Radioactive Decay Calculator

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

Solution

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How It Works

Radioactive decay follows the exponential law N(t) = N₀ e^(−λt). The number of atoms remaining at time t equals the initial number multiplied by an exponential factor whose exponent is the negative product of the decay constant λ and elapsed time t. Larger λ → faster decay; one half-life is the special time at which the exponential factor equals 0.5. Rearrangements solve for initial quantity, decay constant, or elapsed time.

Example Problem

A sample originally contained 1,000 atoms of an isotope with decay constant λ = 0.1386 day⁻¹. How many atoms remain after t = 10 days?

  1. Apply N(t) = N₀ e^(−λt). Substitute N₀ = 1000, λ = 0.1386, t = 10.
  2. Compute the exponent: −λt = −0.1386 × 10 = −1.386.
  3. Evaluate the exponential: e^(−1.386) ≈ 0.2500.
  4. Multiply: N = 1000 × 0.2500 = 250 atoms.
  5. Cross-check with half-life: t½ = ln(2)/0.1386 ≈ 5 days, so 10 days is 2 half-lives, leaving (½)² = 25% = 250 atoms.

Key Concepts

Radioactive decay is the prototype of a memoryless exponential process: each nucleus has a constant per-unit-time probability λ of decaying, independent of how long it has already survived. Over a population this gives the smooth exponential curve, but for any individual atom the decay time is random. The same exponential math governs capacitor discharge, drug elimination, and continuously compounded interest — only the constant changes.

Applications

  • Radiometric dating — measure the current N and known N₀ (or isotope ratio) to solve for t, the sample's age.
  • Medical imaging — calculate how much technetium-99m activity remains at injection time given its 6-hour half-life and time since elution.
  • Reactor decay heat — sum the exponential decay of every fission product to predict cooling pool loads.
  • Forensic chemistry — date paint, paper, and other artefacts through carbon-14 or lead-210 ratios.
  • Smoke detector physics — Am-241's 432-year half-life keeps activity essentially constant over the detector's service life.

Common Mistakes

  • Forgetting the negative sign in the exponent — e^(λt) instead of e^(−λt) gives exponential growth, not decay.
  • Confusing N with N₀ when solving for time — make sure your numerator and denominator inside the natural log are oriented correctly: t = ln(N₀/N)/λ.
  • Mixing units of λ and t — convert both to the same time base before exponentiating.
  • Applying the formula to a mixture of isotopes — each isotope has its own λ and must be decayed independently, then summed.
  • Assuming linear decay — 50% decays per half-life, not per unit of time.

Frequently Asked Questions

How do you calculate radioactive decay?

Apply N(t) = N₀ e^(−λt): multiply the initial number of atoms by e raised to the negative product of the decay constant and elapsed time. Rearrange to solve for N₀, λ, or t depending on which quantity is unknown.

What is the formula for radioactive decay?

N(t) = N₀ e^(−λt), where N₀ is the starting number of atoms, λ is the decay constant, and t is elapsed time. Equivalent forms use half-life: N(t) = N₀ × (½)^(t / t½).

What is the decay constant?

The decay constant λ is the probability per unit time that a given nucleus decays. It's related to the half-life by λ = ln(2) / t½ and to mean lifetime by λ = 1/τ. Larger λ means faster decay.

How do you solve for elapsed time in a decay equation?

Rearrange N = N₀ e^(−λt) to get t = ln(N₀/N) / λ. Plug in the starting and current atom counts (or activities, which scale identically) along with the decay constant to find how long the sample has been decaying.

Is radioactive decay linear or exponential?

Exponential. The number of atoms remaining follows N(t) = N₀ e^(−λt), so each unit of time removes a constant fraction (not a constant amount) of the remaining atoms.

What's the difference between N and activity in decay equations?

N is the number of atoms; activity A = λN is the rate of disintegrations per second. Both follow the same exponential time dependence — A(t) = A₀ e^(−λt) — so the decay equation can be written interchangeably in terms of N or A as long as you compare like to like.

Reference:

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

Worked Examples

Nuclear Medicine

How much F-18 activity remains 3 hours after dose calibration?

