8.2.3 Half-Life

• Half-life is defined as:

The time taken for the initial number of nuclei to halve for a particular isotope

• This means when a time equal to the half-life has passed, the activity of the sample will also half
• This is because the activity is proportional to the number of undecayed nuclei, A ∝ N

When a time equal to the half-life passes, the activity falls by half, when two half-lives pass, the activity falls by another half (which is a quarter of the initial value)

• To find an expression for half-life, start with the equation for exponential decay:

N = N₀ e^–λt

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

• When time t is equal to the half-life t_½, the activity N of the sample will be half of its original value, so N = ½ N₀

• The formula can then be derived as follows:

• Therefore, half-life t_½ can be calculated using the equation:

• This equation shows that half-life t_½ and the radioactive decay rate constant λ are inversely proportional
  • Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

Step 1: Convert the half-life into seconds

• • t_½ = 28 years = 28 × 365 × 24 × 60 × 60 = 8.83 × 10⁸ s

Step 2: Write the equation for half-life

Step 3: Rearrange for λ and calculate

• The half-life of a radioactive substance can be determined from decay curves and log graphs
• Since half-life is the time taken for the initial number of nuclei, or activity, to reduce by half, it can be found by
  • Drawing a line to the curve at the point where the activity has dropped to half of its original value
  • Drawing a line from the curve to the time axis, this is the half-life

Log Graphs

• Straight-line graphs tend to be more useful than curves for interpreting data
  • Nuclei decay exponentially, therefore, to achieve a straight line plot, logarithms can be used

• Take the exponential decay equation for the number of nuclei

N = N₀ e^–λt

• Taking the natural logs of both sides

ln N = ln (N₀) − λt

• In this form, this equation can be compared to the equation of a straight line

y = mx + c

• Where:
  • ln (N) is plotted on the y-axis
  • t is plotted on the x-axis
  • gradient = −λ
  • y-intercept = ln (N₀)

• Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below:

Part (a)

Step 1: Draw lines on the graph to determine the time it takes for technetium to drop to half of its original activity

Step 2: Read the half-life from the graph and convert to seconds

• • t _½ = 6 hours = 6 × 60 × 60 = 21 600 s

Step 3: Write out the half life equation

Step 4: Calculate the decay constant

Part (b)

Step 1: Draw lines on the graph to determine the activity after 24 hours

• • At t = 24 hours, A = 0.5 × 10⁷ Bq

Step 2: Write out the activity equation

A = λN

Step 3: Calculate the number of atoms remaining in the sample