Radioactive Decay Formula — N = N₀(½)^(t/t½)

The radioactive decay law states that every radioactive nucleus has a fixed probability of decaying per unit time, leading to exponential decay. This half life calculator is built on two mathematically identical forms of the radioactive decay equation:

The decay constant λ = ln(2)/t½ = 0.693/t½. The mean lifetime τ = 1/λ = t½/ln(2) = 1.4427×t½ — the average survival time of a nucleus before decaying. Critically, the half-life is fixed for each isotope and completely independent of temperature, pressure, chemical environment, or any external condition — making every decay calculator, decay rate calculator, and radioactivity calculator reliable regardless of the sample's physical state. This half life formula calculator accepts any quantity unit: atoms, grams, moles, Becquerel, or percent.

How to Solve Half-Life Problems — Step-by-Step

Every half life problem involves four variables: N₀, N(t), t, and t½. Any three known quantities determine the fourth. This systematic approach solves all half life problems, half life practice problems, and half life chemistry problems:

• Find N(t) from N₀, t, t½: directly substitute into N=N₀(½)^(t/t½)
• Find t from N₀, N(t), t½: take logarithm — t = t½ × ln(N/N₀)/ln(½)
• Find t½ from N₀, N(t), t: rearrange — t½ = t × ln(½)/ln(N/N₀)
• Find N₀ from N(t), t, t½: divide — N₀ = N(t)/(½)^(t/t½)

The number of half-lives n = t/t½. After n half-lives, (½)^n of the original remains: after 1 t½ → 50%, 2 t½ → 25%, 3 t½ → 12.5%, 10 t½ → 0.098%. These half life example problems and half life questions are solved automatically by entering three known values above.

Carbon-14 Dating — Age = −(t½/ln2) × ln(A/A₀)

Carbon-14 is produced continuously in the upper atmosphere by cosmic ray bombardment of nitrogen-14: ¹⁴N + n → ¹⁴C + p. Living organisms maintain a constant ¹⁴C/¹²C ratio through food and CO₂ exchange. At death, the exchange stops, and ¹⁴C decays with the half-life of carbon-14 = 5730 years.

To determine age: measure the current ¹⁴C activity (A), compare it to the modern standard (A₀ = 13.56 dpm/g carbon = 0.226 Bq/g), and apply the carbon dating calculator formula above. The radiocarbon dating formula t = −(5730/ln2) × ln(A/A₀) is the same as the carbon 14 dating formula taught in every chemistry and archaeology course. The reliable range is 200–50,000 years for this radiocarbon dating calculator. This same formula is the radiometric dating formula used for other isotopes with different half-lives (U-Pb, K-Ar).

Decay Constant λ — The Relationship λ = ln(2)/t½

The decay constant λ is the probability per unit time that a nucleus decays. It is derived from the fundamental differential equation dN/dt = −λN, which integrates to N(t) = N₀e^(−λt). Setting N = N₀/2 and solving gives t½ = ln(2)/λ.

The mean lifetime τ = 1/λ = t½/ln(2) = 1.4427×t½ — always larger than t½ by a factor of 1.4427. This is the time for the population to fall to 1/e ≈ 36.8% of the initial value. The elimination rate constant k_el = ln(2)/t½ in pharmacokinetics follows identical mathematics — the same exponential decay example appears in nuclear physics, drug clearance, RC circuits, and Newton's law of cooling. The decay factor per half-life = 0.5; the decay factor per unit time = e^(−λ).

Types of Radioactive Decay — Alpha, Beta, Gamma

There are five modes of radioactive decay. The three main types of radioactive decay are alpha, beta, and gamma:

• Alpha decay (α): ⁴He nucleus emitted, A decreases by 4, Z decreases by 2. Nuclear equation: ²³⁸U → ²³⁴Th + ⁴He. Stopped by paper.
• Beta-minus decay (β⁻): electron emitted, A unchanged, Z increases by 1 (neutron→proton). Carbon-14 beta decay equation: ¹⁴C → ¹⁴N + β⁻ + ν̄ₑ. The beta decay formula shows Z+1.
• Beta-plus decay (β⁺): positron emitted, A unchanged, Z decreases by 1 (proton→neutron). Used in PET scans — ¹⁸F → ¹⁸O + β⁺ + νₑ.
• Gamma decay (γ): photon emitted, both A and Z unchanged. Nuclear equation: ⁶⁰Co* → ⁶⁰Co + γ. Requires lead shielding.
• Electron capture (EC): ⁴⁰K + e⁻ → ⁴⁰Ar + νₑ. Z decreases by 1, A unchanged.

The golden rule for writing nuclear equations: conserve mass number A and atomic number Z on both sides. This rule instantly solves any nuclear equation for beta decay, alpha radiation equation, or gamma decay nuclear equation.

Half-Life in Pharmacokinetics — Drug Elimination

Drug elimination from the body follows first-order kinetics — mathematically identical to radioactive decay. The elimination rate constant k_el = ln(2)/t½. Steady state is reached after approximately 5 half-lives (96.9% of steady-state concentration achieved). The fraction remaining after n half-lives = (½)^n — the same formula as N=N₀(½)^(t/t½). How many half lives to reach steady state? Answer: 5 half-lives to achieve 96.9% of steady state.

Half-Life Table — Common Radioactive Isotopes

The complete isotope half-life reference is in Table A of the Reference tab above — 25 isotopes organized by application. Key entries: Tc-99m (6 hours, medical imaging), I-131 (8.02 days, thyroid), C-14 (5730 years, carbon dating), U-238 (4.47 billion years, uranium-lead dating), Cs-137 (30.17 years, Chernobyl contamination). The half life of isotopes spans 122 seconds (¹⁵O) to 49.7 billion years (⁸⁷Rb).

Exponential Decay — The Mathematics of Half-Life

The half life formula N=N₀(½)^(t/t½) is an exponential decay function with base ½. The decay factor per half-life = exactly 0.5. The decay factor per unit time = e^(−λ) = (½)^(1/t½). Graph shape: starts at N₀, falls steeply, approaches zero asymptotically — this is the exponential decay curve drawn by the calculator above.

The straight-line test: plotting ln(N) versus t gives a straight line with slope −λ and intercept ln(N₀). This is how half life is experimentally measured from a half life graph or radioactivity graph: plot activity versus time, take ln, and the slope gives −λ, from which t½ = ln(2)/λ.

Common Mistakes in Half-Life Calculations

Worked Examples — 8 Complete Half-Life Problems

Frequently Asked Questions

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