Fluorine-18 (used in FDG PET imaging) has a half-life of 109.7 minutes, giving a decay constant λ = ln(2)/109.7 ≈ 0.00632 min⁻¹. A vial calibrated to 10 mCi at 8:00 AM is administered at 11:00 AM (180 minutes later). What activity does the patient actually receive?

  • Knowns: N₀ = 10 mCi, λ = 0.00632 min⁻¹, t = 180 min
  • N = N₀ × e^(−λt)
  • N = 10 × e^(−0.00632 × 180)
  • N = 10 × e^(−1.1376)

N ≈ 3.21 mCi

Nuclear medicine departments use this decay correction many times a day — pharmacists calibrate to a future time and the nurse must verify the actual dose at the moment of injection.

Radiocarbon Dating

How old is a sample with 25% of its original carbon-14?

A charcoal sample from an archaeological site shows ¹⁴C activity at 25% of a modern reference (N/N₀ = 0.25). Carbon-14's decay constant is λ = ln(2)/5,730 ≈ 1.21 × 10⁻⁴ yr⁻¹ (Cambridge half-life). Estimate the age of the sample.

  • Knowns: N₀ = 1000, N = 250 (relative units), λ = 1.21 × 10⁻⁴ yr⁻¹
  • t = ln(N₀/N) / λ
  • t = ln(4) / 1.21 × 10⁻⁴
  • t = 1.3863 / 1.21 × 10⁻⁴

t ≈ 11,458 yr

Right in line with the "two half-lives" mental check: 2 × 5,730 = 11,460 yr. Real lab work would also apply reservoir corrections and calibration curves (IntCal20) — radiocarbon years aren't quite the same as calendar years past ~12,000 yr.

Radiation Detection

What decay constant does a 50% drop in count rate over 5 minutes imply?

A Geiger counter near an unknown source reads 1,000 counts per minute initially; 5 minutes later it reads 500 cpm. Background has been subtracted. Compute the decay constant of the source so you can match it to a candidate isotope.

  • Knowns: N₀ = 1,000 cpm, N = 500 cpm, t = 5 min
  • λ = ln(N₀/N) / t
  • λ = ln(2) / 5
  • λ = 0.6931 / 5

λ ≈ 0.1386 min⁻¹

A half-life of 5 minutes (since N halved over t = 5 min by definition) is consistent with several short-lived medical isotopes such as O-15 (t½ ≈ 2 min) or some daughter products — combine with gamma-spectroscopy to nail down the species.

Radioactive Decay Formulas

A radioactive sample shrinks exponentially because each atom has the same constant probability per unit time of decaying. The exponential law N(t) = N₀·e−λt can be rearranged to solve for any of its four variables:

N(t) = N₀ × e−λtRemaining quantity after elapsed time t
N₀ = N × eλtInitial quantity from a later measurement
λ = ln(N₀ / N) / tDecay constant from two measurements separated by t
t = ln(N₀ / N) / λElapsed time between two known quantities

Where:

  • N(t) — number of radioactive atoms (or activity, or mass) remaining at time t
  • N₀ — original quantity at time 0 (same unit as N — count, becquerels, grams, etc., as long as both are in the same unit)
  • λ (lambda) — decay constant, probability per unit time that any one atom decays (s⁻¹, min⁻¹, hr⁻¹, day⁻¹, yr⁻¹). Related to half-life by λ = ln(2) / t½.
  • t — elapsed time since N = N₀ (must be in the reciprocal unit of λ — see Half-Life Calculator if you only know t½)
  • e ≈ 2.71828 — base of the natural logarithm; ln is its inverse, used when solving for λ or t

This law applies to any process whose decay rate is proportional to the current population — radioactivity, first-order chemical kinetics, capacitor discharge, and pharmacokinetic elimination all share the same equation. The result for N(t) and N₀ is unitless in the sense that it carries whatever unit you measured the population in; the calculator preserves that — feed it atoms and you get atoms, feed it Bq and you get Bq. The decay constant and time, however, must always be in reciprocal units.

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