Radioactive Half-Life: Formula and Calculation

This calculator finds a radioactive substance’s half-life from the amount present initially and the amount remaining after a measured time. Enter those two amounts in the same unit and the elapsed time in seconds; the result is the half-life in seconds, with the decay constant also shown.

Formula

For exponential radioactive decay, the half-life is T_1/2 = t ln(2) / ln(N₀/N). Here, T_1/2 is half-life in seconds, t is elapsed time in seconds, N₀ is the initial amount, and N is the amount remaining at time t. The two amounts can be mass, number of radioactive nuclei, or activity, provided they use the same unit and represent the same nuclide.

The equivalent decay equation is N = N₀e^−λt, where λ is the decay constant in inverse seconds (s⁻¹). The relationship between the two measures is T_1/2 = ln(2)/λ. The calculator’s second output is λ. The natural logarithm, ln, is used; the amount ratio is dimensionless.

Worked Example

Using the calculator’s defaults: initial amount = 100 g, remaining amount = 50 g, and elapsed time = 10 s.

1. Find the ratio: 100 / 50 = 2.
2. Apply the formula: T_1/2 = 10 × ln(2) / ln(2).
3. Since ln(2) / ln(2) = 1, the half-life is 10 seconds.
4. For the decay constant, λ = ln(100 / 50) / 10 = ln(2) / 10 ≈ 0.0693 s⁻¹.

The calculator therefore returns a half-life of 10.00 s and a decay constant of 0.0693 s⁻¹.

What Half-Life Means

A radioactive half-life is the time required for half the radioactive nuclei in a sample to decay, on average. It is a property of a particular radionuclide, not a countdown for each individual atom: decay of an individual nucleus is probabilistic, while the behavior of a large sample follows a predictable statistical pattern.

After one half-life, half the original radioactive nuclei remain; after two, one quarter; after three, one eighth. More generally, after n half-lives the remaining fraction is (1/2)ⁿ. The amount does not usually reach exactly zero in the mathematical model, but becomes progressively smaller.

Half-life is not the same as the time for a sample to become harmless or undetectable. Those depend on the radionuclide, the original quantity, the radiation emitted, measurement sensitivity, and other conditions. Physical half-life alone does not determine a real-world safety assessment.

When This Calculation Is Useful

This calculation is useful when an initial and later measurement are available and the decay is expected to follow a single exponential process. It can estimate a half-life from experimental data, or help interpret how much of a radionuclide remains after a known interval. For a known half-life, the forward calculation is N = N₀(1/2)^t/T_1/2.

• Use the same kind of measurement at both times: for example, activity at the start and activity later.
• Keep units consistent. If elapsed time is entered in seconds, the calculated half-life is in seconds. Convert minutes, hours, or years to seconds first if using this calculator.
• Use positive amounts, with the remaining amount lower than the initial amount for ordinary decay. If both are equal, there is no measurable decay over that interval and this calculation cannot give a finite half-life.
• Prefer measurements separated enough to show a clear change, while accounting for instrument uncertainty and background signal.

Reading Measurements and Avoiding Errors

Radioactive decay measurements often use activity, measured in becquerels (Bq), where 1 Bq is one decay per second. The older unit curie (Ci) is also encountered; 1 Ci = 3.7 × 10¹⁰ Bq. If activity is proportional to the number of radioactive nuclei and measurement conditions stay consistent, the activity ratio can be used in place of the amount ratio. Do not mix units between the initial and later readings.

Common errors include using the logarithm of the reversed ratio, confusing a percentage remaining with an amount, and rounding intermediate values too early. The formula expects N₀/N, not N/N₀. If 25% remains, enter a remaining amount equal to one quarter of the initial amount; that corresponds to two half-lives, not one. A measurement of zero is also unsuitable for the logarithm: a detector reading below its detection limit does not prove that no radioactive nuclei remain.

Limits and More Complex Decay

The calculation assumes a single radionuclide with a constant decay rate and reliable measurements of the same quantity at both times. Noise, background radiation, changing detector geometry, or a mixture of radionuclides can distort the estimate. In a mixture, each nuclide can have its own half-life, so the total activity may not follow one simple exponential curve.

For improved estimates, take measurements at several times and fit an exponential decay model rather than relying on just two readings. A fitted line of ln(N) against time has slope −λ; its half-life is ln(2)/λ. If the substance is being replenished, physically removed, or transformed into another radioactive substance that contributes to the measured signal, a simple single-nuclide calculation may not describe the data.

Frequently Asked Questions

How do you calculate half-life from a decay constant?
Divide the natural logarithm of 2, approximately 0.693, by the decay constant: T_1/2 = 0.693/λ. Ensure the units are reciprocal: a decay constant in s⁻¹ gives a half-life in seconds.

How do you find half-life if you know the percentage remaining?
Use the remaining fraction in the decay equation. If 25% remains, that is 0.25 of the starting amount, or two half-lives. For a measured elapsed time, the calculator formula is T_1/2 = t ln(2)/ln(N₀/N).

Does half-life mean half the sample’s mass disappears?
It means half the radioactive nuclei have decayed on average. A mass measurement may not fall by exactly half if decay products remain in the sample, or if the sample includes stable material. Activity is often a more direct indicator of the amount of a particular radioactive nuclide.

Can half-life be calculated from activity?
Yes, when both activity readings refer to the same radionuclide and comparable measurement conditions. Substitute initial and later activity for N₀ and N in the formula because activity is proportional to the number of undecayed nuclei.

Can a radioactive half-life change?
For a given radionuclide, half-life is treated as a fixed nuclear property under ordinary conditions. A different apparent result is more often due to measurement uncertainty, a mixture of nuclides, or a process that violates the simple decay model.

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in a0 | Initial amount | g | 100 | 0
in a | Amount remaining | g | 50 | 0
in t | Elapsed time | s | 10 | 0
out h | Half-life | s | t * ln(2) / ln(a0 / a) | 2
out lambda | Decay constant | 1/s | ln(a0 / a) / t | 4