Half-Life Explained: The Formula and Where It Applies

What half-life actually measures

Half-life is the time it takes for half of something to disappear, when the rate at which it disappears is proportional to how much of it is left. That second condition is the important one. Anything that decays in proportion to its current amount — a clump of radioactive atoms, a drug in your bloodstream, the charge on a capacitor leaking through a resistor, a population of excited molecules emitting photons — has a constant half-life. It does not matter whether you start with a tonne of the stuff or a microgram. Half of it will be gone after one half-life, three quarters after two, seven eighths after three. The half-life calculator exists because this single number — the half-life T — is enough to describe the entire trajectory of the decay, and most problems boil down to plugging three of the four numbers (initial amount, remaining amount, elapsed time, half-life) into the formula and solving for the fourth.

The word comes from nuclear physics, where Ernest Rutherford coined it around 1907 to describe how long it took half the atoms in a radium sample to break down. But the maths is much older and much wider than that — it is the closed-form solution of any first-order differential equation, and first-order processes are everywhere. If a quantity decays by a fixed fraction each interval rather than a fixed amount, it has a half-life. If it decays by a fixed amount each interval, it does not — that is linear decay, and the time to zero depends on where you start. Half-life is what you get when "rate" means "fraction of what is currently there".

The half-life equation, derived in one line

The starting point is the rate law: the amount that decays in a short interval is proportional to how much is currently present. Written as a differential equation, that is dN/dt = −λN, where N is the amount at time t and λ is the proportionality constant called the decay constant. The minus sign says the amount is going down, not up. Solving this equation — separate variables, integrate both sides — gives:

N(t) = N₀ × e^(−λt)

That is the natural-exponential form. It is equivalent, but a bit clunky for thinking in half-lives. Substitute λ = ln(2) / T and use the identity e^(−ln(2) × x) = (½)^x to rewrite it as:

N(t) = N₀ × (½)^(t / T)

Now the formula reads in plain English: after one half-life T, the amount is multiplied by ½; after two half-lives, by ¼; after n half-lives, by (½)ⁿ. The two forms — the natural exponential and the powers-of-a-half form — are mathematically identical, and the half-life calculator uses the second one internally because it generalises cleanly to fractional half-lives without losing precision.

Three small companions fall out of the same equation. The decay constant λ = ln(2) / T is the instantaneous fraction lost per unit time. The mean lifetime τ = 1 / λ = T / ln(2) is the average time a single particle survives before decaying — about 1.443 times the half-life, never equal to it. And the half-lives elapsed t / T is just the exponent in the formula, a unit-free count of how many halvings have happened. The calculator shows all three alongside the main answer, because they are usually more interesting than the bare number.

Worked example: carbon-14 dating, both directions

Carbon-14 has a half-life of 5,730 years (the Cambridge half-life, adopted by the National Institute of Standards and Technology as the standard for radiocarbon work). Start with a 100-gram sample. How much is left after 11,460 years?

Plug into the formula: N(t) = 100 × (½)^(11460 / 5730) = 100 × (½)² = 100 × 0.25 = 25 grams. The exponent t / T is 2, so two half-lives have elapsed, and 25 % of the original is still there. The decay constant is λ = ln(2) / 5730 ≈ 1.210 × 10⁻⁴ per year, meaning roughly 0.012 % of the sample decays per year on average. The mean lifetime is τ = 5730 / ln(2) ≈ 8,267 years — the expected lifespan of an individual C-14 atom before it converts to N-14.

Now turn the problem around. A bone fragment is found to contain 25 % of the C-14 it would have had when the animal died. How old is it? Switch the half-life calculator into "Elapsed time" mode and enter N₀ = 100, N(t) = 25, T = 5,730. Algebraically: t = T × log₂(N₀ / N(t)) = 5,730 × log₂(4) = 5,730 × 2 = 11,460 years. The same number, reached from the other end. This back-calculation is the engine of radiocarbon dating — though in practice it uses the C-14 / C-12 ratio rather than the absolute amount, and the result is calibrated against tree-ring chronologies because atmospheric C-14 has not been perfectly constant over time.

