Radioactive Decay Calculator: Half-Life and Calculus

A radioactive decay calculator uses the exponential decay equation to estimate how much of a radioactive sample remains after a chosen time. Enter the starting amount, elapsed time, and half-life to get the remaining amount, the fraction left, and the amount that has decayed.

Formula

The standard model is exponential decay. If a substance loses a constant fraction of its radioactive nuclei per unit of time, its amount follows:

N(t) = N₀e^−λt = N₀(1/2)^t/T_½

• N(t) is the amount remaining after time t, in the same amount unit used for the starting sample.
• N₀ is the starting amount, such as grams, kilograms, or number of nuclei.
• t is elapsed time.
• λ is the decay constant, expressed as inverse time.
• T_½ is the half-life, in the same time unit as t.
• e is the mathematical constant approximately equal to 2.71828.

The calculator uses the half-life form, where λ = ln(2)/T_½. Its first result is N(t) = N₀ exp[−ln(2)t/T_½]. The fraction remaining is N(t)/N₀, and the amount decayed is N₀ − N(t). The same equations apply whether the amount is a mass or a count of nuclei, provided the starting and remaining amounts use matching units.

Worked Example

Using the calculator’s default inputs, the starting amount is 100 g, the elapsed time is 10 years, and the half-life is 5 years.

1. Calculate the number of half-lives: t/T_½ = 10/5 = 2.
2. Find the remaining fraction: (1/2)² = 0.25.
3. Calculate the remaining amount: 100 × 0.25 = 25.00 g.
4. The amount decayed is 100 − 25 = 75.00 g.

After two half-lives, one-quarter remains. This example illustrates why the amount does not decrease by the same fixed number of grams during each equal time interval: each interval removes half of what remains.

How Calculus Describes Radioactive Decay

The calculus model begins with a differential equation: dN/dt = −λN. It says the rate of change of the number of undecayed nuclei is proportional to the number still present. The negative sign indicates that the amount decreases. Solving this equation by separation of variables gives N(t) = N₀e^−λt.

This is a first-order model: the instantaneous decay rate depends on the current amount, not on a fixed quantity lost per year. The half-life is a more convenient way to specify the decay constant because it is directly interpretable: after one half-life, half the original nuclei remain on average. For a pure isotope, λ = ln(2)/T_½.

When to Use the Calculator

Use this calculation when you know the starting amount, the elapsed duration, and the half-life, and want an idealized estimate of what remains. For instance, it can help with textbook exercises, conceptual comparisons, or tracing the mathematical consequences of a stated half-life. It can also calculate the fraction decayed by subtracting the remaining fraction from 1.

Choose units consistently. If the half-life is given in days, enter elapsed time in days; if it is given in seconds, use seconds for both. The amount unit can be grams, milligrams, kilograms, or nuclei, since the equation uses a ratio of amounts. The output uses the same amount unit as the input. US customary mass units such as ounces or pounds also work mathematically, but scientific data are commonly reported in metric units.

Reading Results and Common Mistakes

• Mixing time units: Entering 2 years against a half-life of 100 days is not meaningful until both durations are converted to the same unit.
• Confusing half-life with complete disappearance: A sample does not become exactly zero in this model after a finite number of half-lives. It approaches zero as time increases.
• Assuming a fixed amount disappears each interval: Each half-life halves the amount then present, rather than subtracting half the original amount every time.
• Rounding too early: Keep full precision through intermediate steps and round only the displayed result. A rounded displayed mass does not imply that the underlying prediction is exact.
• Entering an invalid half-life: A half-life must be positive. Zero would make the expression undefined, and a negative value does not represent a physical half-life.

The calculator’s displayed precision is not the same as measurement precision. If the starting quantity or half-life is uncertain, the result is uncertain too; two decimal places merely format the computed value.

Assumptions and Limits

The equation describes radioactive decay under a constant decay probability per nucleus, and it predicts the expected amount remaining. Individual nuclear decays are random. In a small sample, the actual number of decays can differ noticeably from the expected value; a simple calculator does not model that statistical variation.

The calculation also assumes the material and its isotope composition are known and that the supplied half-life applies to the radioactive isotope being considered. It does not identify an isotope from a sample, measure radioactivity, account for detector efficiency, or determine the age of an unknown sample. Real samples can contain mixtures of isotopes, daughter products, or contamination, so a single-half-life calculation may not describe them.

Mass remaining and activity are related but are not interchangeable outputs. For a single isotope, activity is proportional to the number of undecayed nuclei, with A = λN when activity is expressed as decays per unit time. A numerical activity result requires the isotope’s decay constant and a suitable amount or atom count, along with consistent units; this calculator reports remaining amount, not becquerels or curies.

Frequently Asked Questions

What is the radioactive decay formula?
N(t) = N₀e^−λt, where N₀ is the starting amount, λ is the decay constant, and t is elapsed time. If half-life is known, use N(t) = N₀(1/2)^t/T_½.

How much remains after three half-lives?
The fraction remaining is (1/2)³ = 1/8, or 12.5% of the starting amount. The other 87.5% has decayed in the model.

Does radioactive material ever reach zero?
The exponential model approaches zero but does not equal zero at any finite time. In practice, a sample may become too small to detect, which is different from being mathematically zero.

Can I calculate elapsed time if I know the remaining amount?
Yes. Rearranging the model gives t = T_½ × log₂(N₀/N), provided the starting and remaining amounts are positive and measured on the same basis. This calculator is set up to find remaining amount from time, not to perform that inverse calculation.

Is the amount decayed the same as the amount remaining?
No. For a closed sample represented by this model, amount decayed is starting amount minus amount remaining. After one half-life, the two amounts are equal; at other times they generally are not.

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in a | Starting amount | g | 100 | 0
in t | Elapsed time | years | 10 | 0
in h | Half-life | years | 5 | 0
out r | Amount remaining | g | a * exp(-ln(2) * t / h) | 2
out f | Fraction remaining | - | exp(-ln(2) * t / h) | 4
out d | Amount decayed | g | a - r | 2