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

Definition

In nuclear physics, the half‑life of an isotope is the time required for half of the nuclei in a sample to decay. If you start with \$N0\$ atoms, after one half‑life you will have \$N0/2\$ atoms remaining.

Key Formula

The number of atoms remaining after a time \$t\$ is given by:

\$N(t)=N0\left(\frac{1}{2}\right)^{\frac{t}{T{1/2}}}\$

Where \$T_{1/2}\$ is the half‑life of the isotope.

Simple Calculations

Let’s walk through a few examples. 📊

1. Halving each half‑life – If you start with 100 atoms:

2. Using the formula – Carbon‑14 has a half‑life of 5730 years. How many atoms remain after 11460 years (two half‑lives)?

Step 1: Identify \$N_0\$ (any number, say 1000 atoms). Step 2: Plug into the formula:

\$N(11460)=1000\left(\frac{1}{2}\right)^{\frac{11460}{5730}}=1000\left(\frac{1}{2}\right)^2=250\$

So 250 atoms remain. 🎉

3. Real‑world example – A medical isotope used for imaging has a half‑life of 6 hours. If you have 200 µCi at the start, how much activity is left after 18 hours?

Since 18 hours = 3 half‑lives, the activity reduces by a factor of 2³ = 8.

\$200\;\mu\text{Ci} \times \frac{1}{8} = 25\;\mu\text{Ci}\$

Common Isotopes and Their Half‑lives

Quick Recap

• The half‑life is the time for half the nuclei to decay.
• Use the formula \$N(t)=N0(1/2)^{t/T{1/2}}\$ for any time.
• Every half‑life you lose half of what’s left – a simple way to remember the decay.
• Check the table for common isotopes when you need a quick reference.

Remember: Half‑life is a powerful tool for dating fossils, medical imaging, and nuclear power. Keep practicing the calculations, and the numbers will start to feel natural! 🚀