A LevelPhysicsAQARevision NotesNuclear PhysicsRadioactive DecayHalf-Life

Half-Life (AQA A Level Physics) : Revision Note

Katie M

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Katie M

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Half-Life

  • Half-life is defined as:

The average time taken for a given number of nuclei of a particular isotope to halve

  • Since activity A is proportional to the number of undecayed nuclei N, the activity of the sample will also halve

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

When a time equal to the half-life passes, the activity falls by half, when two half-lives pass, the activity falls by another half (which is a quarter of the initial value)

  • To find an expression for half-life, start with the equation for exponential decay:

N space equals space N subscript 0 space e to the power of negative lambda t end exponent

  • Where:

    • N = number of nuclei remaining in a sample

    • N0 = the initial number of undecayed nuclei (when t = 0)

    • λ = decay constant (s-1)

    • t = time interval (s)

  • When time t is equal to the half-life t subscript 1 divided by 2 end subscript, the activity N of the sample will be half of its original value, so N space equals space 1 half N subscript 0

1 half N subscript 0 space equals space N subscript 0 space e to the power of negative lambda t subscript 1 divided by 2 end subscript end exponent

  • The formula can then be derived by first, dividing both sides by N subscript 0:

1 half space equals space e to the power of negative lambda t subscript 1 divided by 2 end subscript end exponent

  • Then, taking the natural log of both sides:

ln space open parentheses 1 half close parentheses space equals space minus lambda t subscript 1 divided by 2 end subscript

  • Finally, applying properties of logarithms:

lambda t subscript 1 divided by 2 end subscript space equals space ln space 2

  • Therefore, half-life t subscript 1 divided by 2 end subscript can be calculated using the equation:

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator lambda end fraction space asymptotically equal to space fraction numerator 0.693 over denominator lambda end fraction

  • This equation shows that half-life t subscript 1 divided by 2 end subscript and the radioactive decay rate constant λ are inversely proportional

    • Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

Worked Example

Strontium-90 is a radioactive isotope with a half-life of 28.0 years. A sample of Strontium-90 has an activity of 6.4 × 109 Bq.

Calculate the decay constant lambda, in s–1, of Strontium-90.

Answer:

Step 1: Convert the half-life into seconds

  • t subscript 1 divided by 2 end subscript = 28 years = 28 × (365 × 24 × 60 × 60) = 8.83 × 108 s

Step 2: Write the equation for half-life

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator lambda end fraction

Step 3: Rearrange for λ and calculate

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator 8.83 cross times 10 to the power of 8 end fraction space equals space 7.85 cross times 10 to the power of negative 10 end exponent space straight s to the power of negative 1 end exponent

Examiner Tips and Tricks

Although you may not be expected to derive the half-life equation, make sure you're comfortable with how to use it in calculations such as that in the worked example.

Half-Life from Decay Curves

  • The half-life of a radioactive substance can be determined from decay curves and log graphs

  • Since half-life is the time taken for the initial number of nuclei (or activity) to reduce by half, it can be found by

    • drawing a line to the curve at the point where the activity has dropped to half of its original value

    • drawing a line from the curve to the time axis, this is the half-life

Log Graphs

  • Straight-line graphs tend to be more useful than curves for interpreting data

    • Nuclei decay exponentially, therefore, to achieve a straight line plot, logarithms can be used

  • Take the exponential decay equation for the number of nuclei

N space equals space N subscript 0 space e to the power of negative lambda t end exponent

  • Taking the natural logs of both sides

ln space N space equals space ln space open parentheses N subscript 0 close parentheses space minus space lambda t

  • In this form, this equation can be compared to the equation of a straight line

y space equals space m x space plus thin space c

  • Where:

    • y-axis variable, y space equals space ln space N

    • x-axis variable, x space equals space t

    • gradient, m space equals space minus lambda

    • y-intercept, c space equals space ln space open parentheses N subscript 0 close parentheses

  • Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below:

Half Life Decay Curves 1, downloadable AS & A Level Physics revision notes
Half Life Decay Curves 2, downloadable AS & A Level Physics revision notes

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:

a) The decay constant for technetium

b) The number of technetium atoms remaining in the sample after 24 hours

Answer:

Part (a)

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 and convert to seconds

  • t subscript 1 divided by 2 end subscript = 6 hours = 6 × 60 × 60 = 21 600 s

Step 3: Write out the half life equation

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator lambda end fraction

Step 4: Calculate the decay constant

t subscript 1 divided by 2 end subscript space equals space fraction numerator ln space 2 over denominator 21 space 600 end fraction space equals space 3.2 cross times 10 to the power of negative 5 end exponent space straight s to the power of negative 1 end exponent

Part (b)

Step 1: Draw lines on the graph to determine the activity after 24 hours

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

  • At t = 24 hours, A = 0.5 × 107 Bq

Step 2: Write out the activity equation

A space equals space lambda N

Step 3: Calculate the number of atoms remaining in the sample

N space equals space A over lambda space equals space fraction numerator 0.5 cross times 10 to the power of 7 over denominator 3.2 cross times 10 to the power of negative 5 end exponent end fraction space equals space 1.56 cross times 10 to the power of 11

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Katie M

Author: Katie M

Expertise: Physics Content Creator

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.

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