Half-Life Calculator

A half life calculator solves the radioactive-decay equation N = N₀ × (1/2)^t/T in three directions: enter any three of initial amount, remaining amount, elapsed time, and half-life, and it returns the fourth. Example: 100 grams of carbon-14 (half-life 5,730 years) drops to 25 grams after 11,460 years - exactly two half-lives.

Need plain percentage math instead of exponential decay? Use the Percentage Calculator for basic percent-of problems, the Percentage Increase Calculator or Percentage Change Calculator to compare two values, and the Discount Calculator for sale prices. Reach for this half-life calculator only when a fixed fraction is lost each period rather than a one-off percentage.

The half-life formula

The remaining quantity after a time t is governed by the exponential decay law:

where N₀ is the initial amount, N is the amount left after time t, and T is the half-life. The exponent t / T is simply the number of half-lives that have elapsed. The same relationship can be written with Euler's number as N = N₀ × e−λt, where the decay constant λ = ln(2) / T. Rearranging the formula lets you solve for whatever you do not know:

• Remaining amount: N = N₀ × (1/2)(t/T)
• Elapsed time: t = T × ln(N₀/N) ÷ ln(2)
• Half-life: T = t × ln(2) ÷ ln(N₀/N)

Notice that the actual amounts never matter on their own - only the ratio N₀/N does. That is why you can express the inputs in grams, atoms, becquerels, milligrams, or percent and still get the right answer, as long as both amounts share the same unit.

A worked example: carbon-14 dating

Suppose a wooden artifact contains 25% of the carbon-14 found in a living tree. Carbon-14 has a half-life of about 5,730 years. Set N₀ = 100 and N = 25, then solve for time:

The artifact is roughly 11,460 years old - exactly two half-lives, which matches the intuition that two halvings (100% → 50% → 25%) get you to a quarter. If instead you knew the age and wanted the remaining fraction, you would run the calculator in "remaining amount" mode with t = 11,460 and T = 5,730 and read back 25 grams. This reversibility is what makes the half-life calculator useful for both dating problems (solve for time) and dosing or shielding problems (solve for the remaining amount).

A second example: drug elimination

Pharmacology uses the identical math under the name first-order elimination. Say a patient takes a 200 mg dose of a medication whose elimination half-life is 4 hours. How much remains after 12 hours? Twelve hours is three half-lives, so the amount is 200 × (1/2)³ = 200 × 0.125 = 25 mg. After 24 hours (six half-lives) only about 3.1 mg is left. Clinicians often estimate that a drug is essentially cleared after four to five half-lives, which is why dosing intervals are frequently tied to the half-life rather than the dose size.

How much of a 200 mg dose remains over time (4-hour half-life)?

This step-by-step table applies N = N₀ × (1/2)^t/T to a 200 mg starting dose with a 4-hour elimination half-life, showing how each 4-hour block halves what is left.

How to use this half-life calculator

Pick the quantity you want to find using the three tabs at the top, then fill in the rest:

1. Remaining amount: enter the initial amount (N₀), the half-life (T), and the elapsed time (t). The calculator returns how much is left.
2. Elapsed time: enter the initial amount, the remaining amount, and the half-life. It returns how much time has passed - ideal for dating problems.
3. Half-life: enter the initial amount, the remaining amount, and the elapsed time. It returns the half-life of the substance - useful when you measure decay in a lab.

Keep your time units consistent (both the half-life and elapsed time in seconds, or both in years), and keep both amounts in the same unit. The result panel also shows the fraction remaining, the number of half-lives elapsed, the decay constant, and the mean lifetime, plus a table tracing the decay through seven half-lives.

Solving for the half-life from lab data

The third mode answers the inverse question scientists actually face in the lab: you measure how much is left after a known time and want the substance's half-life. Suppose you start a sample at 1,000 counts per minute and, after 3 hours, it reads 250 counts per minute. Plug N₀ = 1,000, N = 250, and t = 3 into the half-life formula:

The half-life is 1.5 hours, which checks out: dropping from 1,000 to 250 is two halvings (1,000 → 500 → 250), and two half-lives fit into the 3-hour window. In a real experiment you would take many readings and fit the slope of ln(N) versus time, but the single-point estimate from this calculator is an excellent first approximation and a quick way to verify a fitted value.

Comparing common isotopes

Half-lives span an astonishing range, and the calculator handles all of them with the same formula. A few reference points show why the half-life is the defining property of an isotope:

• Technetium-99m (~6 hours): short enough for medical imaging - it lights up a scan, then clears within a day.
• Iodine-131 (~8 days): used in thyroid treatment; most of it is gone in a couple of months.
• Carbon-14 (~5,730 years): the workhorse of archaeological dating up to roughly 50,000 years.
• Plutonium-239 (~24,100 years): a key reason nuclear waste needs containment over geological timescales.
• Uranium-238 (~4.5 billion years): comparable to the age of the Earth, which is exactly why it is still around.

Drop any of these half-lives into the calculator with a starting amount and an elapsed time to see how dramatically the surviving fraction differs for the same elapsed period.

Half-life, decay constant, and mean lifetime for common isotopes

This quick-reference converts each published half-life into its decay constant λ = ln(2)/T and mean lifetime τ = T/ln(2). Confirm any isotope's accepted half-life against an authoritative nuclear-data table before real work.

