Science

Radioactive Decay Calculator

Calculate remaining radioactive material using N = N₀×e⁻λt. Enter initial amount, half-life or decay constant, and time elapsed. Shows remaining amount, activity, and fraction decayed.

Quick Answer

N = N₀ × e⁻λt, where λ = ln(2)/t½. After each half-life, exactly half of the material remains.

Calculate

Enter the initial amount, decay parameter (half-life or decay constant), and time elapsed.

Presets:
Remaining Amount
250
Amount Decayed
750
Half-Lives Elapsed
2
Fraction Remaining
25.0000%
Current Activity
9.5831e-10 Bq
Initial Activity
3.8332e-9 Bq
Decay Constant (λ)
3.8332e-12 /s

About This Tool

The Radioactive Decay Calculator computes the remaining amount of a radioactive substance after a given time using the exponential decay equation N = N₀ × e⁻λt. It accepts either a half-life or a decay constant as input, automatically converting between them using the relationship λ = ln(2)/t½. The calculator outputs the remaining amount, the amount decayed, the current and initial activities (in becquerels), the number of half-lives elapsed, and the fraction remaining. It supports multiple time units from seconds to years.

The Mathematics of Radioactive Decay

Radioactive decay follows first-order kinetics, meaning the rate of decay is proportional to the number of undecayed nuclei present: dN/dt = -λN. Solving this differential equation gives the exponential decay law: N(t) = N₀ × e⁻λt. The decay constant λ has units of inverse time and represents the probability per unit time that a given nucleus will decay. The half-life t½ = ln(2)/λ is the time for the population to decrease by half. The mean lifetime τ = 1/λ is the average survival time of a single nucleus, approximately 1.443 times the half-life.

Activity and Measurement

Activity (A) measures how many nuclei decay per second: A = λN = λN₀e⁻λt = A₀e⁻λt. The SI unit of activity is the becquerel (Bq), equal to one disintegration per second. The older unit, the curie (Ci), equals 3.7 × 10¹⁰ Bq and was originally defined as the activity of one gram of radium-226. In medical and health physics, activity is crucial for calculating radiation doses. The specific activity (activity per unit mass) depends on both the decay constant and the atomic mass of the isotope.

Common Radioactive Isotopes

Different isotopes span an enormous range of half-lives. Carbon-14 (t½ = 5,730 years) is used for archaeological dating. Iodine-131 (t½ = 8.02 days) is used in thyroid cancer treatment and diagnosis. Technetium-99m (t½ = 6.01 hours) is the most widely used medical imaging isotope. Cobalt-60 (t½ = 5.27 years) is used in radiation therapy and industrial radiography. Uranium-238 (t½ = 4.47 billion years) is used for geological dating. Polonium-214 (t½ = 164 microseconds) is one of the shortest-lived naturally occurring isotopes.

Decay Chains

Many radioactive isotopes do not decay directly to a stable product but instead produce another radioactive isotope, forming a decay chain. For example, uranium-238 decays through a series of 14 steps (including thorium-234, radium-226, radon-222, and polonium-210) before reaching stable lead-206. The Bateman equations describe the time-dependent concentrations of each member in a decay chain. This calculator handles single-step decay; for chain calculations, the secular equilibrium approximation can be used when the parent half-life is much longer than the daughter half-lives.

Applications

Radioactive decay calculations are essential in many fields. In medicine, they determine safe dosages for radiopharmaceuticals and plan radiation therapy schedules. In nuclear power, they predict spent fuel radiation levels and waste storage requirements. In archaeology and geology, they enable radiometric dating techniques (carbon-14, potassium-argon, uranium-lead). In environmental science, they track the dispersion of radioactive contaminants. In security, they help identify nuclear materials through their characteristic decay signatures.

Limitations of This Model

The exponential decay model assumes a single decay mode and a large number of atoms (so that the law of large numbers applies). For very small samples (a few hundred atoms or fewer), the decay becomes noticeably stochastic, and the actual number remaining at any given time fluctuates around the exponential prediction. The model also does not account for decay chains, branching ratios (when a nucleus can decay by more than one pathway), or the production of new radioactive material (as occurs in nuclear reactors). For these more complex scenarios, numerical simulation or the full Bateman equations are required.

Frequently Asked Questions

What is radioactive decay?
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration by emitting radiation (alpha particles, beta particles, or gamma rays). The process follows first-order kinetics, meaning the rate of decay is proportional to the number of undecayed atoms present. The fundamental equation is N = N0 * e^(-lambda*t), where N is the remaining amount, N0 is the initial amount, lambda is the decay constant, and t is time. Each radioactive isotope has a characteristic decay constant that determines how quickly it decays.
What is half-life and how is it related to the decay constant?
Half-life (t1/2) is the time required for exactly half of a radioactive sample to decay. It is related to the decay constant by the equation t1/2 = ln(2)/lambda, or equivalently lambda = ln(2)/t1/2. Half-life is often more intuitive than the decay constant: after one half-life, 50% remains; after two half-lives, 25% remains; after three half-lives, 12.5% remains, and so on. Half-lives range from fractions of a second (polonium-214: 164 microseconds) to billions of years (uranium-238: 4.47 billion years).
What is activity and what units is it measured in?
Activity is the rate at which a radioactive sample decays, measured as the number of disintegrations per second. The SI unit is the becquerel (Bq), where 1 Bq = 1 disintegration per second. The older unit is the curie (Ci), where 1 Ci = 3.7 x 10^10 Bq (originally defined as the activity of 1 gram of radium-226). Activity is calculated as A = lambda * N, where lambda is the decay constant and N is the number of undecayed atoms. Activity decreases over time following the same exponential law as the number of atoms.
What is Carbon-14 dating and how does it work?
Carbon-14 dating uses the radioactive decay of carbon-14 (half-life 5,730 years) to determine the age of organic materials. Living organisms constantly exchange carbon with the environment, maintaining a steady ratio of C-14 to C-12. When an organism dies, it stops taking in new carbon, and its C-14 begins to decay. By measuring the remaining C-14 fraction and applying the decay equation, scientists can calculate when the organism died. The method is reliable for ages up to about 50,000 years (roughly 9 half-lives), beyond which too little C-14 remains to measure accurately.
Can you speed up or slow down radioactive decay?
Under normal conditions, radioactive decay rates are essentially constant and unaffected by temperature, pressure, chemical state, or other external conditions. This is because nuclear decay is governed by the strong and weak nuclear forces, which operate at energy scales far above those of chemical reactions. There are a few extreme exceptions: electron capture rates can be slightly altered by extreme pressure or ionization state, and bound-state beta decay (where the electron is captured into an atomic orbital) depends on the electron configuration. For all practical purposes, however, half-lives are immutable constants of nature.
What is the difference between mean lifetime and half-life?
Mean lifetime (tau) is the average time a single radioactive atom survives before decaying. It equals 1/lambda, where lambda is the decay constant. The relationship between mean lifetime and half-life is: tau = t1/2 / ln(2), or approximately tau = 1.443 * t1/2. Mean lifetime is always longer than half-life because some atoms survive much longer than average, skewing the mean upward. Both quantities fully characterize the decay rate but are used in different contexts: half-life is more common in nuclear physics and chemistry, while mean lifetime is preferred in particle physics.

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