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Half-Life (CIE A Level Physics)

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Half-life definition

  • Half-life is defined as:

The time taken for the initial number of nuclei to reduce by half

  • This means when a time equal to the half-life has passed, the activity of the sample will also half
  • This is because the activity is proportional to the number of undecayed nuclei, A ∝ N

Determining half-life from a graph

Half-Life Graph, downloadable AS & A Level 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)

Calculating half-life

  • The half-life of a radioactive sample can be calculated using the equation:

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

  • Where:
    • t½ = half-life of the sample (s)
    • λ = decay constant of the sample (s-1)
  • This equation shows that:
    • the half-life and the decay constant of a sample are inversely proportional
    • the shorter the half-life, the larger the decay constant and the faster the rate of decay

Derivation of the half-life equation

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

N = N0eλt

  • 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 t½, the activity N of the sample will be half of its original value, so N = ½ N0

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

  • First, divide both sides by N0:

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

  • Then, take the natural log of both sides:

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

  • Finally, apply the properties of logarithms:

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

  • Therefore, half-life t½ can be calculated using the equation:

t subscript 1 half 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

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 λ, in s–1, of strontium-90.

Answer: 

Step 1: Convert the half-life into seconds

28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 108 s

Step 2: Write the equation for half-life

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

Step 3: Rearrange for λ and calculate

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

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Leander

Author: Leander

Expertise: Physics

Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.