• 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
    • NA?= the?Avogadro constant

Step 2: Write down the equation relating to the number of nuclei, mass and molecular massTherefore, the smoke detector is 22.7 years old