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 Objective

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 will not include background radiation).

Definition

The half‑life (\$t_{1/2}\$) of a radioactive isotope is the time required for the number of undecayed nuclei in a sample to be reduced to one‑half of its original amount.

Mathematically, if \$N_0\$ is the initial number of nuclei and \$N\$ is the number remaining after time \$t\$, then for one half‑life:

\$ N = \frac{N0}{2}\quad\text{when}\quad t = t{1/2} \$

Key Points to Remember

• The half‑life is a constant for a given isotope, independent of the amount of material present.
• After \$n\$ half‑lives, the fraction of the original nuclei remaining is \$\left(\frac{1}{2}\right)^n\$.
• Half‑life values can range from fractions of a second to billions of years.
• In IGCSE calculations we ignore background radiation and assume a pure sample.

Using Half‑life in Calculations

1. Identify the half‑life \$t_{1/2}\$ of the isotope from a table or given data.
2. Determine how many half‑lives have elapsed:

   • Count the number of times the sample has been reduced by half.
   • Or calculate \$n = \dfrac{t}{t_{1/2}}\$ where \$t\$ is the elapsed time.

3. Calculate the remaining fraction:

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

4. Find the number of nuclei or mass remaining:

   \$ N = N_0 \left(\frac{1}{2}\right)^n \$

   or

   \$ m = m_0 \left(\frac{1}{2}\right)^n \$

5. Round the answer to the appropriate number of significant figures.

Example Table of Common Isotopes

Worked Example

Problem: A sample contains \$8.0 \times 10^{20}\$ nuclei of \$^{131}\text{I}\$. The half‑life of \$^{131}\text{I}\$ is 8.0 days. How many nuclei remain after 24 days?

Solution:

1. Determine the number of half‑lives elapsed:

   \$ n = \frac{24\ \text{days}}{8.0\ \text{days}} = 3 \$

2. Calculate the remaining fraction:

   \$ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \$

3. Find the remaining number of nuclei:

   \$ N = 8.0 \times 10^{20} \times \frac{1}{8} = 1.0 \times 10^{20}\ \text{nuclei} \$

Therefore, after 24 days, \$1.0 \times 10^{20}\$ nuclei of \$^{131}\text{I}\$ remain.

Interpreting Decay Curves

A decay curve plots the number of nuclei (or activity) against time. The curve is exponential and passes through the point where the quantity has fallen to half its initial value at \$t = t_{1/2}\$.

Key features to identify on a decay curve:

• The point where the curve reaches 50 % of the initial value – this gives the half‑life directly.
• Successive points at 25 %, 12.5 %, etc., correspond to 2, 3, … half‑lives.

Common Pitfalls

• Confusing half‑life with the time taken for a fixed number of nuclei to decay; it is always a fraction (½) of the current amount.
• Using the wrong unit for time; ensure the half‑life and elapsed time are in the same units.
• For non‑integer numbers of half‑lives, remember to use the fractional exponent: \$N = N0 \left(\frac{1}{2}\right)^{t/t{1/2}}\$.

Practice Questions

1. A 5.0 g sample of a radioactive isotope has a half‑life of 10 days. What mass remains after 30 days?
2. The activity of a \$^{14}\text{C}\$ sample is \$2.0 \times 10^5\$ Bq. After how many years will the activity be \$2.5 \times 10^4\$ Bq? (Half‑life of \$^{14}\text{C}\$ = 5,730 years.)
3. From a decay curve, the activity of a sample drops to 25 % of its initial value after 40 minutes. What is the half‑life of the isotope?

Summary

The half‑life is a fundamental concept in nuclear physics, describing the constant time required for half of a radioactive sample to decay. By understanding the definition and applying the simple exponential relationship, students can solve a variety of quantitative problems involving tables of data or decay curves.