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.

Zero-Order Reactions: At the other end sits zero order, where the rate does not depend on concentration at all: t(1/2) = C₀ / (2k). Halve the starting amount and the half-life halves with it, which is the opposite of the first-order behaviour most people expect. This happens when something other than the reactant is the bottleneck, most often a saturated catalyst or enzyme with every active site already occupied. Adding more substrate cannot speed up a machine that is already running flat out.

Alcohol metabolism is the textbook case. Above a fairly low blood concentration the liver enzymes are saturated, so ethanol clears at a roughly constant rate of about one standard drink per hour regardless of how much is in the system. Someone who has had eight drinks does not clear them eight times faster; they simply take eight times as long.

Telling the three orders apart from data

If you have concentration measurements over time, the order falls out of which plot gives a straight line. Concentration against time is linear for zero order. The natural log of concentration against time is linear for first order. The reciprocal of concentration against time is linear for second order. Try all three and take whichever is straightest.

The quicker rule of thumb: measure two successive half-lives. If they are equal, the reaction is first order. If the second is longer than the first, it is second order. If it is shorter, it is zero order.