12.2.6 The Law of Radioactive Decay

• Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which are expected to decay per unit of 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

• 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:

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)

• The law of radioactive decay states:

The rate of decay of a nuclide is proportional to the amount of radioactive material remaining

• 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)

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^–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

Part (a)

Step 1: Write down the known quantities

• • • Mass = 5.1 μg = 5.1 × 10^-6 g
    • Molecular mass of americium = 241
    • N_A = the Avogadro constant

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

Step 3: Calculate the number of nuclei

Part (b)

Step 1: Write down the known quantities

• • • Activity, A = 5.9 × 10⁵ Bq
    • Number of nuclei, N = 1.27 × 10¹⁶

Step 2: Write the equation for activity

Activity, A = λN

Step 3: Rearrange for decay constant λ and calculate the answer

Part (c)

Step 1: Write down the known quantities

• • • Activity, A = 5.9 × 10⁵ Bq
    • Initial activity, A₀ = 6.1 × 10⁵ Bq
    • Decay constant, λ = 4.65 × 10^–11 s^–1

Step 2: Write the equation for activity in exponential form

A = A₀ e^–λt

Step 3: Rearrange for time t

Step 4: Calculate the age of the smoke detector and convert to years

• • • Therefore, the smoke detector is 22.7 years old