Calculate radioactive decay, remaining quantity, elapsed time and decay constant instantly, with step-by-step solutions, formulas, worked examples and practical applications.
Use N = N₀ × (½)^(t / t½), where N₀ is the starting amount, t is the elapsed time and t½ is the half-life. The exponent t / t½ is simply the number of half-lives that have passed, so after 2 half-lives 25% remains, after 3 half-lives 12.5%, and after n half-lives (½)ⁿ remains. To work backwards for time, rearrange to t = t½ × log(N / N₀) ÷ log(½). The equivalent continuous form is N = N₀e^(−λt), where the decay constant λ = ln(2) / t½.
Choose what to solve for, then fill in the rest. Scientific notation works — enter 4.47e9 for 4.47 billion. The half-life and the elapsed time can use different units; the calculator converts for you.
| Isotope | Half-life | Main use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Radiocarbon dating |
| Uranium-238 (²³⁸U) | 4.468 × 10⁹ years | Geological dating, fuel source |
| Potassium-40 (⁴⁰K) | 1.248 × 10⁹ years | K–Ar rock dating |
| Plutonium-239 (²³⁹Pu) | 24,110 years | Nuclear fuel and waste management |
| Radium-226 (²²⁶Ra) | 1,600 years | Historic radiotherapy |
| Caesium-137 (¹³⁷Cs) | 30.05 years | Fallout monitoring, industrial gauges |
| Tritium (³H) | 12.32 years | Hydrology tracing, self-luminous signs |
| Cobalt-60 (⁶⁰Co) | 5.27 years | Sterilisation, radiotherapy |
| Iodine-131 (¹³¹I) | 8.02 days | Thyroid therapy and imaging |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Diagnostic imaging |
| Fluorine-18 (¹⁸F) | 109.77 minutes | PET scanning |
Values are the commonly cited figures used in teaching; published measurements vary slightly in the final digits.
| Metric | Value |
|---|
| Half-lives | Time | Remaining % | Remaining quantity |
|---|
Half-life is the time it takes for half of a quantity to disappear through decay. Start with 100 grams of a radioactive isotope with a half-life of 10 days, and after 10 days you have 50 grams; after 20 days, 25 grams; after 30 days, 12.5 grams. The amount removed shrinks every cycle, but the fraction removed never changes — that is the defining feature of exponential decay.
The most surprising property is that half-life does not depend on how much you started with. Whether the sample is a microgram or a tonne, half of it will be gone after exactly one half-life. It also does not depend on temperature, pressure, chemical form or age. A carbon-14 atom that has sat in a bone for 4,000 years is no more likely to decay in the next second than one created yesterday — radioactive decay has no memory.
That constancy is what makes half-life so useful. Because the rate is fixed and known, the ratio of what remains to what was there originally works as a clock. It is the same principle behind radiocarbon dating an artefact, deciding when a medical scan should be performed, and calculating how long nuclear waste must be isolated.
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing energy and particles as it goes. A nucleus becomes unstable when the balance of protons and neutrons sits outside the range that nuclear forces can hold together comfortably. There are three classic decay modes:
Individual decays are genuinely random: there is no way to predict when a particular nucleus will transform. What makes the process predictable is scale. In any real sample containing enormous numbers of atoms, the proportion decaying per unit time is reliably constant, and that statistical regularity produces the smooth exponential curve.
The exponent t / t½ is the number of half-lives that have elapsed, usually written as n. Because the base is ½, the formula just says "halve it n times". Two equivalent ways of writing the same relationship appear in textbooks: N = N₀ × 2^(−t/t½), and the continuous exponential form below.
Two conditions matter for the result to mean anything. First, t and t½ must be in the same unit before you divide — this calculator handles the conversion, but doing it by hand is where most errors creep in. Second, N and N₀ must be in the same unit as each other, though it can be any unit at all: grams, atoms, becquerels, millicuries or a percentage. Since only the ratio matters, the units cancel.
