Half-Life and Mean Life Explained Simply with Easy Examples
Half-life is the time in which half of the radioactive nuclei in a sample decay. Mean life is the average lifetime of a nucleus before it decays. Together they turn the decay law into numbers we can feel — years for carbon-14, hours for medical tracers, billions of years for uranium.
These ideas matter whenever we handle radioactive material: choosing a medical isotope that decays before it harms the patient, or a waste isotope that must be stored for ten half-lives.
Key Concepts
- Half-life (T½): time for N to drop to N₀/2, in seconds, hours, or years matching the isotope.
- Mean life (τ): average survival time of nuclei, τ = 1/λ.
- Decay constant (λ): in inverse time units.
Formulas
T½ = ln 2 / λ = 0.693/λ. τ = 1/λ = T½ / 0.693 = 1.443 × T½. After n half-lives: N = N₀ (1/2)ⁿ where n = t/T½. Mean life is always longer than half-life because a few nuclei survive very long and pull the average up.
Worked Example
Iodine-131 has a half-life of 8 days. A hospital sample has 6.4 × 10¹⁸ atoms. How many remain after 24 days, and what is the mean life?
- n = 24/8 = 3 half-lives.
- N = 6.4 × 10¹⁸ × (1/2)³ = 6.4 × 10¹⁸ / 8 = 8.0 × 10¹⁷ atoms.
- τ = 1.443 × 8 = 11.5 days.
Common Misconceptions
- Students think half-life means half the mass disappears as energy. The nuclei transform into other nuclei; mass is mostly retained in the products.
- Many think after one half-life the remaining atoms are older and decay faster. Each nucleus's chance stays the same regardless of age.
- Some confuse mean life with half-life. Mean life is about 44 percent longer: τ = 1.443 T½.
Real-World Applications
Doctors pick technetium-99m (6-hour half-life) for scans so activity fades quickly; archaeologists use carbon-14 (5,730 years); geologists use uranium-238 (4.47 billion years) to date rocks; pharmacists calculate expiry of radioactive medicines.
Consider radium-226 with a half-life of about 1,600 years. An old luminous watch dial still contains most of its radium today, because only a tiny fraction of one half-life has passed since it was made. In contrast, fluorine-18 used in PET scans has a half-life of 110 minutes, so hospitals must prepare it close to the patient and use it the same day. Choosing an isotope always starts by matching its half-life to the job's timescale.
In short, half-life halves the sample each period, mean life τ = 1/λ averages each nucleus's lifetime, and the two are linked by T½ = 0.693τ.