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Activity & Half-Life (DP IB Physics): Revision Note

Katie M

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Katie M

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DP Physics: HLDP Physics: SL

Activity & Half-Life

  • The activity of a radioactive sample is defined as:

The number of nuclei which decay in a given time

  • Activity is measured in becquerels (Bq)

    • One becquerel is equivalent to a nucleus decaying every second

  • It is impossible to know when a particular unstable nucleus will decay

  • But the rate at which the activity of a sample decreases can be predicted

    • This is known as the half-life

  • Half-life is defined as:

The time taken for half the undecayed nuclei to decay or the activity of a source to decay by half

  • In other words, the time it takes for the activity of a sample to fall to half its original level

  • Different isotopes have different half-lives, and these can vary from a fraction of a second to billions of years in length

Using Half-life

  • Scientists can measure the half-lives of different isotopes accurately:

  • Uranium-235 has a half-life of 704 million years

    • This means it would take 704 million years for the activity of a uranium-235 sample to decrease to half its original amount

  • Carbon-14 has a half-life of 5700 years

    • So after 5700 years, there would be 50% of the original amount of carbon-14 remaining

    • After two half-lives, or 11 400 years, there would be just 25% of the carbon-14 remaining

  • With each half-life, the amount remaining decreases by half

Half-life Graph, downloadable IGCSE & GCSE Physics revision notes

Graph showing how the activity of a radioactive sample changes over time. Each time the original activity halves, another half-life has passed

  • The time it takes for the activity of the sample to decrease from 100 % to 50 % is the half-life

  • It is the same length of time as it would take to decrease from 50 % activity to 25 % activity

  • The half-life is constant for a particular isotope

  • The proportion of an isotope remaining after n half-lives has passed can be calculated using:

proportion of isotope remaining = open parentheses 1 half close parentheses to the power of n

  • The following table shows that as the number of half-lives increases, both the proportion of the isotope remaining and the activity of the sample halves

Number of half-lives

Proportion of isotope remaining

Activity of sample

0

1

A subscript 0

1

1 half

A subscript 0 over 2

2

1 fourth

A subscript 0 over 4

3

1 over 8

A subscript 0 over 8

4

1 over 16

A subscript 0 over 16

Worked Example

A radioactive sample has a half-life of 3 years. What is the ratio of decayed nuclei to original nuclei after 15 years?

Answer:

Step 1: Calculate the number of half-lives

  • The time period is 15 years

  • The half-life is 3 years

number of half-lives = 15 over 3 = 5

  • Therefore, 5 half-lives have passed

Step 2: Determine the proportion of nuclei remaining

  • The proportion of nuclei remaining is:

open parentheses 1 half close parentheses to the power of 5 space equals space 1 over 32

  • So 1/32 of the original nuclei are remaining

Step 3: Write the ratio correctly

  • If 1/32 of the original nuclei are remaining, then 31/32 must have decayed

  • Therefore, the ratio is 31 decayed : 32 original, or 31:32

Worked Example

A particular radioactive sample contains 2 million un-decayed atoms. After a year, there are only 500 000 atoms left un-decayed.

Determine the half-life of the material.

Answer:

Step 1: Calculate how many times the number of un-decayed atoms has halved

  • There were 2 000 000 atoms to start with

  • 1 000 000 atoms would remain after 1 half-life

  • 500 000 atoms would remain after 2 half-lives

  • Therefore, the sample has undergone 2 half-lives

Step 2: Divide the time period by the number of half-lives

  • The time period is a year

  • The number of half-lives is 2

  • So 1 year represents two half-lives, and 6 months represents one half-life

  • Therefore, the half-life of the sample is 6 months

Decay Curves

  • To calculate the half-life of a sample, the procedure is:

    • Measure the initial activity, A0, of the sample

    • Measure how the activity changes with time

    • Determine the half-life of this original activity

  • The time taken for the activity to decrease to half its original value is the half-life

Worked Example

The radioisotope technetium is used extensively in medicine. The graph below shows how the activity of a sample varies with time.

Worked Example - Half Life Curve, downloadable AS & A Level Physics revision notes

Determine the half-life of this material.

Answer:

Step 1: Draw lines on the graph to determine the time it takes for technetium to drop to half of its original activity

Worked Example - Half Life Curve Ans a, downloadable AS & A Level Physics revision notes

Step 2: Read the half-life from the graph

  • In the diagram above the initial activity, A0, is 8 × 107 Bq

  • The time taken to decrease to 4 × 107 Bq, or ½A0, is 6 hours

  • The time taken to decrease to 2 × 107 Bq is 6 more hours

  • The time taken to decrease to 1 × 107 Bq is 6 more hours

  • Therefore, the half-life of this isotope is 6 hours

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    Katie M

    Author: Katie M

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    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.