Radioactive Isotope Decay Calculator and Formula

A radioactive isotope decay calculator estimates how much of a radionuclide remains after a specified time, using its half-life and starting amount. Enter the initial quantity, elapsed time, and half-life; the calculator returns the remaining quantity and the amount that has decayed.

Formula

The exponential decay law is N = N₀e^−λt = N₀(1/2)^t/T_1/2, where N is the amount remaining, N₀ is the initial amount, t is elapsed time, T_1/2 is the half-life, and λ is the decay constant. The relationship between the two decay descriptions is λ = ln(2)/T_1/2. In this calculator, t and T_1/2 are both entered in years, so their ratio is dimensionless; the amount can be any consistent mass unit, here grams. The amount decayed is N₀ − N, and the fraction remaining is e^{−ln(2)t/T_1/2}.

Worked Example

Using the default inputs, the initial amount is 1 g, elapsed time is 10 years, and half-life is 5 years.

• Elapsed half-lives: t/T_1/2 = 10 years / 5 years = 2.
• Fraction remaining: (1/2)² = 0.25.
• Amount remaining: N = 1 g × 0.25 = 0.2500 g.
• Amount decayed: 1 g − 0.2500 g = 0.7500 g.

The four-decimal amount remaining agrees with the calculator output. The fraction remaining is 0.250000.

Assumptions

• The isotope follows a single, constant radioactive decay rate, so its half-life does not change during the interval.
• The starting amount and remaining amount use the same unit. The elapsed time and half-life use the same time unit.
• “Amount decayed” means the difference between the starting and remaining amount of the isotope. It does not estimate daughter-product mass, radiation dose, or activity.
• The calculation is a continuous statistical model. It predicts an expected amount, not the exact timing of individual nuclear decays.

How to use and interpret the result

Choose the initial quantity and half-life for the specific isotope, then enter the elapsed time in the same unit as that half-life. If a reference gives a half-life in days, convert the elapsed time to days too; if it gives years, use years for both. The calculator accepts a mass as its input, but the same equation also works for numbers of atoms or activity, provided the starting and resulting quantities use the same kind of measurement.

The result is a model estimate for the isotope itself. For example, an initial sample of 8 g after three half-lives has 8 × (1/2)³ = 1 g of the original isotope remaining. The model says 7 g of that isotope has decayed; it does not say that 7 g of the whole sample has vanished. Decay products remain as matter unless they later transform or leave the system.

For activity calculations, use the same decay law on the initial activity: A = A₀(1/2)^t/T_1/2. Activity is the rate of nuclear transformations, commonly expressed in becquerels (Bq), where 1 Bq is one transformation per second. Do not treat activity, isotope mass, absorbed dose, and biological risk as interchangeable quantities.

Half-life, decay constant, and activity

A half-life is the time needed for half the radioactive atoms in a large population to decay, not a deadline when a sample becomes harmless. After one half-life, 50% remains; after two, 25%; after three, 12.5%. The amount approaches zero mathematically but does not abruptly become zero after a set number of half-lives.

The decay constant λ describes the probability of decay per unit time for each nucleus, with units inverse to the time unit used. A larger λ means a shorter half-life. Half-life is a property of a particular radionuclide and, for ordinary radioactive decay, is not changed by storing a sample at a different temperature or by changing its chemical form. A radionuclide can have more than one decay pathway, but its total decay rate is represented by the appropriate overall half-life.

Mass and activity are linked only when the isotope’s identity and nuclear properties are known. In principle, A = λN, where A is activity and N is the number of radioactive atoms. Converting between a weighed mass and atom count also requires the isotope’s molar mass and Avogadro’s constant. A decay calculator using only a mass, time, and half-life cannot determine activity or dose by itself.

Common mistakes and limits

• Mixing time units: dividing hours by a half-life in years gives a meaningless exponent. Convert both values to the same unit before calculating.
• Using the half-life as a linear rate: radioactive decay is exponential. Subtracting the same fixed amount each half-life is incorrect; each interval halves what remains.
• Confusing decay with disappearance: the original nuclide decreases, but its daughter nuclides may remain in the sample and can themselves be radioactive.
• Rounding too early: keep extra digits through intermediate calculations, especially over many half-lives, then round the final result to a useful precision.

This simple model is not a complete description of a chain of radioactive daughters. If daughter nuclides are also radioactive, their amounts depend on each nuclide’s half-life and on how it is produced; a decay-chain model is needed. A measured sample can also differ from a prediction because of counting statistics, measurement uncertainty, contamination, or uncertainty in the input half-life.

Radioactive material and safety

A calculated remaining amount is not a safety assessment. Radiological risk depends on factors such as radiation type and energy, activity, exposure duration, distance, shielding, and whether material can enter the body. This calculator does not determine dose, safe handling limits, storage requirements, or disposal instructions. Follow applicable regulations and institutional radiation-safety guidance; do not use the calculation to handle or dispose of radioactive material.

Frequently Asked Questions

How do I calculate how much radioactive isotope remains?
Use N = N₀(1/2)^t/T_1/2. Divide elapsed time by half-life to find the number of half-lives, then multiply the initial amount by one-half raised to that number.

What remains after three half-lives?
One-eighth, or 12.5%, of the starting amount remains: (1/2)³ = 0.125. The other 87.5% of the original isotope has decayed.

Can I use this calculator for radioactive activity?
Yes, if you enter initial activity in place of initial mass and keep the activity unit consistent. The result is activity remaining, not dose. Activity and mass cannot be converted into one another without isotope-specific information.

Does radioactive material become harmless after ten half-lives?
No universal safety conclusion follows from a number of half-lives. After ten half-lives, about 0.0977% of the initial isotope remains, but whether that residual activity presents a risk depends on the isotope, amount, radiation, exposure, and applicable safety criteria.

Why is a real measurement different from the calculator result?
The formula gives an expected value, while nuclear decays are random events and measurements have uncertainty. Input rounding, uncertain half-life data, radioactive daughter products, and sample contamination can also affect what is measured.

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in n0 | Initial amount | g | 1 | 0
in t | Elapsed time | years | 10 | 0
in h | Half-life | years | 5 | 0
out n | Amount remaining | g | n0 * exp(-ln(2) * t / h) | 4
out d | Amount decayed | g | n0 - n | 4
out f | Fraction remaining | - | exp(-ln(2) * t / h) | 6