Half-Life Calculator

N ＝ N₀ × (½)^(t ÷ T) n ＝ t ÷ T 剩余比例 ＝ N ÷ N₀

N₀

The starting quantity, whatever was there before any of it decayed. It has no unit on this page: grams, becquerels, atoms and concentrations are all the same input, because the decay law only ever asks what fraction is left. It must be greater than zero — a starting quantity of zero makes the remaining share a division of zero by zero rather than a small number

T

The half-life, the time it takes for half of whatever is present to go. It is the property of the substance rather than of the sample, so it does not change when the starting quantity does. It must be entered in the same unit as the elapsed time; the page does not ask which unit that is, because the two times divide into each other and the ratio is unchanged by converting both

t

The elapsed time, the interval you are asking about. It is measured in the same unit as the half-life and it cannot be negative — the formula would happily run backwards, but 'how much was there before' is a different question from the one this panel is set up to answer

n ＝ t ÷ T

The number of half-lives elapsed, the second reading and the one that makes the rest checkable. Half-lives are counted, not decayed in whole steps: 2.5 is a perfectly ordinary value and means the sample is between two and three halvings along. It is printed to four decimals so that 2.9999 and 3 stay distinguishable, since those two mean 'not quite three half-lives' and 'exactly three'

(½)^n

The remaining share, one half multiplied by itself once per half-life. It is the whole of the decay law: no constant, no base other than one half, and no dependence on the substance beyond the half-life. Ten half-lives leave less than a tenth of a percent, and the share never reaches exactly zero no matter how many half-lives pass, though it does eventually underflow to zero in double precision

N ＝ N₀ × (½)^n

The remaining quantity, the primary reading. It is the starting quantity scaled by that share, so it is a proportional answer rather than an absolute one: doubling the starting quantity doubles what is left at every point along the curve, and the percentage is untouched

N ÷ N₀

The remaining share as a percentage, printed with the percent sign already attached. It is the reading to quote when the substance is unknown — 'about 25 percent is left' says everything that '2.5 becquerels' says once the starting activity is on the table, and it is the only one of the three that survives changing the starting quantity

Use it when something is decaying at a rate expressed as a half-life, which is how the decay of anything unstable is written down: a radioisotope, a drug clearing from the body, a pollutant breaking down, a population of things dying off in fixed proportion. It is also the page for the two questions that come up around carbon dating — how much of the original carbon-14 is left after a given number of years, and how many half-lives a measured share corresponds to, the second being the step that turns into an age. Reach for the exponential growth calculator when the base is not one half but is derived from a rate, which is the same law with a different constant. Reach for the doubling time calculator when the question is going the other way and the quantity is growing, since doubling is halving read in the other direction. Reach for the compound interest calculator when the same exponent is being applied to money, where the multiplication happens on a schedule rather than continuously.

1. 100 units with a half-life of 5, after 15

   1. Half-lives elapsed: 15 ÷ 5 = 3
   2. Remaining share: (½)³ = 1 ÷ 8 = 0.125
   3. Remaining quantity: 100 × 0.125 = 12.5
   4. As a percentage: 12.5%

   The row the reference table also carries, and the easiest one to check without the page: three halvings of 100 are 50, then 25, then 12.5. That halving-in-steps reading is exactly what the count of half-lives is for, and it is why the page prints the count next to the amount — the amount alone tells you nothing about how far along the curve you are.

2. Carbon-14 with a half-life of 5,730 years, after 11,460

   1. Half-lives elapsed: 11,460 ÷ 5,730 = 2
   2. Remaining share: (½)² = 0.25
   3. Remaining quantity: 1 × 0.25 = 0.25
   4. As a percentage: 25%

   The case the page was designed around, and the reason the two time fields have no unit selector: the units here are years, and the half-life is far outside anything a dropdown of hours and weeks could hold. The starting quantity is 1 rather than a real measurement, so the answer reads directly as a fraction — after two half-lives, a quarter of the original carbon-14 is left, which is the first half of the reasoning in every radiocarbon date.

3. 200 units with a half-life of 4, after 10

   1. Half-lives elapsed: 10 ÷ 4 = 2.5
   2. Remaining share: (½)^2.5 = 1 ÷ √32 = 0.176776695…
   3. Remaining quantity: 200 × 0.176776695… = 35.355339…
   4. As a percentage: 17.67767%

   The ordinary case, and the one that shows half-lives are not counted in whole steps: two and a half of them have gone by, so the sample sits between a quarter and an eighth of where it started. The share is an irrational number here, which is why the answer carries decimals — it would only be exact if the elapsed time were a whole number of half-lives. Reading 2.5 as the count is doing real work: multiply 12.5 percent by the square root of one half and you have the same 17.68 percent.

