Define the half-life of a particular isotope as the time taken for half the nuclei of that isotope in any sample to decay; recall and use this definition in simple calculations, which might involve information in tables or decay curves (calculations

5.2.4 Half‑life

Learning objectives

• AO1 – State the definition of the half‑life of a radioactive isotope.
• AO2 – Use the definition to carry out simple calculations with tables, decay‑curve data or algebraic manipulation.
• AO3 – Apply practical skills (background correction, plotting, safety) when determining a half‑life in the laboratory.

Definition (AO1)

The half‑life, written t_½, of a particular isotope is the time required for half of the original nuclei in any sample to decay. After one half‑life the number of undecayed nuclei N is

N = N₀/2

After n half‑lives the remaining fraction is \(\bigl(\tfrac12\bigr)^{n}\), so

N = N₀\left(\tfrac12\right)^{n}

Key relationships (AO2)

1. Number of half‑lives elapsed: n = t / t_½
2. Remaining nuclei (or activity) after time t: N = N₀\left(\tfrac12\right)^{t/t_{½}}
3. Re‑arranged to find the half‑life from data (ideal exponential decay):

In the IGCSE exam, when data are given in a table the linear‑interpolation method is usually expected, because the points are rarely perfectly exponential. The logarithmic formula is useful for checking an ideal set of data or for A‑level work.

Quick‑calc box

If the initial activity is A₀, the activity after time t is

A = A₀\left(\tfrac12\right)^{t/t_{½}}

To isolate any quantity, rearrange the equation and use the logarithmic form shown above.

Typical half‑life values (IGCSE context)

Reading a decay curve (AO2)

A decay curve plots activity (or number of nuclei) against elapsed time. It is an exponential decline.

• Draw a horizontal line at ½ A₀.
• The point where this line meets the curve gives the half‑life t_½ on the time axis.

Safety precautions (AO3)

All IGCSE practical work with radioactive sources must follow the ALARA principle (As Low As Reasonably Achievable). Key points are:

• Biological effects – ionising radiation can damage DNA; limit exposure time and keep distance.
• Shielding – use lead bricks, Plexiglas (for β‑particles) or concrete as appropriate.
• Handling tools – tweezers or tongs; never touch a source with bare hands.
• Storage – keep sources in labelled lead containers when not in use.
• Disposal – return spent sources to the supplier or a licensed waste facility.

Using background data (Supplementary)

Although the core syllabus tells you to ignore background radiation in the final calculation, the supplementary material expects you to subtract it when raw counts are given.

1. Measure the background count rate B (source removed).
2. Measure the total count rate R with the source present.
3. Corrected activity: A = R – B.
4. Use the corrected activity in the half‑life formulas above.

Example: Background = 12 cps, measured rate = 212 cps → A = 200 cps, then apply the half‑life equations.

Real‑world connections (AO1)

• Carbon‑14 dating – uses the 5 730 yr half‑life to estimate the age of archaeological samples.
• Tracer studies – isotopes such as ³⁵S allow scientists to follow the movement of substances through a system.
• Medical applications – ⁶⁰Co and ⁹⁹ᵐTc are used in radiotherapy and diagnostic imaging; knowledge of their half‑lives is essential for dose planning.
• Industrial uses – smoke detectors (⁴⁴⁰Am) and radiography (⁶⁰Co) rely on predictable decay rates.

Practical assessment checklist (AO3)

• Plan a series of measurements (e.g., every 2 days for 10 days).
• Record background counts and subtract them from each reading.
• Plot the corrected activity against time (both linear and log A vs t graphs).
• Identify the ½‑point on the linear plot or determine the slope (‑λ) on the log plot.
• Calculate the half‑life using the appropriate method (interpolation or logarithmic formula).
• Estimate uncertainties and comment on the quality of the data.

Worked examples

Example 1 – One half‑life (definition)

A sample contains \(8.0\times10^{6}\) nuclei of ³⁵S. After 87.5 days (its half‑life) how many nuclei remain?

1. Identify \(t{½}=87.5\text{ d}\) and \(N{0}=8.0\times10^{6}\).
2. After one half‑life \(N = N_{0}/2 = 4.0\times10^{6}\) nuclei.

Example 2 – Multiple half‑lives

A carbon‑14 sample has an initial activity of \(A_{0}=200\) Bq. What is the activity after 17 190 years?

1. Number of half‑lives: \(n = \dfrac{t}{t_{½}} = \dfrac{17\,190}{5\,730}=3\).
2. Activity after \(n\) half‑lives: \(A = A_{0}\left(\tfrac12\right)^{3}=200\times\frac18=25\) Bq.

Example 3 – Solving for elapsed time

A ⁶⁰Co source initially has an activity of \(A_{0}=1000\) Bq. After how long will the activity be \(125\) Bq?

1. Set up the decay relation \(125 = 1000\left(\tfrac12\right)^{t/5.27}\).
2. Divide by 1000: \(0.125 = \left(\tfrac12\right)^{t/5.27}\).
3. Recognise \(0.125 = \left(\tfrac12\right)^{3}\) ⇒ \(t/5.27 = 3\).
4. Hence \(t = 3\times5.27 = 15.81\) yr.

Example 4 – Determining the half‑life from a table (interpolation method)

Find the half‑life of the isotope.

1. Initial activity \(A_{0}=800\) Bq; half of this is 400 Bq.
2. 400 Bq lies between the 4‑yr (450 Bq) and 6‑yr (200 Bq) points.
3. Fraction of the interval needed:
   

   \(\displaystyle \frac{450-400}{450-200}= \frac{50}{250}=0.20\).

4. Half‑life ≈ \(4\text{ yr} + 0.20\times(6-4)\text{ yr}=4.4\) yr.

Note: If the data were perfectly exponential you could also use the logarithmic formula

\(t{½}= \dfrac{t}{\log{2}(A_{0}/A)}\) which would give 3 yr for the idealised points (0 yr, 800 Bq) and (6 yr, 200 Bq). The IGCSE exam expects the interpolated answer.

Common pitfalls

• Confusing “half‑life” with “time for a fixed number of decays”. It always refers to half of the original nuclei.
• Mixing units – the time unit used in the calculation must match the unit given for the half‑life.
• Failing to subtract background when raw counts are supplied (required for the supplementary material).
• Assuming the decay curve is linear; only the 50 % point is needed, but the curve itself is exponential.
• Using the logarithmic formula on data that are not perfectly exponential – the exam usually wants linear interpolation.

Quick revision checklist

• Can you state the definition of half‑life in your own words?
• Do you know the formula \(N = N{0}\left(\tfrac12\right)^{t/t{½}}\) (or the activity equivalent)?
• Can you convert between elapsed time and number of half‑lives?
• Are you comfortable reading the half‑life point from a decay curve or interpolating from a table?
• Do you remember to correct for background when raw counts are given?
• Do you know the basic safety rules for handling radioactive sources?

Optional extension – Decay constant (A‑level)

The decay constant \(\lambda\) links the half‑life to the exponential law:

\(\displaystyle \lambda = \frac{\ln 2}{t_{½}}\qquad\text{and}\qquad A = \lambda N\)

This relationship is not required for the IGCSE but is useful for further study.