Radioactive Decay Calculator

Remaining quantity equals initial quantity times e to the power of negative decay constant times time

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Radioactive Decay

The exponential decay law describes how the quantity of a radioactive substance decreases over time. The decay constant λ determines the rate. Larger λ means faster decay.

N(t) = N₀ e^(−λt)

Half-Life

The half-life is the time required for half the atoms in a sample to decay. It is the most commonly quoted measure of decay speed and is unique to each isotope.

t½ = ln(2) / λ

Mean Lifetime

The mean lifetime is the average time an atom survives before decaying. It is always longer than the half-life by a factor of 1/ln(2) ≈ 1.443.

τ = 1 / λ

Activity

Activity measures the number of disintegrations per second. The SI unit is the becquerel (Bq), equal to one decay per second. Higher activity means a more intensely radioactive sample.

A = λ N

How It Works

Radioactive decay follows an exponential law: N(t) = N₀e^(−λt). The decay constant λ determines how quickly a material decays. Related quantities include the half-life (t½ = ln2/λ), mean lifetime (τ = 1/λ), and activity (A = λN). Each isotope has a unique half-life, from microseconds to billions of years.

Example Problem

Carbon-14 has a half-life of 5,730 years. If a sample originally contained 1,000 atoms, how many remain after 11,460 years?

  1. Identify the knowns. Initial quantity N₀ = 1,000 atoms, half-life t½ = 5,730 years, and elapsed time t = 11,460 years.
  2. Identify what we're solving for. We want the remaining number of atoms N(t) after the elapsed time.
  3. Compute the decay constant from the half-life: λ = ln(2) / t½ = 0.6931 / 5,730 ≈ 1.2098 × 10⁻⁴ yr⁻¹.
  4. Write the exponential decay law in symbols: N(t) = N₀ × e^(−λt).
  5. Substitute the known values: N(t) = 1,000 × e^(−1.2098 × 10⁻⁴ × 11,460) = 1,000 × e^(−1.3863).
  6. Simplify the arithmetic: e^(−1.3863) = 0.25, so N(t) = 1,000 × 0.25 = **250 atoms** remain (one quarter of the original sample, as expected after exactly two half-lives).

When to Use Each Variable

  • Solve for Remaining Quantity — when you know the initial quantity, decay constant, and elapsed time, e.g., determining how much of a medical isotope remains after shipping.
  • Solve for Half-Life — when you know the decay constant and want to express the decay rate in the most commonly quoted form.
  • Solve for Mean Lifetime — when you need the average survival time per atom, e.g., calculating expected detection rates in a particle physics experiment.
  • Solve for Activity — when you know the decay constant and number of atoms, e.g., determining the disintegration rate of a radioactive source in becquerels.

Key Concepts

Radioactive decay is a random process governed by the exponential law N(t) = N₀e^(−λt). The decay constant λ is unique to each isotope and determines the half-life (t½ = ln2/λ) and mean lifetime (τ = 1/λ). Activity (A = λN) measures disintegrations per second in becquerels. After n half-lives, the fraction remaining is (½)ⁿ.

Applications

  • Radiometric dating: determining the age of archaeological artifacts (carbon-14) and geological formations (uranium-lead, potassium-argon)
  • Nuclear medicine: calculating dose decay rates for radiopharmaceuticals like technetium-99m used in diagnostic imaging
  • Radiation safety: estimating how long radioactive waste must be stored before activity drops to safe levels
  • Nuclear power: tracking fuel burnup and predicting fission product inventories in reactor cores

Common Mistakes

  • Confusing half-life with mean lifetime — mean lifetime is always 1.443× longer than half-life; using them interchangeably gives wrong decay rates
  • Applying the decay formula to a mixture of isotopes without separating them — each isotope has its own λ and must be decayed independently
  • Assuming activity is constant — activity decreases exponentially along with the number of atoms; a freshly produced medical isotope is far more active than the same sample a few half-lives later

Frequently Asked Questions

What is a half-life?

A half-life is the time it takes for half the atoms in a radioactive sample to decay. After two half-lives, one-quarter of the original atoms remain; after three, one-eighth, and so on.

What is the difference between half-life and mean lifetime?

The mean lifetime τ is the average survival time of an atom. It is always longer than the half-life by a factor of 1/ln(2) ≈ 1.443. Both are determined by the decay constant λ.

How is radioactive decay used in dating?

By measuring the ratio of a radioactive isotope to its decay product, scientists can calculate how long ago the material formed. Carbon-14 dating works for organic materials up to ~50,000 years old; uranium-lead dating covers billions of years.

How do I convert a half-life into the decay constant λ?

Use λ = ln(2) / t½. For carbon-14 with t½ = 5,730 years, λ = 0.6931 / 5,730 ≈ 1.21 × 10⁻⁴ yr⁻¹. The decay constant has units of inverse time (per second, per year, etc.) and is what plugs directly into N(t) = N₀ e^(−λt).

What is a becquerel and how does it compare to a curie?

