Half-Life and Radioactive Decay Calculations

The Random Nature of Radioactive Decay

Imagine rolling millions of dice all at once. You can never predict exactly which die will roll a six, or when it will happen, but you can confidently predict the overall average number of sixes on the first roll.

• This perfectly describes random decay. The process of radioactive decay is spontaneous and unpredictable, occurring without any external influence.
• It is impossible to know when a specific unstable nucleus will decay or which nucleus in a sample will decay next.
• However, because there is a constant probability of decay for each individual nucleus, we can predict the statistical average behavior of millions of radioactive nuclei grouped together.

Understanding Half-Life and Activity

Understanding this constant probability explains why scientists use a statistical average, known as half-life, to measure decay.

• The half-life is the time taken for the number of radioactive nuclei in a sample to exactly halve.
• It can also be measured as the time taken for the activity of a sample to fall to half of its initial level.
• Activity is the rate at which unstable nuclei decay, and it is measured in the becquerel (Bq), where $1\text{ Bq}$ equals 1 decay per second.
• Before calculating a source's true activity, you must record the background count (radiation naturally present in the environment) and subtract it from the total count rate detected by your equipment.

Calculating Net Decline (Higher Tier)

Understanding exactly how fast radiation fades explains why some medical isotopes are safe to inject, while nuclear waste must be buried for millennia.

• After every half-life, the activity of a radioactive sample decreases by half.
• The net decline is the overall reduction in radioactive emission after a specific period of time, often expressed as a ratio.
• For a given number of integral half-lives (whole numbers like 1, 2, or 3), the fraction remaining of the original sample is calculated using:

$\text{Fraction Remaining} = \left(\frac{1}{2}\right)^n$

Where:

• $n$ = the number of integral half-lives

Worked Example

A radioactive isotope used in hospitals has a half-life of 6 hours. Calculate the net decline in emission after 24 hours, expressing your answer as a ratio.

1. Find the number of integral half-lives ($n$):

   • $n = 24 \div 6 = 4$ half-lives

2. Calculate the fraction remaining using the formula:

   • $\text{Fraction Remaining} = (1/2)^4 = 1/16$

3. Express the net decline as a ratio of final activity to initial activity:

   • $\text{Ratio} = 1:16$

Interpreting Activity-Time Graphs

Why do graphs of radioactive decay never actually reach zero?

• When you plot the activity of a sample against time, it forms an exponential decay curve on an activity-time graph.
• Because decay is a random statistical process, the curve constantly halves but never truly touches the x-axis (it is asymptotic).
• You can interpret these graphs to find the half-life by identifying the initial activity on the y-axis (where time is zero) and dividing it by two to find the half-value.
• First, draw a horizontal dashed line from this half-value on the y-axis until it meets the decay curve.
• Then, draw a vertical dashed line down from that intersection point to the x-axis—the value on the x-axis is your half-life.
• If the tail end of the graph levels off as a flat horizontal line above zero, this represents the background radiation, which must be subtracted from the total count rate before halving.