The continuous version of the same physics uses the natural exponential:
Both formulas describe the identical curve; they differ only in whether you count time in half-lives or use the instantaneous decay rate. The half-life form is easier for mental arithmetic and homework, while the exponential form is what appears in physics derivations and in the differential equation dN/dt = −λN, which says the decay rate at any moment is proportional to how much is left. Solving that equation is where the exponential comes from.
The decay constant λ (lambda) is the probability per unit time that any given nucleus will decay. A large λ means a fast decay and a short half-life; a small λ means the opposite. Its units are reciprocal time — per second, per day, per year — so it must always be quoted with the unit attached.
A related quantity is the mean lifetime τ = 1/λ = t½ / ln(2) ≈ 1.4427 × t½. This is the average survival time of a nucleus, and it is always longer than the half-life — because although half the sample is gone by t½, the survivors can persist for a very long time and pull the average up. For carbon-14 the mean lifetime is about 8,267 years against a half-life of 5,730.
The two are inversely related through the natural logarithm of 2, and converting between them is a one-line calculation. The relationship falls out of setting N = N₀/2 in the exponential formula: ½ = e^(−λt½), so ln(½) = −λt½, and therefore λt½ = ln(2).
| Quantity | Symbol | Formula | Meaning |
|---|---|---|---|
| Half-life | t½ | ln(2) / λ | Time for half the sample to decay |
| Decay constant | λ | ln(2) / t½ | Probability of decay per unit time |
| Mean lifetime | τ | 1 / λ = t½ / ln(2) | Average survival time of one nucleus |
| Activity | A | λN | Decays per second, measured in becquerels |
| Number of half-lives | n | t / t½ | How many halvings have occurred |
Activity deserves a note because it is what a Geiger counter actually measures. Since A = λN and N decays exponentially, activity decays with exactly the same half-life as the number of atoms. One becquerel is one decay per second; the older unit, the curie, equals 3.7 × 10¹⁰ Bq.
| Half-lives elapsed | Fraction remaining | Percent remaining | Percent decayed |
|---|---|---|---|
| 0 | 1 | 100% | 0% |
| 1 | 1/2 | 50% | 50% |
| 2 | 1/4 | 25% | 75% |
| 3 | 1/8 | 12.5% | 87.5% |
| 4 | 1/16 | 6.25% | 93.75% |
| 5 | 1/32 | 3.125% | 96.875% |
| 7 | 1/128 | 0.781% | 99.219% |
| 10 | 1/1024 | 0.098% | 99.902% |
| 20 | 1/1,048,576 | 0.000095% | >99.9999% |
The same mathematics applies to every isotope, but the half-life decides what each one is good for. These four cover the range from minutes to millennia.
| Isotope | Half-life | Main application | Why that half-life suits it |
|---|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Archaeology and radiocarbon dating | Long enough to date human history, short enough to still be measurable |
| Iodine-131 (¹³¹I) | 8.02 days | Thyroid treatment | Delivers a sustained therapeutic dose over days, then clears |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Medical imaging | A scan finishes within a shift and activity is gone within days |
| Fluorine-18 (¹⁸F) | 109.77 minutes | PET scanning | Minimal patient dose, but must be made near the scanner |
| Uranium-238 (²³⁸U) | 4.468 billion years | Geological dating | Matches the age of rocks and of the Earth itself |
| Caesium-137 (¹³⁷Cs) | 30.05 years | Fallout monitoring | Long enough to persist for generations after a release |
Each bar shows how much of the original sample survives after each successive half-life. The pattern is identical for every isotope — only the length of one step changes.
Notice that the bars never reach zero. Halving repeatedly gets arbitrarily close to nothing without ever arriving, which is why decay is described by a curve that approaches the axis rather than meeting it.
Radiocarbon dating is the best-known application of half-life. Cosmic rays continually produce carbon-14 in the upper atmosphere, and living organisms take it in along with ordinary carbon-12 through food and photosynthesis, holding a roughly constant ratio while alive. At death, intake stops and the carbon-14 begins decaying with its 5,730-year half-life while the carbon-12 stays put. Measuring the surviving ratio gives the elapsed time.
