Half-Life Calculator

Half-life is the time it takes a quantity to fall to half its starting value. The term comes up most often with atoms undergoing radioactive decay, but it describes any decay process that removes a fixed proportion per unit of time rather than a fixed amount.

Carbon-14 dating is the best known application. The half-life of carbon-14 is about 5,730 years, which makes it reliable for dates back to roughly 50,000 years. The method was developed by William Libby and rests on the fact that carbon-14 is constantly being made in the atmosphere. It enters plants through photosynthesis and then animals when they eat the plants. Once the plant or animal dies, no new carbon-14 comes in and what is there decays, so measuring how much is left says when the organism died.

Definition and formula

Three equivalent formulas describe exponential decay:

$$\left(1\right) N\left(t\right)=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$$ $$\left(2\right) N\left(t\right)=N_0{e}^{-\frac{t}{\tau }}$$ $$\left(3\right) N\left(t\right)=N_0{e}^{-\lambda t}$$

where:

• N₀ is the initial quantity
• N_t is the quantity remaining after time t
• t_1/2 is the half-life
• τ is the mean lifetime
• λ is the decay constant

They describe the same curve using three different constants, which is why the second calculator on this page converts freely between them. Which one you reach for is a matter of what you already know.

A worked example

Suppose an archaeologist finds a fossil holding 25% as much carbon-14 as a living sample. The age follows from rearranging equation 1, since N_t, N₀ and t_1/2 are all known:

$$t=\frac{t_{1/2}·\ln \left(\frac{N_t}{N_0}\right)}{-\ln \left(2\right)}$$ $$t=\frac{5730·\ln \left(\frac{25}{100}\right)}{-0.693}=11460$$

The fossil is 11,460 years old. That one is easy to sanity-check: 25% is exactly two halvings, and two carbon-14 half-lives is 2 × 5,730 = 11,460 years. Most real measurements do not land on a whole number of half-lives, which is what the calculator above is for.

The relationship between the three constants

The three forms can be tied together, which is what lets any one of t_1/2, τ and λ produce the other two.

(1) Set the three decay factors equal, since all three describe the same curve:

$${\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}=e^{-\frac{t}{\tau }}=e^{-\lambda t}$$

(2) Take the natural log of each part:

$$\frac{t}{t_{1/2}}·\ln \left(\frac{1}{2}\right)=-\frac{t}{\tau }=-\lambda t$$

(3) Since ln(1/2) = −ln 2, the left side becomes −t·ln2 / t_1/2. Divide every part by −t:

$$\frac{\ln \left(2\right)}{t_{1/2}}=\frac{1}{\tau }=\lambda$$

(4) Which rearranges to the relationship the second calculator uses:

$$t_{1/2}=\tau ·\ln \left(2\right)=\frac{\ln \left(2\right)}{\lambda }$$

(5) And the decay constant is simply the reciprocal of the mean lifetime, λ = 1/τ.

What the mean lifetime actually means

The half-life is the point where half the sample is gone. The mean lifetime is the average time an individual atom survives, and it is always longer, by a factor of 1 / ln 2 = 1.442695. For carbon-14, a half-life of 5,730 years gives a mean lifetime of about 8,267 years.

The reason it is longer is that the distribution has a long tail. Half the atoms are gone within one half-life, but the survivors keep going, and a few last many times longer than average. Averaging over all of them pulls the figure past the halfway point.

At exactly one mean lifetime, 1/e of the sample is left, which is 36.79% rather than 50%. That 1/e point is where the exponential form of the equation is at its most natural, which is why physics tends to work in τ and λ while chemistry and medicine tend to work in half-lives.

Why decay never quite finishes

Each half-life removes half of what is left, not half of what you started with, so the amount remaining falls by a constant proportion rather than a constant amount.

Half-lives elapsed	Fraction left
1	50%
2	25%
3	12.5%
4	6.25%
5	3.125%
10	0.0977%

After ten half-lives less than a tenth of one percent remains, which is why ten half-lives is a common rule of thumb for treating a sample as spent. Mathematically it never reaches zero, though in practice a sample eventually contains a countable number of atoms and the last one decays at some particular moment.

This is also why the phrase "half-life" tells you nothing on its own about how dangerous something is. A short half-life means intense activity over a short window; a long one means weak activity spread over ages. Technetium-99m, at 6.01 hours, is short enough that a patient scanned in the morning has under 7% of it left the next day.

Half-lives worth knowing

Isotope	Half-life	Used for
Technetium-99m	6.01 hours	medical imaging
Radon-222	3.82 days	indoor air testing
Iodine-131	8.02 days	thyroid treatment
Cobalt-60	5.27 years	sterilisation, radiotherapy
Caesium-137	30.08 years	fallout monitoring
Carbon-14	5,730 years	radiocarbon dating
Uranium-235	703.8 million years	reactor fuel, dating
Potassium-40	1.248 billion years	potassium-argon dating
Uranium-238	4.468 billion years	dating the oldest rocks

The spread here is what makes the technique so useful. Carbon-14 covers human history and a little beyond; potassium-40 and uranium-238 reach back to the formation of the Earth. Picking the wrong isotope for a job gives useless answers, since a sample much older than about ten half-lives has nothing measurable left, and one much younger has barely changed.

Beyond radioactivity

Any process that removes a fixed proportion per unit time has a half-life. In pharmacology the biological half-life describes how long a drug takes to fall to half its concentration in the bloodstream. Caffeine sits at roughly five hours in a healthy adult, which is why an afternoon coffee still has a quarter of its caffeine working at bedtime.

Drug dosing intervals are built around this. A medicine with a short half-life needs frequent doses to stay in its effective range; a long one can be taken once a day, and takes correspondingly longer to clear after the last dose.

The same maths turns up in capacitor discharge, in the cooling of an object toward room temperature, and in any decline where the rate depends on how much is currently there. Where the calculator is used, the units simply have to be consistent: time and half-life in the same unit, the two quantities in the same unit as each other.

Which box to leave empty

The first calculator takes any three of the four values and returns the fourth, so the question is which one you are missing.

Leave half-life empty when you have measured a decline and want to characterise the substance. This is the laboratory case: a known starting amount, a known amount now, and a known gap between the two readings.

Leave time empty when the substance is known and you want a date or an age. Carbon dating works this way, and so does asking how long a stored sample has left before it drops below a usable level.

Leave quantity remains empty to forecast: given what you have now and how long you plan to wait, how much will be there at the end. That is the everyday case for scheduling doses or planning when a source needs replacing.

Leave initial quantity empty to work backwards from a current reading to what must have been there originally, which is the reconstruction case in contamination and forensic work.

Whichever is missing, the other three fix the curve completely, because two points and a decay law leave nothing free.

Common mistakes

The first is treating decay as linear. Two half-lives leave 25%, not 0%. Halving twice is not the same as subtracting half twice.

The second is mixing units. If the half-life is in days, the elapsed time has to be in days too. The quantities are more forgiving, since only their ratio matters: grams against grams, counts against counts, or percentages against percentages all work equally well, and entering 25 and 100 gives the same answer as entering 0.25 and 1.

The third is confusing the mean lifetime with the half-life. They differ by 44%, so using one where the other belongs is a large error, not a rounding difference.