Radioactive Decay Half-Life Formula and Calculator

This calculator finds a radioactive isotope’s half-life from its decay constant, using the relationship between the two. Enter the decay constant in inverse seconds; the result is the half-life in seconds, with the mean lifetime also shown for context.

Formula

The half-life is the natural logarithm of 2 divided by the decay constant:

t_1/2 = ln(2) / λ

• t_1/2 is the half-life, measured here in seconds (s).
• λ (lambda) is the decay constant, measured in inverse seconds (s⁻¹).
• ln(2) is approximately 0.693147, a dimensionless constant.

The calculator’s second result is the mean lifetime, τ = 1 / λ, also in seconds. The decay constant and time unit must match: if λ is reported in min⁻¹, calculate the result in minutes, or convert λ to s⁻¹ before using this calculator. For example, 1 min⁻¹ equals 1/60 s⁻¹.

Worked Example

Using the calculator’s default decay constant, λ = 0.0001 s⁻¹:

1. Write the half-life equation: t_1/2 = ln(2) / λ.
2. Substitute the values: t_1/2 = 0.69314718056 / 0.0001 s⁻¹.
3. Calculate: t_1/2 = 6931.4718056 s, or 6931.472 s to the calculator’s three decimal places.

The corresponding mean lifetime is τ = 1 / 0.0001 s⁻¹ = 10000 s. The half-life is shorter because it is ln(2), or about 0.693, times the mean lifetime.

Assumptions

• The sample follows single-exponential radioactive decay with a constant decay probability per unit time.
• The input is a positive decay constant expressed in s⁻¹; seconds are the output time unit.
• The calculation describes an idealized isotope population and does not account for measurement uncertainty, mixtures of isotopes, or changing conditions in a detector.

How Radioactive Decay Works

Radioactive decay is a random process: no one can predict exactly when a particular unstable nucleus will decay. For a large population of identical nuclei, however, the fraction expected to decay in a fixed interval is described statistically by a constant decay rate. If N₀ is the starting number of radioactive nuclei and N(t) is the number remaining after time t, the exponential decay law is:

N(t) = N₀e^−λt

Here, t must use the same unit that defines λ. At one half-life, N(t) equals N₀/2. Substituting that condition gives 1/2 = e^−λt_1/2. Taking natural logarithms and rearranging yields t_1/2 = ln(2)/λ.

“Half-life” is a statistical description, not a timer for each atom. After one half-life, the expected amount of the parent isotope is half the initial amount; after two, one quarter; after three, one eighth. For small samples, the actual number of decays can fluctuate around those expected fractions.

Using the Formula and Reading the Result

This version of the calculator is useful when a decay constant is known and you want the equivalent half-life. It works for radioactive isotopes whose decay is adequately described by a single constant λ. The output is a time, not an activity or a mass: convert it to minutes, days, or years only after checking the scale.

For instance, a result of 6931.472 seconds is approximately 115.5245 minutes, or 1.924 days. These conversions do not change the underlying half-life. A longer half-life corresponds to a smaller decay constant, while a larger decay constant means faster decay.

The formula also works in reverse when half-life is known: λ = ln(2)/t_1/2, provided both quantities use compatible time units. A separate common calculation is remaining quantity after a specified duration. That uses N(t) = N₀e^−λt, or equivalently N(t) = N₀(1/2)^t/t_1/2. The duration t and half-life must be in the same units.

Related Quantities and Common Misconceptions

Half-life and mean lifetime describe the same exponential decay in different ways. The mean lifetime is τ = 1/λ, while the half-life is t_1/2 = τ ln(2). Do not treat them as equal: the mean lifetime is about 1.4427 times the half-life. Neither quantity means every nucleus survives for that duration.

Half-life is also distinct from activity. Activity is the number of decays per unit time, commonly expressed in becquerels (Bq), where 1 Bq is one decay per second. For a sample with N radioactive nuclei, activity A = λN when λ is in s⁻¹; as the number of parent nuclei falls, the activity falls too. A half-life alone cannot tell you a sample’s activity without knowing how many parent nuclei are present.

In a mixture of radionuclides, each isotope can have its own decay constant and half-life. The combined signal may not follow one simple exponential curve. Likewise, daughter products can themselves be radioactive, so a measured signal may include contributions from more than the original parent isotope.

Measurement, Precision, and Limits

Decay constants may be obtained from a published nuclear-data reference or estimated from measurements of activity or parent-isotope abundance over time. An experiment needs suitable detection, background correction, and enough observations to distinguish a decay trend from random counting variation. The calculator performs the conversion once λ is supplied; it does not validate the source or uncertainty of that input.

• Check the unit: a decay constant in day⁻¹ produces a result in days unless converted. Mixing day⁻¹ with seconds without conversion gives an error by a factor of 86,400.
• Use the natural logarithm: the constant is ln(2), not log₁₀(2). A base-10 logarithm requires an additional conversion factor.
• Keep significant figures sensible: the displayed decimals are formatting, not evidence that the input is known that precisely. Report uncertainty when the decay constant is measured or quoted with uncertainty.
• Do not interpret the model as individual certainty: exponential decay predicts average population behavior, with statistical fluctuations in finite samples.

The basic formula assumes a constant decay rate and one radioactive parent population. It does not model chemical separation, changing isotope mixtures, daughter-product buildup, detector efficiency, or uncertainty propagation. Nuclear decay rates are generally not changed by ordinary chemical conditions, but a particular measurement can still be affected by sample composition and how it is observed.

Frequently Asked Questions

What is the formula for radioactive decay half-life?
When the decay constant λ is known, the half-life is t_1/2 = ln(2)/λ. Use compatible units so the result has the intended time unit.

How do I calculate half-life from a decay constant?
Divide 0.693147, the value of ln(2), by the decay constant. For λ = 0.0001 s⁻¹, the half-life is about 6931.472 seconds.

How do I find the decay constant from half-life?
Rearrange the equation: λ = ln(2)/t_1/2. If the half-life is in years, the resulting decay constant is in year⁻¹ unless you convert the time to another unit.

Does a radioactive sample disappear after one half-life?
No. One half-life leaves, on average, half of the original parent nuclei. After each further half-life, half of the remaining parent nuclei decay; the amount approaches zero but the model does not reach exactly zero in a finite number of half-lives.

Are half-life and mean lifetime the same?
No. Mean lifetime is 1/λ, and half-life is ln(2)/λ. The half-life is about 69.3% of the mean lifetime for a single exponential decay process.

Can I use half-life to calculate remaining material?
Yes, if the sample follows exponential decay and you know the elapsed time and initial amount. Use N(t) = N₀(1/2)^t/t_1/2, with elapsed time and half-life expressed in the same units.

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in lambda | Decay constant | s^-1 | 0.0001 | 0
out t | Half-life | s | ln(2) / lambda | 3
out tau | Mean lifetime | s | 1 / lambda | 3