The technique is practical out to roughly 50,000 years — about nine half-lives, beyond which too little carbon-14 remains to measure reliably. Results are also calibrated against tree rings and other records, because atmospheric carbon-14 has not been perfectly constant: nuclear weapons testing in the mid-twentieth century raised it sharply, and fossil fuel burning dilutes it.
Nuclear medicine depends on choosing an isotope whose half-life matches the clinical task. It must last long enough to be produced, delivered and used, yet decay quickly enough that the patient's radiation dose stays low. Technetium-99m, at 6.01 hours, is the workhorse of diagnostic imaging for exactly that reason — a scan can be completed within a shift, and activity is negligible within a couple of days.
Therapeutic isotopes work differently. Iodine-131, with an 8.02-day half-life, is taken up by thyroid tissue and delivers a sustained dose over days, which is what makes it effective against thyroid cancer and hyperthyroidism. Australia's supply of medical radioisotopes comes largely from ANSTO's facility at Lucas Heights, which produces technetium-99m generators and other isotopes; because the material starts decaying the moment it is made, the logistics are built around half-life arithmetic.
PET scanning uses positron emitters with very short half-lives, most commonly fluorine-18 at 109.77 minutes. The tracer is usually attached to a glucose analogue that accumulates in metabolically active tissue, and the annihilation photons from positron decay are detected in pairs to reconstruct an image. Such a short half-life means the tracer must be produced in a cyclotron close to the scanner and used within hours — after two hours roughly 47% remains, and after a working day almost none.
Half-life also determines how long a patient remains mildly radioactive after a procedure, which is why discharge advice sometimes includes short-term limits on close contact with young children or pregnant women. Those intervals are set from the effective half-life, which combines physical decay with the body's own excretion of the substance and is therefore always shorter than the physical half-life alone.
Two isotopes with the same activity can present very different long-term problems, and half-life is what separates them. Short-lived material is intensely active but disappears quickly; long-lived material is weakly active but persists. Waste management strategies split along that line: short-lived hospital waste is simply stored until it has decayed, while long-lived material requires engineered disposal.
The rule of thumb used in laboratories is that after ten half-lives activity has fallen below 0.1% of its starting value, which for technetium-99m is under three days and for iodine-131 about eleven weeks. For caesium-137, at just over 30 years, ten half-lives is three centuries — which is why it dominated the long-term contamination picture after major reactor accidents.
Decay clocks are used throughout earth and environmental science. Potassium-40 dating (1.248 billion years) puts ages on volcanic rock; uranium-lead dating on zircon crystals reaches back billions of years and underpins the accepted age of the Earth. Tritium, at 12.32 years, traces how recently groundwater entered an aquifer — valuable in Australia, where knowing whether bore water is decades or millennia old determines how sustainably it can be drawn. Lead-210 dating of lake and estuary sediments reconstructs pollution and erosion histories over the past century.
Beyond nuclear physics, exponential decay with a characteristic half-life turns up wherever a quantity falls at a rate proportional to its own size. Excited atomic states, unstable particles in accelerator experiments, and the discharge of a capacitor through a resistor all follow the same mathematics — the RC circuit's time constant is a mean lifetime, and its half-life equivalent is RC × ln(2). Recognising the pattern means one formula covers many apparently unrelated systems.
In chemical kinetics, half-life behaves differently depending on reaction order, and this is a favourite exam point. For a first-order reaction the half-life is constant and equals ln(2)/k, exactly like radioactive decay. For a zero-order reaction it is [A]₀/2k, which shrinks as the reaction proceeds. For a second-order reaction it is 1/(k[A]₀), which grows as concentration falls. So a constant half-life measured across several cycles is itself evidence that a reaction is first-order.
