Half-Life Calculator – Exponential Decay, Remaining Amount & Time | SILDIL
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Half-Life Calculator

Calculate the remaining amount, initial amount, elapsed time, or half-life of a substance undergoing exponential decay. See the number of half-lives, decay constant, mean lifetime, step-by-step calculation, and decay schedule.

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Quick examples

Display options

Remaining Amount
MeasurementValue

Exponential Decay Over Time

Exponential Decay Over Time

The amount remaining follows N = N₀ × (1/2)t/t₁⁄₂ — a smooth exponential curve, not a straight line.

Remaining amount

Decay Schedule

Half-LivesTimeRemaining% Remaining

How to Calculate Half-Life and Exponential Decay

Step-by-step method to find the half-life, remaining amount, elapsed time, or initial amount of a decaying substance.

  1. 1. Choose what to solve for: Select whether you want to calculate the remaining amount, initial amount, elapsed time, or the half-life itself.
  2. 2. Enter the known values: Input the three known variables and select your preferred time and quantity units from the dropdown menus.
  3. 3. Use the target lookup (optional): Expand the target section to find the exact time required to reach a specific remaining amount or percentage.
  4. 4. Review results and decay schedule: See the calculated value, decay constant, mean lifetime, and a step-by-step breakdown of the decay over time.

What Is Half-Life?

Half-life is the time required for a quantity undergoing exponential decay to fall to half of its initial amount. It shows up whenever something shrinks by the same proportion, again and again, over equal steps of time — most famously radioactive decay, but also drug elimination, capacitor discharge, and other first-order processes.

Half-Life Formulas

Remaining Amount

N = N₀ × (1/2)t/t₁⁄₂ — example: N₀ = 100, t₁⁄₂ = 8 hours, t = 24 hours → 3 half-lives → N = 100 × (1/2)³ = 12.5.

Initial Amount

N₀ = N × 2t/t₁⁄₂ — example: N = 25 g, t = 10 hours, t₁⁄₂ = 5 hours → N₀ = 25 × 2² = 100 g.

Elapsed Time

t = t₁⁄₂ × log₂(N₀/N) — example: N₀ = 100 g, N = 25 g, t₁⁄₂ = 5 years → t = 5 × log₂(4) = 10 years.

Half-Life

t₁⁄₂ = t × ln(2) / ln(N₀/N) — example: N₀ = 100 g, N = 25 g, t = 20 years → t₁⁄₂ = 20 × ln(2)/ln(4) = 10 years.

Equivalent Exponential-Decay Form

N = N₀e−λt, where λ = ln(2)/t₁⁄₂ — mathematically identical to the half-life form above.

How Many Half-Lives Have Passed?

n = t / t₁⁄₂. After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after 4, 6.25%; after 5, 3.125%. The amount never reaches exactly zero after a fixed number of half-lives — it keeps halving, indefinitely, approaching zero without ever quite touching it.

Half-Life and Decay Constant

λ = ln(2)/t₁⁄₂ and t₁⁄₂ = ln(2)/λ. A larger decay constant means a shorter half-life — the substance decays faster.

Half-Life and Mean Lifetime

τ = t₁⁄₂ / ln(2) and t₁⁄₂ = τ ln(2). Mean lifetime is the average time a single particle or unit survives before decaying — always somewhat longer than the half-life.

Common Half-Life Calculation Mistakes

  • Treating decay as linear. Half-life describes exponential decay, not a steady straight-line decrease.
  • Mixing time units. Half-life and elapsed time must be expressed in the same, or a clearly converted, time unit.
  • Rounding too early. Use full precision throughout the calculation; round only the final displayed value.
  • Assuming zero is reached. Standard exponential decay approaches zero asymptotically — it never reaches exactly zero at a finite time.
  • Confusing half-life with decay constant. They are related by λ = ln(2)/t₁⁄₂ but are not the same quantity.

Frequently Asked Questions

Q: What is half-life?

The time it takes for a decaying quantity to fall to half of its starting value.

Q: What is the half-life formula?

N = N₀ × (1/2)^(t/t₁/₂), where N is the remaining amount after time t.

Q: How do you calculate half-life?

t₁/₂ = t × ln(2) / ln(N₀/N), using any two known amounts and the elapsed time between them.

Q: How do you calculate remaining amount using half-life?

Divide elapsed time by half-life to get the number of half-lives, then multiply the initial amount by (1/2) raised to that power.

Q: How many half-lives have passed?

n = t / t₁/₂ — this value is not rounded, since 1.5 half-lives is a perfectly valid, meaningful result.

Q: What happens after 3 half-lives?

12.5% of the original amount remains (100% → 50% → 25% → 12.5%).

Q: How are half-life and decay constant related?

λ = ln(2) / t₁/₂ — a shorter half-life means a larger decay constant.

Q: What is the difference between half-life and mean lifetime?

Half-life is the time to reach 50% remaining; mean lifetime (τ = t₁/₂/ln(2)) is the average survival time of an individual unit, and is always somewhat longer.

Q: Does half-life apply only to radioactive decay?

No — the same mathematics describes any first-order exponential-decay process, such as capacitor discharge or drug elimination, though this calculator does not provide medical dosing guidance.

Q: Can a substance reach exactly zero after a finite number of half-lives?

No. Standard exponential decay only approaches zero as time grows, and never reaches it exactly at any finite time.

Q: Do time units have to match?

Yes — half-life and elapsed time must use the same time unit (or be converted consistently) before dividing one by the other.

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