A third pass: suppose you measure both endpoints and want the half-life. A 50-gram sample of an unknown isotope drops to 6.25 grams over 30 days. Then T = t × ln(2) / ln(N₀ / N(t)) = 30 × ln(2) / ln(8) = 30 × 0.6931 / 2.0794 = 10 days. The ratio of 8 is three halvings (2 × 2 × 2), so 30 days divided into 3 half-lives gives 10 days each — and the algebra confirms it. The calculator's "Half-life T" mode does exactly this rearrangement.

Where half-life shows up

Radioactive decay

The original use. Unstable nuclei emit alpha, beta or gamma radiation and convert into a different element or isotope. Each individual nucleus decays at a random moment, but the population as a whole obeys the equation exactly because the number of nuclei is enormous (a gram of carbon-14 contains about 4 × 10²² atoms). Half-lives range across more than 50 orders of magnitude: polonium-214 has a half-life of 164 microseconds, uranium-238 has a half-life of 4.47 billion years, tellurium-128 has a half-life of 2.2 × 10²⁴ years. Anything with a half-life much longer than the age of the universe is, for practical purposes, stable.

Pharmacokinetics

Most drugs are eliminated by first-order kinetics over normal dosing ranges, which means each has an effective half-life in the body. Caffeine is roughly 5 hours, ibuprofen about 2 hours, fluoxetine 1 to 4 days for the parent compound, and amiodarone a notorious 25 to 100 days. The "five half-lives" rule of thumb — used by clinicians to estimate when more than 95 % of a dose has been cleared — falls directly out of the equation: after five half-lives, (½)⁵ = 1/32 ≈ 3.1 % remains. The maths is the same as in the radioactive case, but the constants are set by liver and kidney function rather than by nuclear physics. The calculator works for both; the only difference is which unit you choose for time.

RC circuits and capacitor discharge

A capacitor discharging through a resistor loses charge in proportion to the charge remaining, so its voltage follows V(t) = V₀ × e^(−t / RC). The time constant τ = RC is the mean lifetime, so the half-life of the voltage is T = RC × ln(2) ≈ 0.693 × RC. The capacitor energy calculator handles the stored energy side of the same physics. Electrical engineers often quote time constants instead of half-lives, but the two are linked by the same ln(2) factor that connects λ and T.

Chemistry and reaction kinetics

Any first-order chemical reaction — including many pseudo-first-order conditions where one reactant is in large excess — has a constant half-life that depends only on the rate constant k. The relationship is identical: T = ln(2) / k. The decomposition of hydrogen peroxide catalysed by an enzyme, the racemisation of amino acids in fossil teeth, and the photodissociation of ozone in the upper atmosphere all behave this way under the right conditions. Second-order and zero-order reactions do not have constant half-lives, which is one quick way to tell which regime you are in: measure the time for successive halvings, and if they are equal, the reaction is first-order.

How to think about successive half-lives

A practical trick: most people overestimate how long it takes for "almost all" of a substance to vanish. After ten half-lives, less than 0.1 % is left; after twenty, less than one part per million. The exponent calculator makes the maths obvious: (½)¹⁰ = 1/1024 ≈ 0.000977. A useful sanity check: log₂ of any ratio gives the number of half-lives. Reducing a quantity by a factor of 1,000 takes log₂(1000) ≈ 9.97 half-lives; reducing by a factor of a million takes about 19.93. The geometry of halving is universal even when the physics is not — caffeine, uranium and a labelled protein all follow the same curve, just at different time scales. The logarithm calculator handles the log₂ conversion if you want to do this kind of arithmetic on tap.

Common mistakes

Mixing units between t and T

The single most common error. The formula is dimensionless in the exponent — t / T must be unitless — which means the elapsed time and the half-life have to be in the same unit. If the half-life is in years and you enter the time in days, the calculator will return nonsense. Iodine-131 has a half-life of 8.02 days; ask how much remains after one year and you need to either convert the year to 365.25 days or the 8.02 days to 0.02195 years. The amounts N₀ and N(t) are also unit-flexible but must match each other — grams and grams, atoms and atoms, counts and counts.

Using half-life on a non-first-order process

If the decay rate does not scale with the current amount, the process does not have a constant half-life and this formula does not apply. Examples include alcohol metabolism above roughly 0.05 % BAC (mostly zero-order — the liver clears it at a fixed grams-per-hour rate independent of concentration), some chemical reactions at high concentration, and any process with feedback or threshold behaviour. Half-life is a property of first-order systems specifically.

