Half-Life Calculator
Remaining quantity after decay from initial amount, half-life and elapsed time.
Remaining quantity
12.5
12.5% left
Decayed amount
87.5
Half-lives elapsed
3
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How the Half-Life Calculator works
The Half-Life Calculator determines how much of a substance remains after radioactive or exponential decay, given an initial quantity, the substance's half-life, and the elapsed time. It's an essential tool for students, physicists, chemists, medical professionals, and anyone working with radioactive isotopes, carbon dating, or pharmacokinetics.
At its core, this calculator applies the exponential decay formula to find the remaining quantity of a substance after a specified period. You input three values: the initial amount (N₀), the half-life (t½) of the substance, and the total elapsed time (t). The calculator then computes how much of the original quantity persists. Because radioactive decay is a probabilistic, memoryless process, the fraction that decays in any given half-life period is always exactly one-half of whatever is currently present — not one-half of the original amount. This distinction is critical for accurate results.
The concept of a half-life is rooted in the physics of nuclear instability. Every radioactive isotope has a characteristic half-life that ranges enormously — from fractions of a microsecond for highly unstable nuclei to billions of years for stable isotopes like uranium-238. The half-life is constant for a given isotope under normal conditions and is unaffected by temperature, pressure, or chemical state, which makes it an exceptionally reliable clock. This property underpins techniques like radiocarbon dating, where the known half-life of Carbon-14 (approximately 5,730 years) allows scientists to estimate the age of organic materials.
A common mistake users make is confusing the half-life with the 'time for complete decay.' No matter how many half-lives pass, exponential decay means the quantity approaches zero asymptotically — it never reaches absolute zero mathematically. After 10 half-lives, roughly 0.1% of the original substance remains; after 20 half-lives, about 0.0001% remains. For practical purposes this is negligible, but it is never truly zero. Another frequent error is mismatching units: if your half-life is in days but your elapsed time is in years, you must convert to the same unit before calculating, or results will be wildly inaccurate.
Beyond nuclear physics, the same exponential decay model applies to pharmacokinetics (drug elimination from the body), capacitor discharge in electronics, and even the cooling of hot objects described by Newton's law of cooling. In medicine, a drug's biological half-life tells clinicians how frequently a dose must be administered to maintain a therapeutic level. This calculator handles all these scenarios identically because the underlying mathematics — exponential decay — is universal. Always verify that your half-life value matches the specific context: nuclear half-life and biological half-life for the same substance can differ significantly.
Formula
N = N₀ · (1/2)^(t / T½)
Pro tips
- Always use consistent units for half-life and elapsed time — convert both to the same unit (seconds, days, years) before entering values, as a mismatch is the single most common source of error.
- To find how many half-lives have elapsed, simply divide your elapsed time by the half-life; this integer or decimal tells you immediately how much decay has occurred without needing the full formula.
- When working with very short or very long half-lives (e.g., milliseconds or millions of years), use scientific notation for your inputs to avoid rounding errors that compound exponentially.
- For pharmacokinetics, remember to distinguish between the drug's nuclear half-life and its biological half-life — the latter accounts for metabolism and excretion and is the clinically relevant figure.
- If you know the remaining quantity and initial quantity but need to find the elapsed time, rearrange the formula: t = t½ × log₂(N₀ / N(t)), which is equally useful for carbon dating calculations.
Key terms
- Half-Life (t½)
- — The time required for exactly half of a given quantity of a radioactive or decaying substance to transform into another substance or form.
- Initial Quantity (N₀)
- — The amount of the substance present at the starting point of measurement, before any decay has occurred.
- Decay Constant (λ)
- — A substance-specific constant representing the probability of decay per unit time, mathematically equal to ln(2) divided by the half-life.
- Exponential Decay
- — A mathematical process in which a quantity decreases at a rate proportional to its current value, producing a characteristic curved graph that never reaches zero.
- Remaining Quantity N(t)
- — The amount of the original substance still present after a specified elapsed time t, as calculated by the decay formula.
- Radioactive Isotope
- — An unstable variant of a chemical element whose nucleus spontaneously loses energy by emitting radiation, causing it to decay into a different element or isotope over time.