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

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.