Half Life Calculator

Half life calculator explained

Half‑life is the time it takes for an amount to reduce to half under first‑order (exponential) decay. This model describes radioactive decay, drug elimination, chemical degradation, and many natural attenuation processes. Use this calculator to solve for remaining quantity, time to a target, half‑life, decay constant, or the number of half‑lives instantly, in any consistent unit system.

Key formulas

• Exponential decay law: Q(t) = Q₀ · e^(−λt)
• Half‑life: t₁/₂ = ln(2) / λ
• Remaining amount: Q(t) = Q₀ · (1/2)^(t / t₁/₂)
• Time to target: t = t₁/₂ · log(Q(t)/Q₀) / log(1/2)
• Number of half‑lives: n = t / t₁/₂

When to use each mode

• Remaining amount: You know Q₀, t₁/₂, and elapsed time t.
• Time to target: You know Q₀, t₁/₂, and a target Q(t).
• Half‑life: You know Q₀, measured Q(t), and elapsed time t.
• Decay constant: You know t₁/₂ and want λ for modeling with e^(−λt).
• Number of half‑lives: You want an intuitive “how many halvings occurred?”

Worked examples

• Remaining: Q₀ = 100 g, t₁/₂ = 6 h, t = 12 h → Q(t) = 100 · (1/2)^(12/6) = 25 g.
• Time: Q₀ = 80 mg, t₁/₂ = 4 h, Q(t) = 10 mg → t = 4 · log(10/80)/log(1/2) ≈ 12 h.
• Half‑life: Q₀ = 50 Bq, Q(t) = 6.25 Bq after 8 h → t₁/₂ = 8 / (log(6.25/50)/log(1/2)) = 4 h.

Best practices

• Match units: Use consistent time units for t and t₁/₂; the calculator converts internally.
• Use scientific notation: Enter tiny or huge values with 3.2e‑5 to reduce typing errors.
• Interpret λ: Larger λ means faster decay; λ and t₁/₂ are inversely related via ln(2).
• Check reasonableness: Fraction and percent remaining help verify whether results make sense.

Who uses this calculator?

• Students: Physics, chemistry, and pharmacokinetics problem sets and labs.
• Professionals: Environmental remediation, radiological safety, stability studies.
• Researchers: Parameter estimation from decay experiments.
• Educators: Demonstrate exponential decay and half‑life intuition.

Frequently asked questions

HOW TO calculate remaining amount after a given time?

Enter Q₀, t₁/₂, and time t, then select “Remaining amount.” The calculator returns Q(t), fraction remaining, percent remaining, λ, and n.

HOW TO find the time to reach a target amount?

Enter Q₀, t₁/₂, and target Q(t), then choose “Time.” The tool solves t using the logarithmic rearrangement of the decay law.

HOW TO determine half‑life from measurements?

Enter Q₀, measured Q(t), and elapsed time t, then select “Half‑life.” We compute t₁/₂ directly from the ratio Q(t)/Q₀.

HOW TO compute the decay constant λ?

Provide t₁/₂ and choose “Decay constant.” The calculator applies λ = ln(2) / t₁/₂ and shows λ in your chosen time unit.

HOW TO interpret the number of half‑lives?

n = t / t₁/₂ tells you how many halvings occurred. For n = 3, 12.5% remains; for n = 1.5, about 35.4% remains.

HOW TO enter values in different units?

Select time units for t and t₁/₂ (s, min, h, d, y). Amount units are optional and used for labeling outputs only.