Understanding Half-Life and Radioactive Decay
The half-life of a substance is the time required for half of its original quantity to decay or transform. This concept is used across physics, chemistry, pharmacology, and nuclear engineering. It’s essential in determining how long a substance remains active or detectable in a system.
Radioactive decay is a first-order process, meaning the rate of decay is proportional to the quantity of substance present. The mathematical representation is Nt = N0 × (1/2)^(t / t₁/₂).
Common Uses of Half-Life
- Carbon dating in archaeology and geology.
- Determining radioactive safety levels in nuclear power plants.
- Pharmacokinetics — measuring how long drugs stay active in the body.
- Environmental decay of isotopes and pollutants.
Mathematical Derivation
Decay Law: Nt = N0 × e^(-λt) Half-life: when Nt = N0/2 => 1/2 = e^(-λt₁/₂) => ln(2) = λ × t₁/₂ => t₁/₂ = ln(2)/λ
Here, λ (lambda) represents the decay constant. The relationship between mean lifetime (τ), half-life, and decay constant is given as:
τ = 1/λ and t₁/₂ = τ × ln(2).
Half-Life in Medicine
In pharmacology, the biological half-life of a drug determines dosing schedules. Drugs with shorter half-lives are administered more frequently. Understanding this concept ensures effective therapeutic results while avoiding toxicity.