CIE A Level Physics (9702) 2019-2021

Revision Notes

28.2.2 Activity & The Decay Constant

Activity & The Decay Constant

  • Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which are expected to decay per unit time
    • This is known as the average decay rate
  • As a result, each radioactive element can be assigned a decay constant
  • The decay constant λ is defined as:

 The probability that an individual nucleus will decay per unit of time

  • When a sample is highly radioactive, this means the number of decays per unit time is very high
    • This suggests it has a high level of activity
  • Activity, or the number of decays per unit time can be calculated using:

Activity & The Decay Constant equation 1

  • Where:
    • A = activity of the sample (Bq)
    • ΔN = number of decayed nuclei
    • Δt = time interval (s)
    • λ = decay constant (s-1)
    • N = number of nuclei remaining in a sample
  • The activity of a sample is measured in Becquerels (Bq)
    • An activity of 1 Bq is equal to one decay per second, or 1 s-1
  • This equation shows:
    • The greater the decay constant, the greater the activity of the sample
    • The activity depends on the number of undecayed nuclei remaining in the sample
    • The minus sign indicates that the number of nuclei remaining decreases with time – however, for calculations it can be omitted

Worked example: Decay constant

Activity___The_Decay_Constant_Worked_Example_-_Decay_Constant_Question, downloadable AS & A Level Physics revision notes

Part (a)

Step 1:            Write down the known quantities

Mass = 5.1 μg = 5.1 × 10-6 g

Molecular mass of americium = 241

Step 2:            Write down the equation relating number of nuclei, mass and molecular mass

Activity & The Decay Constant Worked Example equation 1

where NA is the Avogadro constant

Step 3:            Calculate the number of nuclei

Activity & The Decay Constant Worked Example equation 2

Part (b)

Step 1:            Write the equation for activity

Activity, A = λN

Step 2:            Rearrange for decay constant λ and calculate the answer

Activity & The Decay Constant Worked Example equation 3

The Exponential Nature of Radioactive Decay

  • In radioactive decay, the number of nuclei falls very rapidly, without ever reaching zero
    • Such a model is known as exponential decay
  • The graph of number of undecayed nuclei and time has a very distinctive shape

Equations for Radioactive Decay

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

N = N0e–λt

  • Where:
    • N0 = the initial number of undecayed nuclei (when t = 0)
    • 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 be represented in exponential form by the equation:

A = A0e–λt

  • 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 = C0e–λt

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: Exponential decay

The_Exponential_Nature_of_Radioactive_Decay_Worked_Example_-_Exponential_Decay_Question, downloadable AS & A Level Physics revision notes

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 = A0e–λt

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

Author: Katie

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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