• 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:

• 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

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

where N_A is the Avogadro constant

Step 3:            Calculate the number of nuclei

Part (b)

Step 1:            Write the equation for activity

Activity, A = λN

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

• 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 e^x
• The inverse function of e^x is ln(y), known as the natural logarithmic function
  • This is because, if e^x = y, then x = ln(y)

Worked example: Exponential decay

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

Step 4:            Calculate the ratio A/A₀

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