Half-Life Calculator

What the Calculator Solves

Four quantities describe a decaying sample: how much you started with, how much remains, how long has passed, and how long the half-life is. Supply any three and the fourth follows from a single equation. The calculator also converts between half-life, decay constant, and mean lifetime, which are three ways of writing the same physical property.

The same mathematics applies far beyond radioactivity. A drug clearing from plasma, a pollutant breaking down in soil, a capacitor discharging, and a radioactive tracer in a scanner all follow a curve that halves in equal intervals. That shared shape is why a tool built for nuclear physics is useful in pharmacology and environmental science.

What the calculator cannot do is decide which model fits your data. A measured curve that halves once and then flattens is not a single exponential; it is at least two components with different half-lives. Fitting a straight line through it on a log plot will look convincing and be wrong. Treat the output as arithmetic, and treat the choice of model as the scientific question.

Definition and the Halving Formula

Half-life is the time required for half of the atoms present to decay, or more generally for a quantity to fall to half its current value. The definition is recursive: after one half-life, half remains; after the next, half of that half remains. The fraction left after n half-lives is (1/2)ⁿ, so 50%, 25%, 12.5%, 6.25%, and so on.

The compact form is N_t = N₀ × (1/2)^{t / t_1/2}. Here N₀ is the starting amount, N_t the amount remaining after elapsed time t, and t_1/2 the half-life. The exponent is dimensionless because time is divided by time, which is a useful check when you are unsure whether a unit conversion is needed.

The concept emerged from the study of radioactivity in the early twentieth century, when Ernest Rutherford and Frederick Soddy recognized that decay rates are proportional to the number of atoms present rather than to elapsed time. That insight turned a collection of observed decay curves into a predictive law, and it is the reason a half-life is a constant for a given isotope rather than a property that changes with temperature or chemical form.

Three Equivalent Forms of the Decay Law

The same curve can be written with a base of one-half, a base of e, or a mean lifetime: N_t = N₀(1/2)^t/t_1/2 = N₀e^−λt = N₀e^−t/τ. The first is the most intuitive for quick estimates, the second is the most convenient for calculus, and the third appears in particle physics and in some medical literature.

The exponential form comes from the differential equation dN/dt = −λN, which says the rate of loss is proportional to the amount present. Solving it gives a pure exponential, and the half-life is the time at which the exponent equals −ln 2. That is where the 0.693 appears in every conversion: it is not a magic number, it is the natural logarithm of two.

Because the decay constant has units of inverse time, the product λt is dimensionless, just like t/t_1/2. This is worth checking before you trust a result. If you enter a half-life of 5,730 years and an elapsed time of 12,000 days, the ratio is meaningless until both are in the same unit; the calculator will not warn you, and the answer will look perfectly plausible.

Half-Life, Decay Constant, and Mean Lifetime

Each of these three constants determines the other two. The decay constant is the probability per unit time that a given atom will decay, so a larger λ means a faster process and a shorter half-life. The mean lifetime is the average time an atom survives, which is longer than the half-life because a small fraction of atoms persist for many half-lives and pull the average up.

The conversions are λ = ln 2 / t_1/2, τ = 1/λ, and t_1/2 = τ ln 2 ≈ 0.693τ. The factor 1.4427 in the other direction is simply 1/ln 2. A useful sanity check is that the mean lifetime is always about 44% longer than the half-life, never shorter, because the exponential tail is long.

Fields differ in which constant they favor. Nuclear physicists often quote λ, radiochemists quote half-life, and particle physicists quote lifetime because the exponential distribution falls out of the theory naturally. When comparing published values, convert to a common representation before drawing conclusions; a factor-of-1.44 difference between two sources is usually a units-or-definition mismatch rather than a discovery.

Carbon-14 Dating

Carbon-14 is produced in the upper atmosphere when cosmic-ray neutrons collide with nitrogen-14. Living plants take it up through photosynthesis, animals take it up by eating plants, and the ratio of carbon-14 to carbon-12 in a living organism stays close to the atmospheric ratio. When the organism dies, uptake stops and the carbon-14 already present decays with a half-life of about 5,730 years.

