Calculating Half-Life

Section 1 of 7The half-life rule

Definition

• Definition The half-life is the time taken for the number of undecayed nuclei of a radioactive isotope to halve.
  • It is also the time for the activity or count rate to halve.
• Each isotope has its own fixed half-life, from fractions of a second to billions of years.

Repeated halving

• Repeated halving After every half-life the amount remaining is multiplied by one half, so the pattern is exponential, not a straight line.

Section 2 of 7Amount remaining

Number of half-lives	Fraction remaining	Percentage remaining	Starting with 800 nuclei
0	1	100%	800
1	1/2	50%	400
2	1/4	25%	200
3	1/8	12.5%	100
4	1/16	6.25%	50

Section 3 of 7Finding a half-life from data

1

Question: a sample contains 2 000 000 undecayed atoms. After 1 year only 500 000 remain. Find the half-life.

2

Count the halvings: 2 000 000 to 1 000 000 is one half-life, and 1 000 000 to 500 000 is a second half-life, so two half-lives have passed.

3

Divide the time by the number of half-lives: 1 year ÷ 2 = 6 months.

4

State the answer: the half-life of the sample is 6 months.

Section 4 of 7Carbon dating

1Living things take in carbon

• Living things take in carbon While an organism is alive, the proportion of radioactive carbon-14 to stable carbon-12 in its body stays constant because carbon is constantly replaced.

2Death stops the replacement

• Death stops the replacement After death no new carbon-14 is taken in, and the carbon-14 already present decays.

3Compare with living tissue

• Compare with living tissue Measure how much carbon-14 is left compared with a living sample.
  • Carbon-14 has a half-life of about 5700 years.

4Calculate the age

• Calculate the age Work out the number of half-lives that have passed and multiply by the half-life to estimate the age.

Section 5 of 7Worked example: age of wood

1

Question: ancient wood contains 12.5% of the carbon-14 found in living wood. The half-life of carbon-14 is 5730 years. Estimate its age.

2

Halve repeatedly: 100% → 50% → 25% → 12.5%, so three half-lives have passed.

3

Multiply: age = 3 × 5730 years = 17 190 years.

4

This is an estimate, so it is sensible to round it, for example to about 17 000 years.

Section 6 of 7Key terms

Term	Meaning
Half-life	The time taken for the number of undecayed nuclei, or the activity, of a radioactive sample to fall to half its starting value.
Activity	The number of nuclei in a sample that decay each second.
Carbon dating	A method of finding the age of once-living material by measuring how much carbon-14 remains.
Carbon-14	A radioactive isotope of carbon with a half-life of about 5700 years.
Isotope	Atoms of the same element with different numbers of neutrons.

Section 7 of 7Exam technique

1

Thinking that after two half-lives the whole sample has gone. Half of the remaining half is left, which is a quarter.

2

Dividing the starting number by the time instead of counting halvings.

3

Forgetting to multiply the number of half-lives by the length of one half-life when finding an age.

4

Believing the half-life depends on the size of the sample. It is a fixed property of the isotope.

5

Treating a carbon-dating result as exact. It is an estimate.