4. 80 units with a half-life of 12, after exactly 12

   1. Half-lives elapsed: 12 ÷ 12 = 1
   2. Remaining share: (½)¹ = 0.5
   3. Remaining quantity: 80 × 0.5 = 40
   4. As a percentage: 50%

   One half-life, which is the definition of the quantity being checked rather than a case of it. This is the row to use when a half-life has been quoted and you want to confirm it means what you think it means: whatever you start with, half of it is gone after that long, and nothing about the starting quantity changes the answer.

This page answers one direction only: given a starting quantity, a half-life and an interval, how much is left. The reverse questions — how long it takes to reach a given amount, and what the half-life is given a starting and an ending quantity — are not fields on this panel, though both follow from the same equation and both are worked through in the formula block and the questions below. There is no calculator page for either of them in this batch, so a reader who needs one should rearrange the exponent rather than expect a second set of inputs. The two time fields share a unit and the page never asks which one: entering a half-life in days and an elapsed time in hours gives an answer that is wrong by a factor of twenty-four, and it comes back looking entirely normal, which is the one mistake on this page that cannot be caught from the inside. The starting quantity is unlabelled, so the page cannot convert between units of mass or activity — grams in, grams out. There are no grades: 'how much is left' is a continuous reading and the page prints it rather than judging it. The share never reaches zero, but it does underflow: past roughly a thousand half-lives the floating-point result is exactly zero, and that is a limit of the arithmetic rather than a statement about the substance. Finally, this is a model of decay by whole halvings and it assumes the rate is constant, which is what a half-life means and is not true of everything that fades.

How do I work out how much is left after a given time?

Divide the elapsed time by the half-life to get the number of half-lives, then halve the starting quantity that many times. With a half-life of 5 and an elapsed time of 15, three half-lives have passed, so the remaining share is 1 ÷ 8 and a starting quantity of 100 leaves 12.5. The page does the same thing in one expression: the starting quantity times one half raised to the power of elapsed time over half-life.

How long until only a certain amount is left?

Turn the equation around. Divide the amount you want to reach by the starting quantity to get the share, take the logarithm of that share to the base one half, and multiply by the half-life. Wanting an eighth of the original left means three half-lives, whatever the substance; wanting a quarter of it left means two, so a 5,730-year half-life puts that answer at 11,460 years. The count of half-lives is the part worth doing first, because the rest is one multiplication.

What unit should the half-life and the elapsed time be in?

Any unit, as long as both are in the same one. The two times divide into each other, so a half-life of 5 days with an elapsed time of 15 days gives exactly the same answer as 120 hours with 360 hours. The page deliberately has no unit selector, because the units people actually use span from seconds to billions of years and no list would hold them all. The cost is that mixing units is not detected: days against hours gives an answer twenty-four times too large, and it looks completely normal.

Does the answer ever reach zero?

Not as a matter of mathematics — every half-life removes half of what is there, and half of a positive number is always positive, so the remaining share approaches zero without arriving. In practice it does arrive: a computer stores a number with finite precision, and past roughly a thousand half-lives the result underflows and comes back as exactly zero. That is the arithmetic running out rather than the substance disappearing, and it happens far beyond any range where the model is being used seriously.

What is the difference between half-life and the decay constant?

They describe the same decay in two ways. The decay constant is the rate per unit time in the exponent, and the half-life is the time for half of it to go; the two are related by a factor of the natural logarithm of two, with the half-life equal to that logarithm divided by the decay constant. This page uses the half-life because that is how the numbers are published — 5,730 years for carbon-14, a few hours for a drug — while the decay constant is what appears if you write the same law using the number e as the base.

Does the starting quantity change how long the decay takes?

No. That is the whole point of a half-life: it is a property of the substance, not of the sample. Starting with 200 units rather than 100 means there is 200 times two to the minus n left instead of 100 times two to the minus n, so every quantity along the curve is doubled and the percentages are identical. The count of half-lives and the remaining share do not move at all when you change the starting quantity.

• Half-Life — the definition of the half-life and the exponential decay law that this page evaluates, including the relation between the number of half-lives and the fraction remaining — Wolfram MathWorld (United States)
• Exponential Decay — the general law of which this page is the half-life case, with the decay constant and the half-life related by a factor of the natural logarithm of two — Wolfram MathWorld (United States)
• Radiocarbon dating — how the remaining fraction of carbon-14 in a sample is turned into an age, which is the step that follows the number this page prints — Wikipedia