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?
- Apply N(t) = N₀ e^(−λt). Substitute N₀ = 1000, λ = 0.1386, t = 10.
- Compute the exponent: −λt = −0.1386 × 10 = −1.386.
- Evaluate the exponential: e^(−1.386) ≈ 0.2500.
- Multiply: N = 1000 × 0.2500 = 250 atoms.
- 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.
Related Calculators
- Half-Life Calculator — t½ = ln(2)/λ — convert between half-life and decay constant
- Mean Lifetime Calculator — τ = 1/λ — average atom survival time
- Activity Calculator — A = λN — disintegration rate in becquerels
- Radioactive Material Calculator — the full four-equation hub for decay problems
- Einstein Equation Calculator — E = mc² — energy released by mass conversion
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