Half-Life / Radioactive Decay

Calculating Exponential Decay with the Half-Life Formula

A half-life is the time it takes for exactly half of a quantity to decay — whether that’s a radioactive isotope, a drug in the body, or any other exponentially-decaying amount. Enter the initial quantity and half-life, plus either the elapsed time or the remaining amount, and this calculator finds the other value.

The Formula

$$\vB{N(t)} = \vA{N_0} \times 0.5^{t / T}$$ $$t = T \times \log_2\left(\frac{\vA{N_0}}{\vB{N_t}}\right)$$

where $N_0$ is the initial quantity, $T$ is the half-life, $t$ is the elapsed time, and $N_t$ is the remaining quantity at time $t$.

Worked Example

A substance with an initial quantity of 100 g and a half-life of 10 days:

1. After 20 days (2 half-lives): $\vB{N(t)} = 100 \times 0.5^{20/10} = 100 \times 0.25 \approx \vB{25}$ g remains.
2. That’s $\vC{25\%}$ of the initial quantity, after exactly 2 half-lives have elapsed.

Key Factors to Consider

• A substance is never fully “gone” under pure exponential decay — the quantity only ever approaches zero. This is a genuine mathematical property of exponential decay: the formula produces an ever-smaller but never-exactly-zero remaining amount — in practice, a quantity becomes negligible (rather than mathematically zero) after several half-lives, commonly considered effectively depleted after around 5-7 half-lives.
• Different substances have vastly different half-lives, from fractions of a second to billions of years. Radioactive isotopes used in medicine often have half-lives of hours to days, while some used in geological dating (like uranium-238) have half-lives measured in billions of years — the same formula applies at every scale, only the half-life value changes.
• Drug elimination half-life in the body follows the same math, but real biological elimination can be more complex. Many drugs approximate simple exponential elimination reasonably well, but some follow more complex multi-phase elimination patterns — a drug’s actual clinical half-life should come from pharmacological data specific to that drug, not assumed to be a perfect single exponential.
• Carbon dating and similar techniques work by measuring the remaining fraction and solving for elapsed time. This is exactly the “solve for time” direction of this calculator — knowing the original and current amount of a radioactive isotope lets you calculate how long decay has been occurring, the basic principle behind radiometric dating methods.

Common Mistakes

• Assuming a substance is completely gone after a small number of half-lives. Exponential decay never truly reaches zero — after 2 half-lives, 25% remains, not 0%; after 5 half-lives, about 3% remains. “Effectively gone” is a practical approximation, not a mathematical certainty.
• Thinking decay removes a fixed amount each half-life rather than a fixed fraction. Each half-life removes half of whatever quantity remains at that point, not half of the original amount every time — 100 g becomes 50 g, then 25 g, then 12.5 g, never dropping by a flat 50 g each period.
• Mixing time units between the half-life and the elapsed time. If the half-life is entered in days, the elapsed time (or vice versa) needs to be in the same unit — entering one in days and the other in years silently produces a meaningless result.
• Using this calculator’s simple single-exponential model for a substance with a genuinely multi-phase decay or elimination pattern. As noted above, a real drug’s clinical half-life data (or a complex nuclear decay chain) may not follow one clean exponential curve — check substance-specific data before treating this simple model as exact for a real-world case.

Useful to Know

Radiocarbon dating has a practical age limit, driven directly by carbon-14’s own half-life. After roughly 8-10 half-lives (around 50,000-60,000 years, since carbon-14’s half-life is about 5,730 years), so little carbon-14 remains in a sample that measuring it reliably becomes impractical — which is exactly why radiocarbon dating isn’t used for anything older than about 50,000 years. Older samples are dated with isotopes that have much longer half-lives instead, like potassium-40 or uranium-238, using this same exponential-decay math.