Measuring HalfLife
 Halflife 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 halflife 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 halflife passes, the activity falls by half, when two halflives pass, the activity falls by another half (which is a quarter of the initial value)
 To find an expression for halflife, 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 (s1)
 t = time interval (s)
 When time t is equal to the halflife t½, 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, halflife t½ can be calculated using the equation:
 This equation shows that halflife t½ and the radioactive decay rate constant λ are inversely proportional
 Therefore, the shorter the halflife, the larger the decay constant and the faster the decay
 The halflife of a radioactive substance can be determined from decay curves and log graphs
 Since halflife 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 halflife
Measuring Long HalfLives
 For nuclides with long halflives, 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 = nNA
 Measuring the total activity A of the sample using the counts collected by a detector
 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 HalfLives
 For nuclides with short halflives, 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 halflife 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
 Straightline 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 yaxis
 t is plotted on the xaxis
 gradient = −λ
 yintercept = ln (N0)
 Halflives 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: