Half-Life Calculator — decay constant, mean lifetime, amount remaining

The half-life calculator above links the four quantities that describe exponential decay: the half-life t½, the decay constant λ, the mean lifetime τ, and the amount left after a given elapsed time. Choose which one you want and supply the rest. It handles radioactive decay and first-order chemical kinetics equally, because both obey the same equation, and it converts activity as well as quantity since activity falls on exactly the same curve.

Arb Digital publishes these calculators to take the arithmetic out of the way so the reasoning stays visible. This page is specifically the half-life form of exponential decay, the one used for radioactive nuclides and first-order reactions; for decay described by an arbitrary rate rather than a halving time, the exponential growth calculator is the general case, and our content decay calculator is a marketing tool about traffic loss on published pages, not a physical one. Everything here describes how the calculation works, nothing more.

What This Half-Life Calculator Does

Half-life is the time for a quantity to fall to half its value. That definition alone fixes everything else, because exponential decay has only one free parameter. Once you know the half-life, you know the decay constant, the mean lifetime, and the fraction remaining at any future moment. The calculator moves between all of them.

In the default mode it computes the remaining amount from an initial quantity, a half-life and an elapsed time. Switch the selector and it will instead solve for the elapsed time that produced an observed remaining amount — the direction used in radiometric dating — or for the half-life itself, which is what you get from a measured decay curve when the nuclide or reaction is not yet characterised.

Alongside the headline number the result grid reports the percent remaining, the number of half-lives that have elapsed, the decay constant λ, and the mean lifetime τ. If you supply an initial activity in becquerels, the sub-line under the headline number reports the activity at time t, which decays by exactly the same factor as the number of atoms.

How to Use It

1. Choose what to solve for. Remaining amount, elapsed time, or half-life. The field for the chosen quantity is overwritten with the answer.
2. Set the time unit once. The selector applies to both the half-life and the elapsed time, which removes the most common source of error in this calculation.
3. Enter the initial quantity. Any unit works — grams, moles, atoms, counts per minute — as long as the remaining quantity uses the same one, because only the ratio enters the maths.
4. Enter the half-life and elapsed time. They do not need to be whole multiples of one another; fractional half-lives are handled directly.
5. Add an initial activity if you have one. It is optional and does not affect any other output; set it to zero to leave it out.

The Formula and How It's Calculated

Exponential decay says the rate of loss is proportional to how much is present. Integrating that gives the working equation:

• N(t) = N₀ e^−λt = N₀ × (1/2)^t/t½
• λ = ln(2) / t½, so t½ = ln(2) / λ
• τ = 1 / λ = t½ / ln(2) ≈ 1.4427 × t½
• t = t½ × log₂(N₀ / N) = ln(N₀ / N) / λ

The two forms of N(t) are the same statement written differently. The exponential form is the one that falls out of the calculus; the halving form is the one that is easier to reason about. ln(2) ≈ 0.693147 is the bridge between them, and it is the only constant in the whole subject.

Take the default values as a worked check. Carbon-14 has a half-life of 5,700 years, and the elapsed time is set to 11,400 years, which is exactly two half-lives. Starting from 100 units, the first half-life leaves 50 and the second leaves 25, so N(t) = 25 and 25% remains. The exponential route must agree: λ = 0.693147 / 5,700 = 1.2160 × 10⁻⁴ per year, λt = 1.2160 × 10⁻⁴ × 11,400 = 1.38629, and e^−1.38629 = 0.25000. It does. The mean lifetime is 5,700 / 0.693147 = 8,223 years, and an initial activity of 1,000 Bq falls to 250 Bq over the same interval.

Decay Constant and Mean Lifetime Are Not the Same Thing

The decay constant λ is a probability per unit time. For a single unstable nucleus, λ dt is the chance it decays in the next small interval dt, and that chance never changes with age. A nucleus that has already survived ten half-lives is exactly as likely to decay in the next second as a freshly created one. This is the property that makes decay memoryless, and it is why the curve is exponential rather than any other shape. The NIST radiation physics programme maintains the measurement standards that underpin the activity units used alongside it.

The mean lifetime τ is the average survival time of a nucleus, and it is not the half-life. It is longer — about 44% longer, since τ = t½ / ln 2 ≈ 1.4427 t½. The reason is that the distribution of lifetimes has a long tail: most nuclei decay early, but a few survive far past the half-life, and those stragglers drag the arithmetic mean above the median. The half-life is precisely that median. Physics papers and nuclear data tables frequently quote τ where a chemistry text would quote t½, and mistaking one for the other introduces a 44% error that is easy to miss because the two numbers are the same order of magnitude.

A third convention appears in reactor and dosimetry work: the decay constant expressed in reciprocal seconds regardless of how long the half-life is. A nuclide with a half-life of a million years has a λ of about 2.2 × 10⁻¹⁴ s⁻¹. The scientific notation converter is useful when moving between those representations without losing an exponent.

Why First-Order Kinetics Uses the Same Equation

Radioactive decay is the best-known case, but it is a special case of first-order kinetics. Any process whose rate is proportional to the amount of one reactant present follows the same equation, and therefore has a constant half-life independent of starting concentration. First-order chemical reactions, many drug-elimination processes, and some isomerisations all qualify, and the IUPAC Gold Book definition of half-life is written generally enough to cover all of them.

The constant-half-life property is diagnostic, and it is the practical test for whether a reaction is genuinely first order. Plot the natural logarithm of concentration against time: if the process is first order the plot is a straight line whose slope is −λ, often written −k in kinetics. If the plot curves, the order is something else, and the concept of a fixed half-life does not apply. For a second-order reaction the half-life doubles every time it elapses, because the rate depends on the square of the concentration and falls away far faster than the amount does.

This distinction is what makes half-life meaningful in the first place. Saying "the half-life is 3 hours" is only a complete description when the underlying kinetics are first order. Otherwise the half-life is a property of one particular starting condition, not of the substance, and quoting it without that condition is meaningless. Where a reaction's spontaneity rather than its rate is in question, the Gibbs free energy calculator covers the thermodynamic side; note that thermodynamics says nothing about how fast anything happens.

The Carbon-14 Example, and Why the Number Has Changed

Carbon-14 is the standard illustration. The evaluated nuclear data compiled by the National Nuclear Data Center at Brookhaven National Laboratory gives a half-life of 5,700 ± 30 years, and that is the value loaded as the default above. Two half-lives, 11,400 years, leaves a quarter of the original carbon-14; five half-lives, 28,500 years, leaves about 3.1%; ten half-lives, 57,000 years, leaves under a tenth of a percent, which is roughly where the technique runs out of measurable signal.

You will also see 5,730 years quoted very widely, and 5,568 years in older literature. These are not errors so much as different vintages of the same measurement, refined as counting statistics improved. The 5,568-year figure, called the Libby half-life, is still used deliberately in some radiocarbon reporting so that dates published across many decades stay mutually comparable, with the correction applied at the calibration stage instead. The lesson generalises: always take the half-life and the convention from the same source rather than mixing a modern constant into an older framework.

The uncertainty of ± 30 years matters as much as the central value. It is about 0.5%, and it propagates into any age you calculate from it — a 20,000-year date inherits roughly a hundred years of uncertainty from the constant alone, before any measurement error on the sample itself. Any answer this calculator returns carries the uncertainty of the half-life you typed in, which the tool has no way of knowing. Our percentage calculator is a quick way to propagate that relative uncertainty through to the result.

Reading the Curve: What "Ten Half-Lives" Really Means

Because the loss is multiplicative, intuition trained on straight lines fails badly here. After one half-life half is gone; after two, three quarters; after three, seven eighths. The absolute amount lost in each successive half-life shrinks even though the fraction lost is identical, so the curve flattens into a long tail that approaches zero without ever reaching it. Mathematically nothing ever fully decays; practically, ten half-lives leaves 1/1024 of the original, roughly 0.098%, which is the rule of thumb behind the common convention of treating ten half-lives as effectively complete.

Fractional half-lives follow the same rule and are worth being comfortable with. Half a half-life leaves 2^−0.5 ≈ 70.7% rather than 75%, which is what linear intuition predicts. A quarter of a half-life leaves about 84%. The gap between the exponential answer and the linear guess is largest in the first half-life and closes only because the linear guess would have hit zero and stopped.

The reverse direction is where the arithmetic earns its keep. If a sample shows 12% of its original activity, the number of half-lives elapsed is log₂(1/0.12) = 3.06, so the age is 3.06 × t½. That is the entire basis of radiometric dating, and it is why the tool reports half-lives elapsed as a first-class output rather than burying it. The unit converter handles any conversion between time units that falls outside the five offered here.

Common Mistakes to Avoid

• Mixing time units. A half-life in days with an elapsed time in years is the single most common error; the unit selector here applies to both together for exactly that reason.
• Confusing mean lifetime with half-life. τ is 1.4427 times t½, and swapping them silently inflates or deflates every result by about 44%.
• Assuming decay is linear. Two half-lives leaves 25%, not 0%. Extrapolating a straight line through the first data points always overshoots.
• Applying half-life to a non-first-order process. If the half-life changes as the reaction proceeds, the kinetics are not first order and this equation does not describe them.
• Quoting a half-life without its source. Carbon-14 appears in the literature as 5,568, 5,700 and 5,730 years; the value and the convention it belongs to must come from the same place.

Related Free Tools From Arb Digital

For growth or decay at an arbitrary rate rather than a halving time, use the exponential growth calculator. Keep very large and very small decay constants readable with the scientific notation converter, convert between time units with the unit converter, and work out fractions remaining with the percentage calculator. On the chemistry side, the Gibbs free energy calculator covers spontaneity rather than rate. The full free online tools hub lists everything else.

Frequently Asked Questions

This tool provides educational estimates of exponential decay arithmetic. It is not laboratory, radiological, or safety guidance, and any real measurement should be interpreted against a validated method and source data.