Confusing half-life with mean lifetime

These are different numbers. Half-life T is the median lifetime of a particle — half of any large sample will have decayed by then. Mean lifetime τ is the arithmetic mean — the average time a particle survives. Because the exponential distribution is skewed (a few long-lived particles drag the average up), τ is always longer than T by a factor of 1 / ln(2) ≈ 1.443. Physicists often work with τ because it falls out of the differential equation cleanly; the public is more familiar with T because halving is easier to picture. The half-life calculator reports both.

Assuming the half-life of a parent decays the chain

If a radioactive isotope decays into a daughter that is itself radioactive, the daughter's half-life is independent of the parent's. Uranium-238 has a 4.47-billion-year half-life and decays through a chain of fourteen intermediate isotopes — including radon-222 (3.8 days), polonium-214 (164 microseconds) and lead-210 (22 years) — before reaching stable lead-206. The single-isotope formula does not capture chain dynamics; for those, you need the Bateman equations. The calculator on this page is for one decay step at a time.

When to seek expert input

The maths is exact for first-order processes; the real-world interpretation often is not. Medication dosing should follow prescribing information and clinical judgement, not a back-of-the-envelope sum from a published half-life — interpatient variability, active metabolites and organ function all shift the effective number. Radiation-safety calculations for occupational or environmental exposure belong to qualified health physicists and the relevant regulator (IAEA, EPA, UKHSA). Radiocarbon dates need calibration curves and laboratory standards beyond the bare formula. The calculator is a teaching and sanity-checking tool, not a substitute for domain expertise.

Frequently asked questions

What is the formula for half-life?

N(t) = N₀ × (½)^(t / T), where N₀ is the initial amount, N(t) is the amount remaining after time t, and T is the half-life. Equivalently, N(t) = N₀ × e^(−λt) with λ = ln(2) / T. The two forms are mathematically identical — the first is easier to reason about in half-lives, the second drops out of the differential equation dN/dt = −λN.

Is half-life the same for all amounts of a substance?

Yes. That is the defining feature of a first-order process. A tonne of cobalt-60 has the same half-life (5.27 years) as a microgram, because each individual nucleus decays independently with the same probability per unit time. The total number of decays per second is larger in the bigger sample, but the fraction decaying per second is the same.

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

The time constant τ (used in physics and engineering for things like RC circuits) is the mean lifetime, equal to 1 / λ. The half-life T is the median lifetime, equal to ln(2) / λ. They differ by a constant factor: T = τ × ln(2) ≈ 0.693 × τ. Both describe the same exponential decay; they just emphasise different points on the curve.

How many half-lives until "all gone"?

Strictly speaking, never — the formula approaches zero asymptotically but never reaches it. In practice, "five half-lives" is the conventional cutoff in pharmacology because (½)⁵ ≈ 3.1 % is below most thresholds of detection or clinical relevance. For "one part in a million" you need about 20 half-lives, and for one part per billion about 30.

Can the half-life of a substance change?

For pure radioactive decay, no, to an extraordinarily good approximation — the half-life is a property of the nucleus and is not affected by temperature, pressure, chemical environment or magnetic fields under normal conditions. The handful of known exceptions (some electron-capture decays show sub-percent shifts in extreme chemical or ionisation states) are research curiosities, not engineering concerns. For drug half-lives, RC half-lives and chemical-kinetic half-lives, the answer is yes — they depend on physiology, component values and temperature, and can drift with conditions.

How do I calculate when only the percentage remaining is given?

Set N₀ = 100 and N(t) = the percentage remaining (so 25 % becomes N(t) = 25), then enter the elapsed time and solve for T. Or vice-versa: enter T and solve for t. The formula only cares about the ratio N(t) / N₀, so any consistent units work — you do not need to know the absolute amounts.

Related calculators

For the inverse process — exponential growth rather than decay — the compound interest calculator runs the same maths with a positive rate constant. The exponent calculator handles the (½)^n step directly if you want to compute the fraction remaining for a particular number of half-lives. The logarithm calculator is what you need for the reverse direction — going from a ratio to a count of half-lives via log₂. And the pH calculator is another worked example of the log-scale thinking that runs through half-life problems.