Radioactive Decay Equations (OCR A Level Physics)

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Katie M


Katie M



Radioactive Decay Equations

  • In radioactive decay, the number of undecayed nuclei falls very rapidly, without ever reaching zero
    • Such a model is known as exponential decay

  • The graph of number of undecayed nuclei against time has a very distinctive shape:

Exponential Decay Graph, downloadable AS & A Level Physics revision notes

Radioactive decay follows an exponential pattern. The graph shows three different isotopes each with a different rate of decay

  • The key features of this graph are:
    • The steeper the slope, the larger the decay constant λ (and vice versa)
    • The decay curves always start on the y-axis at the initial number of undecayed nuclei (N0)

Equations for Radioactive Decay

  • The number of undecayed nuclei N can be represented in exponential form by the equation:

N = N0 e–λt

  • Where:
    • N0 = the initial number of undecayed nuclei (when t = 0)
    • N = number of undecayed nuclei at a certain time t
    • λ = decay constant (s-1)
    • t = time interval (s)

  • The number of nuclei can be substituted for other quantities.
  • For example, the activity A is directly proportional to N, so it can also be represented in exponential form by the equation:

A = A0 e–λt

  • Where:
    • A = activity at a certain time t (Bq)
    • A0 = initial activity (Bq)

  • The received count rate C is related to the activity of the sample, hence it can also be represented in exponential form by the equation:

C = C0 e–λt

  • Where:
    • C = count rate at a certain time t (counts per minute or cpm)
    • C0 = initial count rate (counts per minute or cpm)

The exponential function e

  • The symbol e represents the exponential constant
    • It is approximately equal to e = 2.718

  • On a calculator, it is shown by the button ex
  • The inverse function of ex is ln(y), known as the natural logarithmic function
    • This is because, if ex = y, then x = ln(y)

Worked example

Strontium-90 decays with the emission of a β-particle to form Yttrium-90.The decay constant of Strontium-90 is 0.025 year -1. Determine the activity A of the sample after 5.0 years, expressing the answer as a fraction of the initial activity A0.

Step 1: Write out the known quantities

    • Decay constant, λ = 0.025 year -1
    • Time interval, t = 5.0 years
    • Both quantities have the same unit, so there is no need for conversion

Step 2: Write the equation for activity in exponential form

A = A0 e–λ

Step 3: Rearrange the equation for the ratio between A and A0

The Exponential Nature of Radioactive Decay Worked Example equation 1

Step 4: Calculate the ratio A/A0

The Exponential Nature of Radioactive Decay Worked Example equation 2

Therefore, the activity of Strontium-90 decreases by a factor of 0.88, or 12%, after 5 years

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Katie M

Author: Katie M

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.