Who this calculator is for

• Students in chemistry, physics, and biology checking homework on decay, nuclear reactions, or pharmacokinetics.
• Teachers and tutors generating worked examples with clean, verifiable numbers.
• Lab researchers estimating the activity of a radioactive source after storage, or back-calculating a half-life from measured counts.
• Healthcare and pharmacy learners reasoning about how long a drug stays in the body.
• Curious readers who want to understand carbon dating, nuclear waste timelines, or why isotopes are described by their half-lives.

Key terms explained

• Half-life (T): the time for half of a quantity to decay. It is constant for a given isotope or process, independent of the amount present.
• Decay constant (λ): the fraction that decays per unit time, equal to ln(2)/T ≈ 0.693/T. Larger λ means faster decay.
• Mean lifetime (τ): the average survival time of one atom, equal to T/ln(2) ≈ 1.443 × T, or 1/λ.
• Activity: the number of decays per second (measured in becquerels or curies). It falls off with the same half-life as the number of atoms.
• Number of half-lives (n): the elapsed time divided by the half-life, t/T. The fraction remaining is simply (1/2)ⁿ.

How the decay unfolds over time

Because each half-life removes half of what is left, the amount falls quickly at first and then more slowly - an exponential, not a straight line. The pattern is worth memorizing:

• After 1 half-life: 50% remains.
• After 2 half-lives: 25% remains.
• After 3 half-lives: 12.5% remains.
• After 4 half-lives: 6.25% remains.
• After 7 half-lives: about 0.78% remains.
• After 10 half-lives: about 0.098% remains - often treated as "effectively gone."

How much is left after n half-lives?

This scenario matrix reads straight off (1/2)ⁿ, then scales it to a 100-unit sample and a 200 mg dose so you can eyeball any decay problem without typing.

Crucially, the material never reaches exactly zero in this model; it halves forever. That is why long-lived isotopes like plutonium-239 (half-life ~24,100 years) stay measurably radioactive for tens of thousands of years, while short-lived medical tracers like technetium-99m (half-life ~6 hours) clear in a day or two.

What changes the result the most

Two inputs dominate every half-life problem:

• The ratio of time to half-life (t/T): this exponent is the whole story. Doubling the elapsed time at a fixed half-life squares the surviving fraction (e.g., 50% becomes 25%).
• The half-life itself: a longer half-life means slower decay, so more survives at any given time. Comparing two isotopes is really comparing their half-lives.
• The starting amount scales the answer linearly but never changes the fraction remaining - twice as much material still decays to half in one half-life.

Tips for getting accurate answers

• Match your units before you type. A half-life in years and a time in days will give an answer that is off by a factor of 365.
• Use percentages when you only know the fraction remaining: set N₀ = 100 and N to the percent left. The math depends only on the ratio.
• Sanity-check with whole half-lives. If your remaining fraction is 50%, 25%, or 12.5%, the elapsed time should be 1, 2, or 3 half-lives exactly.
• For dating, work in the substance's native half-life (5,730 years for carbon-14, 1,600 years for radium-226) so the answer comes out directly in those time units.

Why decay is random yet predictable

For any single atom, decay is genuinely unpredictable - quantum mechanics gives only a probability, never a schedule. So how can a half-life calculator be so precise? The answer is the law of large numbers. A single gram of carbon contains on the order of 10²² atoms, and averaging over that many independent random events produces a smooth, deterministic exponential curve with vanishingly small statistical noise. This is why the formula N = N₀ × (1/2)^t/T is treated as exact for macroscopic samples even though the underlying physics is probabilistic.

Limitations and assumptions

• It models simple first-order decay with a single, constant half-life. Decay chains (where a daughter product is itself radioactive) and branching decays need more detailed modeling.
• It ignores counting statistics and measurement error, which matter when you infer a half-life from real lab data.
• For drugs, true biological half-life can vary with absorption, distribution, organ function, and dose - this is the idealized pharmacokinetic estimate.
• The model never reaches zero, so "fully decayed" is a practical threshold (often ~10 half-lives), not a mathematical one.
• This tool is for education and planning. For nuclear safety, dosimetry, or clinical decisions, rely on qualified experts and authoritative reference data.

Related calculators

If your problem is about exponents and powers rather than decay, the Exponent Calculator evaluates (1/2)ⁿ directly, and the Logarithm Calculator handles the ln and log₂ steps when you rearrange the formula by hand. For growth instead of decay - like compound interest - see the Scientific Calculator, which also covers e^x and natural logs in one place.

About this formula

The exponential decay law N = N₀ × (1/2)^t/T is a standard, deterministic mathematical identity taught in every introductory physics and chemistry course; it requires no external data source - it follows directly from the definition of a half-life as the time for a quantity to halve. The decay constant relationship λ = ln(2)/T and the mean lifetime τ = T/ln(2) are algebraic consequences of that same definition. The specific half-life figures used in the examples above (carbon-14 ~5,730 years, technetium-99m ~6 hours, plutonium-239 ~24,100 years) are widely published nuclear-data values; always confirm an isotope's accepted half-life against an authoritative nuclear data table before using it in real work.