Measuring Half-Life (DP IB Physics: HL)

• 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

• 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

A linear decay curve. This represents the relationship:

Measuring Long Half-Lives

• For nuclides with long half-lives, on the scale of years, this can be measured by:
  • Measuring the mass of the nuclide in a pure sample
  • Determining the number of atoms N in the sample using N = nN_A
  • Measuring the total activity A of the sample using the counts collected by a detector
  • Determining the decay constant using 
  • Calculating half-life using 

• Note: The sample must be sufficiently large enough in order for a significant number of decays to occur per unit time so that an accurate measure of activity can be made

Measuring Short Half-Lives

• For nuclides with short half-lives, on the scale of seconds, hours or days, this can be measured by:
  • Measuring the background count rate in the laboratory (to subtract from each reading)
  • Taking readings of the count rate against time until the value equals that of the background count rate (i.e. until all of the sample has decayed)
  • Plotting a graph of activity, A, against time, t (as corrected count rate ∝ activity, A)
  • Making at least 3 estimates of half-life from the graph and taking a mean

OR 

• • Plotting a graph of ln N against time, t (as corrected count rate ∝ number of nuclei in the sample, N)
  • Finding the gradient of this graph, which gives –λ
  • Calculating half-life using 

• Straight-line graphs tend to be more useful than curves for interpreting data
  • Due to the exponential nature of radioactive decay logarithms can be used to achieve a straight line graph

• 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:

A logarithmic graph. This represents the relationship:

• Note: experimentally, the measurement generally taken is the count rate of the source
  • Since count rate ∝ activity ∝ number of nuclei, the graphs will all take the same shapes when plotted against time (or number of half-lives) linearly or logarithmically  

Part (a)

Step 1: List the known quantities

• • Half-life, t_½ = 28 years

Step 2: Write the equation for half-life

Step 3: Rearrange for λ and calculate

= 0.025 year⁻¹

Part (b)

Step 1: List the known quantities

• • Decay constant, λ = 0.025 year⁻¹
  • Time passed, t = 50 years

Step 2: Write the equation for exponential decay

Step 3: Rearrange for  and calculate

= 0.287

• • Therefore, 28.7% of the sample will remain after 50 years

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

Step 5: Draw lines on the graph to determine the activity after 24 hours

• • At t = 24 hours, A = 0.5 × 10⁷ Bq

Step 6: Write out the activity equation

A = λN

Step 7: Calculate the number of atoms remaining in the sample