| Reaction order | Half-life formula | Depends on starting concentration? |
|---|---|---|
| Zero-order | t½ = [A]₀ / 2k | Yes — halves each cycle |
| First-order | t½ = ln(2) / k | No — constant, like radioactive decay |
| Second-order | t½ = 1 / (k[A]₀) | Yes — doubles each cycle |
Pharmacokinetics uses the same first-order framework for drug elimination. A drug's half-life sets the dosing interval, and repeated dosing reaches a steady state after roughly four to five half-lives — the same arithmetic that says 93.75% to 96.875% of a single dose has cleared by then.
| Term | Definition |
|---|---|
| Isotope | A form of an element with the same number of protons but a different number of neutrons. Isotopes of one element behave identically in chemistry but can differ completely in nuclear stability. |
| Radioisotope | An isotope whose nucleus is unstable and therefore decays, emitting radiation as it transforms. |
| Activity | The number of nuclear decays occurring per unit time in a sample, calculated as A = λN. It falls with the same half-life as the number of atoms. |
| Becquerel (Bq) | The SI unit of activity: one decay per second. The older curie (Ci) equals 3.7 × 10¹⁰ Bq, and millicuries still appear in medical practice. |
| Alpha decay | Emission of a helium nucleus (2 protons, 2 neutrons), reducing the atomic number by 2. Heavily ionising but stopped by paper or skin. |
| Beta decay | Conversion of a neutron to a proton with emission of an electron, or the reverse with a positron. More penetrating than alpha but stopped by a few millimetres of aluminium. |
| Gamma radiation | High-energy photons released as a nucleus sheds excess energy without changing composition. Highly penetrating and requires dense shielding such as lead. |
| Decay constant (λ) | The probability per unit time that a given nucleus decays, equal to ln(2) ÷ t½. Units are reciprocal time. |
| Mean lifetime (τ) | The average survival time of a nucleus, equal to 1/λ or about 1.4427 × t½. Always longer than the half-life. |
| Effective half-life | The combined effect of physical decay and biological excretion in a living body. Always shorter than the physical half-life alone. |
| Exponential decay | Any process losing a constant fraction of itself per unit time, described by N = N₀e^(−λt). |
| Parent and daughter | The original decaying nuclide is the parent; the nuclide it becomes is the daughter. Daughters are often radioactive themselves, forming a decay chain. |
| Mistake | Why it goes wrong | How to avoid it |
|---|---|---|
| Mixing time units | Dividing hours by days gives a meaningless exponent | Convert t and t½ to the same unit first, or let the calculator do it |
| Multiplying instead of exponentiating | Decay is repeated halving, not a fixed amount removed per period | Use (½)^n, never N₀ − n × something |
| Thinking two half-lives means nothing is left | Two halvings leave a quarter, not zero | Track the sequence: 50%, 25%, 12.5%, 6.25% |
| Using log base 10 and ln inconsistently | Mixing bases within one calculation corrupts the result | Either base works for a ratio, but be consistent; λ specifically needs ln(2) |
| Confusing λ with t½ | They are reciprocally related, not equal | λ = ln(2)/t½, and λ carries units of 1/time |
| Assuming a longer half-life is more hazardous | Longer half-life means lower activity for the same quantity | Judge risk by activity, decay type and exposure route |
| Expecting half-life to change with conditions | Radioactive decay is independent of temperature, pressure and chemistry | Treat t½ as a fixed property of the isotope |
| Applying constant half-life to any reaction | Only first-order kinetics has a concentration-independent half-life | Check the reaction order before assuming |
| Rounding the number of half-lives early | The exponent amplifies small rounding errors | Keep full precision until the final answer |
N = N₀ × (½)^(t/t½)N = N₀ × e^(−λt)n = t / t½(½)^nt = t½ × log(N/N₀) / log(½)t½ = t × log(½) / log(N/N₀)N₀ = N / (½)^(t/t½)λ = ln(2) / t½ ≈ 0.693147 / t½τ = 1/λ ≈ 1.4427 × t½A = λN (becquerels = decays per second)t½ = ln(2) / k1 → 50% 2 → 25% 3 → 12.5% 10 → 0.098%
1) (½)³ = 12.5% 2) n = 3, so 200 × 0.125 = 25 g 3) 4 half-lives (½⁴ = 6.25%) 4) n = 2, so 40 × 0.25 = 10 mCi 5) 32
1) λ = 0.693147 ÷ 5,730 = 1.2097 × 10⁻⁴ per year 2) n = log(0.12) ÷ log(0.5) = 3.0589, so t½ = 30 ÷ 3.0589 ≈ 9.81 minutes 3) n = 24 ÷ 6.01 = 3.9933, so (½)^3.9933 ≈ 0.0628, about 6.28% 4) n = log(0.225) ÷ log(0.5) = 2.152, so age ≈ 2.152 × 5,730 ≈ 12,331 years 5) τ = 5,730 ÷ ln 2 ≈ 8,267 years; it is longer because the surviving nuclei can persist indefinitely, and those long survival times pull the average above the halfway point
What is half-life?
