Formula & Calculator
Half-Life Relation
Relationship between a radioactive isotope's half-life and its decay constant.
Interpretation
t₁/₂ = ln2 / λ. Half‑life is the time for half of a radioactive sample to decay. Inversely proportional to decay constant. Used to characterise radioisotopes. Essential for dating and safety.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| t₁/₂ | Half-life | s |
| λ | Decay constant | 1/s |
What it means
The half‑life (t₁/₂) is the time required for the activity or number of nuclei of a radioactive isotope to decrease by half. It is directly related to the decay constant λ by t₁/₂ = ln(2)/λ ≈ 0.693/λ. Half‑lives range from microseconds (e.g., ²¹⁴Po) to billions of years (e.g., ²³⁸U). This parameter is critical for selecting isotopes for medical, industrial, and research applications. It determines how long a radioactive source remains active, affects the design of nuclear waste disposal, and is used in geological and archaeological dating. The half‑life is independent of initial quantity and environmental conditions (temperature, pressure). Understanding this relation is fundamental for radiation protection, nuclear medicine, and nuclear engineering.
Worked example
Half‑Life – Two Examples
Real‑World| Parameter | Value |
|---|---|
| λ | 0.001 s⁻¹ |
| Parameter | Value |
|---|---|
| λ | 1.21×10⁻⁴ yr⁻¹ |
Common mistakes
- Half‑life t₁/₂: The time for half of the radioactive nuclei to decay – independent of initial amount.
- Decay constant λ: In 1/time – t₁/₂ = ln(2)/λ.
- ln(2): Approximately 0.693 – use the exact value for precision.
- Units: If λ is in s⁻¹, half‑life is in seconds – convert to hours, days, years as needed.
- Multiple half‑lives: After n half‑lives, the remaining fraction is (1/2)ⁿ – do not simply divide by n.
Applications
The half‑life relation, t₁/₂ = ln2/λ, links the half‑life of a radioactive isotope to its decay constant. It is a fundamental parameter for identifying radioisotopes, for planning experiments and medical procedures, and for managing nuclear materials. Half‑life values range from fractions of a second to billions of years, determining the usefulness of isotopes for various applications: short half‑lives are ideal for medical imaging (e.g., technetium‑99m), while long half‑lives are critical for geological dating (e.g., uranium‑238). Nuclear engineers use half‑life to estimate the longevity of nuclear waste and to design storage solutions. By knowing the half‑life, professionals can predict the activity over time and implement appropriate safety measures for handling and disposal.
- Identification and characterisation of radioactive isotopes
- Scheduling of diagnostic and therapeutic nuclear medicine procedures
- Design of long‑term nuclear waste storage and disposal
- Geochronology and archaeological dating
- Safety planning for radiation emergencies and decommissioning
Frequently Asked Questions
The half‑life (T₁/₂) is the time required for half of a given number of radioactive nuclei to decay. It is defined by N(T₁/₂) = N₀ / 2. From the decay law, this gives e^(–λ T₁/₂) = 1/2, so T₁/₂ = ln(2)/λ.
They are inversely related: λ = ln(2) / T₁/₂. A short half‑life means a large decay constant (rapid decay); a long half‑life means a small decay constant (slow decay).
- Carbon‑14: 5,730 years.
- Uranium‑238: 4.47×10⁹ years.
- Uranium‑235: 7.04×10⁸ years.
- Iodine‑131: 8.02 days.
- Technetium‑99m: 6.01 hours.
After n half‑lives, the fraction remaining is f = (1/2)^n. For example, after 1 half‑life, 0.5; after 2, 0.25; after 3, 0.125. This is useful for quick mental estimates.
The average lifetime is τ = 1/λ. Since λ = ln(2)/T₁/₂, we have τ = T₁/₂ / ln(2) ≈ 1.44 × T₁/₂. The average lifetime is longer than the half‑life.
Half‑life is more intuitive and easier to understand. It gives a direct sense of how long a radioactive material remains hazardous or useful. The decay constant is a less intuitive probability per unit time.
Measure the activity (or number of decays) over time, plot ln(A) vs. t, and obtain the slope (–λ). Then compute T₁/₂ = ln(2) / λ. Alternatively, measure the time for the activity to drop to half, but this is less precise.
For imaging, isotopes with short half‑lives (e.g., Tc‑99m, 6 hours) are preferred to minimise patient radiation dose. For therapeutic applications, longer half‑lives may be used to deliver dose over a longer period.
Half‑life is a physical decay property. Equivalent dose (in Sv) is a measure of biological effect. The half‑life determines how long the source remains active, which influences the total dose received over time.
- Using half‑life instead of average lifetime in formulas requiring τ.
- Forgetting to convert half‑life to consistent units when using λ = ln(2)/T.
- Applying the half‑life concept to first‑order chemical reactions without noting the analogy.