Radioactive Decay Calculations: Formula and Examples

Radioactive decay calculations estimate how much of a radioactive isotope remains after a given time, using its half-life or decay constant. Enter the initial quantity, elapsed time, and half-life to calculate the remaining quantity and the fraction that has decayed.

Formula

For a single radioactive isotope decaying under the usual first-order model, the remaining quantity is N = N₀e^−λt = N₀(1/2)^t/T_½. The calculator uses the equivalent expression N = N₀ exp(−ln(2) × t / T_½).

• N is the quantity remaining, in the same units as the starting quantity.
• N₀ is the initial quantity, in this example grams.
• t is elapsed time, in years here.
• T_½ is the half-life, in years here.
• λ is the decay constant, here in inverse years: λ = ln(2) / T_½.

The input time and half-life must use the same unit. The calculator’s fraction-decayed output is 100 × (1 − N/N₀), expressed as a percentage. This is the percentage of the original radioactive atoms that have decayed, not necessarily a percentage reduction in measured radiation from a real object.

Worked Example

Using the defaults, the sample begins with 10 g of an isotope, has a half-life of 5730 years, and is observed after 5730 years.

1. Calculate the decay constant: λ = ln(2) / 5730 years = 0.0001209681 per year.
2. Calculate the exponent: −λt = −(ln(2) / 5730) × 5730 = −ln(2) ≈ −0.6931472.
3. Calculate the amount remaining: N = 10 g × exp(−ln(2)) = 10 g × 0.5 = 5.0000 g.
4. Calculate the fraction decayed: 100 × (1 − 0.5) = 50.00%.

The result is one half of the starting amount because the elapsed time equals one half-life. This is an expected-value calculation: individual atoms decay randomly, while the exponential law describes the statistical behavior of a large population.

Assumptions

• The sample contains one isotope whose decay follows a constant half-life over the time considered.
• The initial and remaining quantities refer to that isotope alone, not the total mass of a mixture or an object.
• Elapsed time and half-life are entered in matching units. The calculator uses years for both.
• The calculated amount is a mathematical estimate; it does not account for measurement uncertainty, sample loss, or daughter products unless those are modeled separately.

How Radioactive Decay Works

An unstable atomic nucleus can transform spontaneously, emitting radiation or particles as it changes into a more stable state. The probability of decay for any one nucleus during a short interval is described by a constant rate. Since the number of undecayed nuclei falls over time, the number of decays per unit time also generally falls.

Half-life is the time required for half the radioactive nuclei in a large sample to decay. It is not the time until every atom disappears. After each additional half-life, half of what remained is left: 100% becomes 50%, then 25%, then 12.5%. In principle, the amount approaches zero without reaching exactly zero in a finite number of half-lives.

Half-life is a property of the isotope and its decay mode, not the size or shape of an ordinary sample. Changing the sample’s mass changes the starting number of radioactive atoms, but it does not change the isotope’s half-life under normal conditions.

Using the Calculation and Choosing Inputs

Use this calculation when the starting quantity, elapsed time, and isotope’s half-life are known and you want an idealized remaining quantity. The quantity can be expressed as mass, number of atoms, or another measure proportional to the number of parent atoms, provided the initial and final quantities use the same unit.

• Find the half-life for the exact isotope, not merely the element. Isotopes of one element can have very different half-lives.
• Use a documented reference value and keep the same time unit for elapsed time and half-life. For example, convert days to years before entering a half-life expressed in years.
• Enter the initial amount of the radioactive parent isotope, rather than the total mass of a sample containing other isotopes, minerals, or contaminants.
• For time before the stated initial measurement, this calculator is not appropriate: its elapsed-time input is nonnegative and the model is intended to project forward.

For a changing quantity, a useful rearrangement is t = T_½ × log₂(N₀/N), where log₂ means logarithm base 2. This estimates elapsed time if the initial and present amounts are known and the same single-isotope model applies. It becomes unreliable when the remaining amount is near a measurement limit or when the sample’s history is uncertain.

Common Misconceptions and Limits

A 50% reduction in parent atoms does not automatically mean a 50% reduction in every measured radiation reading. Activity—the number of nuclear decays per unit time—is proportional to the number of parent atoms for a given isotope, so it follows the same exponential decrease in an ideal measurement. But a detector reading can also depend on geometry, shielding, radiation type, background, and contributions from other radionuclides.

Many real samples contain decay chains: a parent isotope decays into a radioactive daughter, which may itself decay. The amount of a daughter can first increase and later decrease. The single-isotope equation does not calculate those changing daughter quantities; a chain model needs the relevant isotopes, decay constants, and starting amounts.

Other limits include uncertainty in the half-life, measurement error, chemical separation or loss of material, and the possibility that a sample is a mixture. Rounding can also matter over very long periods or when the remaining fraction is extremely small. The calculator reports a numerical result, not evidence that a sample is safe or a substitute for radiological measurement and specialist interpretation.

Frequently Asked Questions

What is the formula for radioactive decay?
For one isotope, N = N₀e^−λt, where N₀ is the starting quantity, t is elapsed time, and λ is the decay constant. Since λ = ln(2)/T_½, the same formula can be written N = N₀(1/2)^t/T_½.

How much remains after two half-lives?
One quarter remains: after the first half-life 50% is left, and after the second half-life half of that remains, or 25%. In general, after k half-lives the remaining fraction is (1/2)^k.

Does radioactive decay depend on the starting mass?
The half-life does not depend on the starting mass. Starting with more of the same isotope gives a proportionally larger amount remaining at any later time, while the fraction remaining is determined by elapsed time and half-life.

Can a radioactive substance reach zero?
The ideal exponential model approaches zero but does not reach exactly zero after a finite time. A real sample can, however, contain so few parent atoms that none are detected, or none remain after the last atoms decay.

Can I use this formula for a radioactive decay chain?
Not by itself. This calculation follows the parent isotope only. Calculating radioactive daughter products requires a decay-chain model that includes their production and decay rates.

Is the calculated remaining amount a safety assessment?
No. It estimates the amount of parent isotope under stated assumptions. Assessing radiation exposure or whether material is safe requires appropriate measurements, isotope identification, and qualified radiological guidance.

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in n0 | Initial quantity | g | 10 | 0
in t | Elapsed time | years | 5730 | 0
in h | Half-life | years | 5730 | 0
out n | Quantity remaining | g | n0 * exp(-ln(2) * t / h) | 4
out d | Fraction decayed | % | 100 * (1 - exp(-ln(2) * t / h)) | 2
out lam | Decay constant | 1/years | ln(2) / h | 6