How this instrument works
A half-life is the time it takes for exactly half of a radioactive (or otherwise exponentially decaying) quantity to disappear, no matter how much you started with. Wait that same interval again and it halves once more, leaving a quarter of the original; wait a third time and an eighth remains. That doubling-in-reverse pattern is what N(t) = N0 x (1/2)^(t / half-life) describes: N0 is the starting amount, t is however long has elapsed, and dividing t by that interval tells the formula how many halvings to apply, including fractional ones.
The pattern holds because radioactive decay is memoryless — every remaining atom has exactly the same fixed probability of decaying in the next instant, regardless of how long it has already survived. That constant per-atom decay probability is what produces a smooth exponential curve at the scale of trillions of atoms, rather than some other shape, and it is why a single number, the half-life, fully characterizes the decay of any given isotope.
Half-lives span an enormous range depending on the isotope: polonium-214's is measured in microseconds, while uranium-238's stretches roughly to the age of the universe. Carbon-14, with a value close to 5,730 years, sits in a range useful for dating organic material from the last several tens of thousands of years — after about ten of these intervals, under a thousandth of the original carbon-14 remains, which is roughly the practical limit of radiocarbon dating.
- Enter your starting amount into Initial quantity (N0) — any unit works (grams, atoms, activity in becquerels) as long as you stay consistent.
- Enter the isotope's half-life into Half-life (same time unit as t) — for example, 5730 for carbon-14 measured in years.
- Enter how much time has passed into Elapsed time (same unit as half-life) — it must use the identical time unit as the half-life field.
- Read Remaining quantity, N(t) — the amount left after that much time has decayed away.
- To find how much has decayed rather than what remains, subtract N(t) from Initial quantity (N0) yourself; the instrument reports what's left, not what's gone.
Worked example — carbon-14 after one half-life
Enter 100 into Initial quantity (N0), 5730 into Half-life (same time unit as t), and 5730 into Elapsed time (same unit as half-life) — carbon-14's widely cited value, with exactly one full interval elapsed. Remaining quantity, N(t) reads 50.0000: the exponent t / half-life equals 5730/5730 = 1 exactly, so N(t) = 100 x (1/2)^1 = 50.
That answer holds by definition regardless of the exponential formula's fine print — one full half-life must always leave exactly half of whatever you started with. Push Elapsed time to 11460, two full intervals later, and Remaining quantity, N(t) drops to a quarter of the start, 25; a fractional elapsed time, like 2865 (a quarter of that same span), lands between 50 and 100 rather than on a clean fraction, since the decay is continuous, not stepwise.
Questions
What does it mean for a half-life to be a fixed fraction, not a fixed amount?
It means the same span of time always removes half of whatever remains, not a fixed amount of the original. Starting from 100, the first interval leaves 50, a drop of 50; the next leaves 25, a drop of only 25, even though the same amount of time passed both times. The quantity remaining keeps shrinking, so the amount lost each round keeps shrinking too — only the fraction removed, one half, stays constant.
Why is carbon-14's half-life sometimes given as 5,730 years and sometimes 5,568?
5,568 years is the 'Libby half-life,' the value Willard Libby's lab settled on shortly after he developed radiocarbon dating in the late 1940s. By the early 1960s, more precise measurements had converged on 5,730 years, the more accurate 'Cambridge half-life,' closer to today's best value near 5,700 years. But at the 1962 Radiocarbon Conference in Cambridge, scientists agreed to keep reporting ages on the older 5,568-year Libby scale for consistency with dates already published, rather than switch. That convention stuck: labs still quote raw ages on the Libby half-life today and correct for the gap with calibration curves, which is why both numbers show up in different sources.
Can I use any time unit — seconds, days, years?
Yes, as long as Half-life (same time unit as t) and Elapsed time (same unit as half-life) use the identical unit. The formula only ever evaluates the ratio t / half-life, so entering both in years, both in seconds, or both in any other matching unit gives the same result; mixing units, like a half-life in years against an elapsed time in days, silently produces a wrong answer.
How many half-lives until almost nothing is left?
By ten half-lives, (1/2)^10 = 1/1024, so under 0.1% of the original quantity remains — the conventional cutoff radiocarbon dating and radiation-safety planning both use as 'practically gone.' By twenty half-lives that fraction drops below one part in a million. The decay never mathematically reaches exactly zero, since it's a continuous exponential, but it becomes undetectable well before that.
Does this formula work for anything other than radioactivity?
Yes — any process with a constant proportional decay rate follows the same N(t) = N0 x (1/2)^(t/half-life) shape, including drug elimination from the bloodstream (pharmacokinetic half-life) and capacitor voltage discharge in an electrical circuit. The physics behind each process differs, but the mathematics of halving on a fixed schedule is identical, which is why 'half-life' shows up well outside nuclear chemistry.
Why does the instrument require the half-life to be greater than zero?
The half-life sits in the denominator of the exponent, t / half-life, so a value of zero would make that division undefined, and a negative half-life describes growth rather than decay, which this formula isn't built to represent. The instrument blanks the reading and states the problem on the formula line rather than returning a nonsensical figure.