=30

years), how much remains after

120

years?

Example 36

hard

A sample contains

80\%

parent and

20\%

daughter atoms (no daughter initially). How many half-lives have passed (to nearest)?

Example 37

hard

After how many half-lives does an isotope reach less than

0.1\%

of its original amount?

Example 38

hard

If an isotope has

t_{1/2}=10

days and you want

75\%

to have decayed, how long must you wait?

Example 39

challenge

A sample originally containing

1\text{ mol}

of isotope (

t_{1/2}=10

min) is measured at

0.125\text{ mol}

. The decay produces a stable daughter. How many moles of daughter are present and how long has passed?

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Related Concepts

Radioactivity

Background Knowledge

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

radioactivity