The becquerel (Bq) is the SI unit for activity, equal to one disintegration per second. The older non-SI curie (Ci) is defined as exactly 3.7 × 10¹⁰ Bq — roughly the activity of one gram of radium-226. Nuclear medicine doses are typically quoted in MBq or mCi (1 mCi = 37 MBq).

How long does radioactive waste stay dangerous?

Activity decreases exponentially, but "safe" depends on the isotope and exposure pathway. A common rule of thumb is that an isotope's hazard is largely gone after about 10 half-lives, when activity has dropped to ~0.1% of the original. Fission products like Cs-137 (t½ ≈ 30 yr) become low-hazard within centuries; transuranic isotopes like Pu-239 (t½ ≈ 24,000 yr) require geological-timescale storage.

Does the decay law depend on temperature, pressure, or chemistry?

No — to extremely high precision, radioactive decay is unaffected by ordinary chemical or environmental conditions because it is a nuclear process. Tiny effects (less than a part in 10⁻⁴) have been measured for some electron-capture decays under extreme pressure, but for engineering and clinical work the rate is treated as a fundamental property of the isotope.

Reference:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Worked Examples

Nuclear Medicine

What is the half-life of Fluorine-18 from its decay constant?

Fluorine-18 is the workhorse PET-imaging isotope: a cyclotron makes it on-site, the radiochemist binds it to glucose to form FDG, and the patient is scanned before too much of the dose has decayed. Compute t½ from λ ≈ 1.053 × 10⁻⁴ s⁻¹.

  • Knowns: λ = 1.053 × 10⁻⁴ s⁻¹
  • Formula: t½ = ln(2) / λ
  • t½ = 0.6931 / (1.053 × 10⁻⁴)

t½ ≈ 6,582 s ≈ 109.7 minutes

F-18's ~110-min half-life is short enough to limit patient dose and long enough to allow synthesis and scanning, which is why it dominates clinical PET — other PET isotopes like ¹¹C have even shorter half-lives that require an on-site cyclotron and rushed chemistry.

Radiocarbon Dating

How old is an artifact with 25% of its carbon-14 remaining?

An archaeologist measures that a wooden tool retains 25% of the carbon-14 a living tree would contain. Using the C-14 decay constant λ ≈ 3.84 × 10⁻¹² s⁻¹, find the elapsed time since the tree died.

  • Knowns: N₀ = 1 (normalized initial C-14), N = 0.25 (current fraction), λ = 3.84 × 10⁻¹² s⁻¹
  • Formula (decay, solved for time): t = ln(N₀ / N) / λ
  • t = ln(1 / 0.25) / (3.84 × 10⁻¹²)
  • t = ln(4) / (3.84 × 10⁻¹²) = 1.3863 / (3.84 × 10⁻¹²)

t ≈ 3.61 × 10¹¹ s ≈ 11,440 years

25% remaining means two half-lives have elapsed, so the age is roughly 2 × 5,730 ≈ 11,460 years — consistent with our exact answer to within rounding of the decay constant.

Nuclear Medicine

What is the activity of an iodine-131 sample containing 10¹⁶ nuclei?

Iodine-131 is used for thyroid ablation therapy. With λ ≈ 9.98 × 10⁻⁷ s⁻¹ (half-life ≈ 8.02 days) and N = 10¹⁶ nuclei in a dose vial, compute the activity A = λN.

  • Knowns: λ = 9.98 × 10⁻⁷ s⁻¹, N = 1 × 10¹⁶ nuclei
  • Formula: A = λ × N
  • A = (9.98 × 10⁻⁷) × (1 × 10¹⁶)

A ≈ 9.98 × 10⁹ Bq ≈ 9.98 GBq

1 GBq corresponds to 10⁹ disintegrations per second; in older units this is about 270 mCi. Therapeutic I-131 doses are typically in the 1–10 GBq range, so this sample is in the clinical-dose ballpark.

Radioactive Material Formulas

Four related equations describe how a radioactive isotope decreases in number, how long the decay takes, and how active the sample is at any moment:

N(t) = N₀ × e^(−λ × t)Exponential decay law
t½ = ln(2) / λHalf-life
τ = 1 / λMean lifetime
A = λ × NActivity (disintegrations per second)

Where:

  • N₀ — initial number of radioactive atoms (unitless count)
  • N(t) — number of atoms remaining after elapsed time t
  • λ — decay constant, in inverse time (s⁻¹, yr⁻¹, etc.)
  • t — elapsed time since t = 0, in seconds, days, or years
  • t½ — half-life: time for half of the sample to decay, in seconds, days, or years
  • τ — mean lifetime: average survival time per atom (= 1.443 × t½)
  • A — activity, in becquerels (Bq = 1 disintegration/s); 1 curie (Ci) = 3.7×10¹⁰ Bq
  • N — current number of radioactive atoms in the sample

The decay law assumes a single isotope decaying via a single mode. For mixtures (parent + daughter chains or multi-isotope samples) each species must be decayed independently with its own λ and then summed. The exponential form is a fundamental property of nuclear decay — it does not depend on temperature, pressure, or chemical environment to any measurable degree under ordinary conditions.

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