Half-life is the time required for half of a quantity to decay. After one half-life 50% remains, after two 25%, after three 12.5%, and after n half-lives the fraction remaining is (½)ⁿ. It is a fixed property of each radioactive isotope and does not depend on how much you started with, so half of a microgram sample and half of a tonne both disappear in the same time.
How do you calculate half-life?
To find the remaining quantity, use N = N₀ × (½)^(t/t½): divide the elapsed time by the half-life to get the number of half-lives, raise ½ to that power, and multiply by the starting amount. To find the half-life itself from measured data, use t½ = t × log(½) ÷ log(N/N₀). For example, a sample falling from 80 g to 10 g in 6 days has lost 3 half-lives, so t½ = 6 ÷ 3 = 2 days.
What is the half-life formula?
The standard form is N = N₀ × (½)^(t/t½), where N is the quantity remaining, N₀ the initial quantity, t the elapsed time and t½ the half-life. It can also be written N = N₀ × 2^(−t/t½), or in continuous form as N = N₀e^(−λt) with the decay constant λ = ln(2)/t½. All three describe the same curve.
What is radioactive decay?
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, releasing energy and particles. The three main modes are alpha decay (emitting a helium nucleus), beta decay (a neutron and proton converting with emission of an electron or positron) and gamma emission (releasing energy as a photon without changing composition). Individual decays are random, but in a sample of many atoms the proportion decaying per unit time is constant, producing an exponential curve.
What is the decay constant?
The decay constant λ is the probability per unit time that any given nucleus will decay, calculated as λ = ln(2) ÷ t½ ≈ 0.693147 ÷ t½. Its units are reciprocal time, so it must always be quoted with a unit — carbon-14's decay constant is about 1.2097 × 10⁻⁴ per year. A large λ means fast decay and a short half-life.
How do you calculate remaining mass after decay?
Divide the elapsed time by the half-life to get the number of half-lives n, then multiply the starting mass by (½)ⁿ. A 500 mg dose with a 6-hour half-life measured 24 hours later has passed 4 half-lives, so 500 × (½)⁴ = 31.25 mg remains. The units of mass do not matter as long as both quantities use the same one, because only the ratio enters the formula.
Why is half-life important?
Because the decay rate is fixed and known, half-life works as a clock and as a planning tool. It dates archaeological and geological samples, sets the timing of medical imaging and therapy, determines how long radioactive waste must be isolated, fixes drug dosing intervals, and tells safety officers when short-lived material has decayed to a negligible level.
What is the half-life of carbon-14?
Carbon-14 has a half-life of about 5,730 years. Living organisms maintain a roughly constant ratio of carbon-14 to stable carbon-12; when they die, intake stops and the carbon-14 decays while the carbon-12 remains. Measuring the surviving ratio dates the sample, reliably out to roughly 50,000 years — about nine half-lives, beyond which too little carbon-14 remains to measure accurately.
How is half-life used in medicine?
The isotope is chosen so its half-life matches the task. Technetium-99m, at 6.01 hours, is ideal for diagnostic imaging because a scan finishes within a shift and activity falls to negligible levels within a couple of days. Fluorine-18, at 109.77 minutes, is used for PET scans and must be produced close to the scanner. Iodine-131, at 8.02 days, delivers a sustained therapeutic dose to thyroid tissue. Drug half-lives set dosing intervals, with steady state reached after about four to five half-lives.
