Radioactive Decay Simulator

Enter an initial quantity, half-life, and elapsed time to see how many atoms remain. The exponential decay curve marks each half-life, and the atom grid shows the proportion of decayed vs undecayed atoms at a glance. Presets cover common isotopes from Carbon-14 to Technetium-99m.

Decay Curve

Parameters

Initial quantity (N₀)

Half-life (t½)

Elapsed time (t)

Time unit

Presets

Results

Remaining (N)

250

Decayed

750

Fraction remaining

25.00%

Percent decayed

75.00%

Half-lives elapsed

2.00

Decay constant (λ)

1.210e-4 per year

Activity (λN)

0.03024 per year

Atom Grid

Undecayed (25.0%) Decayed (75.0%)

Step-by-Step Calculation

1. Decay formula

N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/t_{1/2}}

2. Substitute values

N(11460) = 1000 \times \left(\frac{1}{2}\right)^{11460/5730}

3. Half-lives elapsed

\frac{t}{t_{1/2}} = \frac{11460}{5730} = 2

4. Remaining quantity

N = 1000 \times 0.25 = 250

5. Decay constant

\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.6931}{5730} = 1.210e-4 \text{ per year}

6. Activity

A = \lambda N = (1.210e-4)(250) = 0.03024

Understanding Radioactive Decay Simulator

Radioactive atoms have unstable nuclei. Their protons and neutrons are arranged in a state with excess energy or an unbalanced nuclear force. A decay happens when the nucleus changes into a more stable arrangement and releases particles or energy.

No outside action can make one particular nucleus decay at a chosen moment. Heat, pressure, light, and ordinary chemical reactions usually do not alter its decay rate.

This is why radioactive decay behaves differently from processes such as burning fuel or cooling water. It is governed by probability at the level of individual atoms.

For a large sample, that random behavior produces a very reliable pattern. During each equal time interval, the same fraction of the nuclei present is likely to decay. It is not the same number each time, because fewer undecayed nuclei remain later.

After one half life, one half remains. After two half lives, one quarter remains. After three half lives, one eighth remains.

The curve becomes less steep over time but does not reach exactly zero in the mathematical model. A simulator represents averages, while a small real sample may show noticeable random variation.

The decay constant describes the chance that a nucleus will decay in a short time. A larger decay constant means a shorter half life and a faster fall in the number of undecayed nuclei. Activity measures how many decays occur each second, not how much material is present by mass.

Its SI unit is the becquerel, meaning one decay per second. A sample can have little mass yet high activity if its nuclei decay quickly. Activity decreases as nuclei disappear, so the radiation from a sealed sample generally becomes weaker with time.

Different decays release different forms of radiation. Alpha particles are heavy and can be stopped by paper or skin, though they are dangerous if inhaled or swallowed. Beta particles travel farther and need suitable shielding such as plastic or thin metal.

Gamma rays are high energy electromagnetic radiation and often need dense shielding such as lead or concrete. In medicine, a short half life can limit how long a tracer remains active in the body. In archaeology, a long half life makes carbon dating useful over thousands of years.

When interpreting results, keep units consistent and distinguish elapsed time, number of nuclei, activity, and radiation dose. Dose depends on the radiation type, energy, exposure time, and where the source is located.

Reference Guide

Exponential Decay Law

Radioactive decay is a random process, but with large numbers of atoms it follows a predictable exponential pattern.

N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/t_{1/2}}

where $N_0$ is the initial quantity, $t_{1/2}$ is the half-life, and $t$ is elapsed time. An equivalent form uses the decay constant: $N(t) = N_0 e^{-\lambda t}$ .

Half-Life

The half-life is the time it takes for exactly half of the remaining atoms to decay. After one half-life, 50% remain. After two, 25%. After three, 12.5%.

t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}

Half-lives range from fractions of a second (Polonium-214) to billions of years (Uranium-238). The half-life is an intrinsic property of each isotope and cannot be changed by temperature, pressure, or chemical reactions.

Decay Constant and Activity

The decay constant $\lambda$ gives the probability per unit time that any single atom will decay. Activity measures the total number of decays per unit time.

A = \lambda N(t)

Activity is measured in becquerels (1 Bq = 1 decay/second) or curies (1 Ci = 3.7 × 10¹⁰ decays/second). As atoms decay, the activity decreases at the same exponential rate.

Applications

Radiocarbon dating

Carbon-14 (t½ = 5,730 years) is used to date organic materials up to about 50,000 years old by measuring the remaining C-14.

Medical imaging

Technetium-99m (t½ = 6 hours) is the most widely used isotope in diagnostic scans. Its short half-life limits radiation exposure.

Cancer treatment

Cobalt-60 and Iodine-131 are used in radiation therapy. Their half-lives determine treatment scheduling and dose planning.

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