First-Order Reaction Half-Life Calculator

How Chemical Half-Life Is Calculated

Half-life is the time required for the concentration of a substance to decrease to half its initial value. It applies to radioactive decay, drug metabolism, and first-order chemical reactions.

First-Order Half-Life Formula: t(1/2) = ln(2) / k = 0.693 / k

Where:

• t(1/2) = half-life (any time unit)
• k = rate constant (same time unit, s⁻¹, min⁻¹, hr⁻¹, etc.)
• ln(2) = natural log of 2 ≈ 0.6931

Remaining Concentration Formula: C(t) = C₀ × (0.5)^(t / t(1/2))

Or equivalently: C(t) = C₀ × e^(−kt)

Worked Example: A pesticide with a half-life of 14 days is applied at 500 mg/kg soil. How much remains after 42 days?

• Number of half-lives = 42 / 14 = 3
• C(42) = 500 × (0.5)³ = 500 × 0.125 = 62.5 mg/kg

Half-Life Reference Values:

• Caffeine in bloodstream: ~5 hours
• Aspirin: ~3–4 hours
• Diazepam (Valium): 20–70 hours
• DDT in soil: ~2–15 years
• Carbon-14 (radioactive): 5,730 years
• Plutonium-239: 24,100 years

After Multiple Half-Lives:

• 1 half-life: 50% remains
• 5 half-lives: ~3.1% remains (often considered “cleared” in pharmacology)
• 10 half-lives: ~0.1% remains

Second-Order Reactions: Not all substances follow first-order kinetics. For second-order reactions, half-life depends on the initial concentration: t(1/2) = 1 / (k × C₀). This means the half-life changes as the reaction proceeds — a key distinction from radioactive decay, which is always first-order.