What is exponential decay?
Exponential decay describes any quantity that decreases at a rate proportional to its current size, so it loses the same fraction — not the same amount — in each equal time interval. Mathematically it satisfies dN/dt = −λN, whose solution is N = N₀e^(−λt). Radioactive decay is the classic example, but capacitor discharge, drug elimination and first-order chemical reactions follow the same curve.
Can half-life change?
For practical purposes no. Radioactive half-life is a fixed property of each isotope and is unaffected by temperature, pressure, chemical form, magnetic fields or how long the sample has already existed. A few exotic exceptions exist in nuclear physics — electron capture rates can shift very slightly when the electron environment changes, as in fully ionised atoms — but these are laboratory curiosities and never relevant to normal calculations. Note that the effective half-life of a substance in the body is shorter than its physical half-life, because excretion removes material as well.
How do scientists measure half-life?
For short half-lives, they measure the activity of a sample at intervals with a detector and fit the exponential curve, or plot the logarithm of activity against time and take the slope, which equals −λ. For very long half-lives such as uranium-238's 4.468 billion years, waiting is impossible, so they instead count the absolute number of atoms present and the number of decays per second, then use A = λN to solve for λ and convert to a half-life.
What is the relationship between half-life and the decay constant?
They are inversely related through the natural logarithm of 2: λ = ln(2)/t½ and t½ = ln(2)/λ, with the product λ × t½ always equal to about 0.693147. The relationship comes from setting N = N₀/2 in N = N₀e^(−λt), which gives ln(½) = −λt½. A third related quantity is the mean lifetime τ = 1/λ, which equals t½ ÷ ln(2) ≈ 1.4427 × t½.
Why does radioactive decay follow an exponential curve?
Because each nucleus has the same constant probability of decaying in any given moment, independently of the others and of how long it has already existed. That means the number of decays per unit time is proportional to the number of nuclei still present, which is the differential equation dN/dt = −λN. Its solution is an exponential, so the fraction remaining halves over every equal interval rather than the amount falling by a fixed step.
What are common examples of half-life?
Carbon-14 at 5,730 years for dating organic material; uranium-238 at 4.468 billion years and potassium-40 at 1.248 billion years for geological dating; caesium-137 at 30.05 years in fallout monitoring; cobalt-60 at 5.27 years for sterilisation; iodine-131 at 8.02 days for thyroid treatment; technetium-99m at 6.01 hours for imaging; and fluorine-18 at 109.77 minutes for PET scans. Outside nuclear science, drug half-lives range from minutes to days.
Does a longer half-life mean more dangerous?
No — it means slower decay. For the same number of atoms, a longer half-life gives fewer decays per second, so lower activity. The trade-off is duration: short-lived isotopes are intensely active but vanish quickly, while long-lived ones are weakly active but persist for centuries. Actual risk depends on activity, the type and energy of radiation emitted, and whether the material can be inhaled or ingested.
Is half-life the same in chemistry as in radioactive decay?
Only for first-order reactions, where t½ = ln(2)/k and the half-life is constant regardless of starting concentration, exactly as in radioactive decay. A zero-order reaction has t½ = [A]₀/2k, which shrinks as the reaction proceeds, and a second-order reaction has t½ = 1/(k[A]₀), which grows. A half-life measured as constant across several cycles is itself evidence that a reaction is first-order.
Does this half-life calculator show the steps?
Yes. Choose whether to solve for the remaining quantity, elapsed time, half-life or initial quantity, and the calculator shows the substituted formula plus each step — the number of half-lives, the fraction remaining, the result, the decay constant and the mean lifetime. It also plots the decay curve, lists a decay table for the first ten half-lives, accepts scientific notation such as 4.47e9, allows different units for the half-life and the elapsed time, includes fifteen isotope presets, and copies results to the clipboard.