A sample with 25% of the modern ratio has been through two half-lives, so its age is roughly 11,460 years. A sample with 12.5% has been through three, or about 17,190 years. The arithmetic is simple; the calibration is not. Atmospheric carbon-14 has varied over time, so radiocarbon ages are converted to calendar ages using calibration curves built from tree rings, corals, and other independently dated archives.

Willard Libby developed the method in the late 1940s and received the Nobel Prize in Chemistry in 1960. Practical limits sit near 50,000 years, where the remaining carbon-14 is a tiny fraction of the original and contamination by modern carbon becomes dominant. A fingerprint, a preservative, or a fragment of root can shift a result by centuries, which is why sample pretreatment is as important as the counting statistics.

For a worked check, use the calculator with an initial amount of 100, a remaining amount of 25, and a half-life of 5,730 years. The elapsed time should come out at 11,460 years. If it does not, the units or the input fields were mismatched, not the physics.

Half-Life in Nuclear Medicine

Medical imaging depends on isotopes that are active long enough to reach the scanner and short enough to leave the patient quickly. Technetium-99m, with a half-life of about six hours, is the workhorse of single-photon emission imaging; it is produced on site from a molybdenum-99 generator and used for bone, cardiac, thyroid, and renal scans. Fluorine-18, at about 110 minutes, is the standard positron emitter for PET imaging with fluorodeoxyglucose.

Therapy uses longer-lived isotopes. Iodine-131, with a half-life of about eight days, concentrates in thyroid tissue and is used to treat hyperthyroidism and differentiated thyroid cancer. Lutetium-177, at roughly 6.6 days, is used in targeted radionuclide therapy, where a peptide or antibody carries the isotope to a tumor. The choice of half-life is matched to the biology: long enough for uptake, short enough to limit dose to the rest of the body.

Dose calculations are a chain of decay corrections. A vial calibrated at 8:00 a.m. contains less activity by the time a patient is injected at 10:00 a.m., and the difference is computed with the same exponential law used for any other sample. Residual activity in syringes, tubing, and shielding is tracked the same way, which is why nuclear medicine departments keep decay tables and dose calibrators rather than relying on arithmetic alone.

Drug Half-Life and Steady State

Plasma half-life is the time for the concentration of a drug in blood to fall by half after absorption and distribution are complete. It is a summary of clearance and volume of distribution, not a property of the drug in isolation: liver function, kidney function, body size, age, and interactions that inhibit or induce metabolic enzymes all change it. Two patients on the same dose can have half-lives that differ by a factor of three.

Repeated dosing approaches a steady state after roughly four to five half-lives, when the amount eliminated between doses equals the amount absorbed. A drug with a 24-hour half-life takes about four days to reach that plateau; one with a 4-hour half-life gets there in less than a day. A loading dose can shorten the wait, which is why some antibiotics and antiarrhythmics are started with a larger first dose.

Short half-lives usually mean frequent dosing, and long half-lives allow once-daily or weekly schedules. Ibuprofen, at roughly two hours, is taken every four to six hours; fluoxetine and its active metabolite persist for days, which smooths missed doses but also prolongs side effects after stopping. In renal or hepatic impairment, the same logic predicts accumulation, and dose adjustment is based on measured clearance rather than on the label's average.

One caution: plasma half-life is not the same as duration of effect. A drug can leave the plasma quickly while its effect persists through irreversible binding, active metabolites, or downstream changes in gene expression. The number describes the concentration curve, not the clinical timeline.

Half-Lives Worth Knowing

The range spans more than thirty orders of magnitude. The values below are physical half-lives, which are unaffected by temperature, pressure, or chemical bonding for decay modes that occur in the nucleus. Electron capture is a minor exception, with a small dependence on the electron density at the nucleus.

A long half-life does not automatically mean low hazard, and a short one does not mean safe. Hazard depends on the radiation type, the energy per decay, the chemical behavior of the element in the body, and the amount present. Radon-222 has a short half-life but is a leading cause of lung cancer because it is inhaled as a gas and its decay products deposit in the airway.

Reading a Decay Curve

Plotted on linear axes, an exponential decay looks like a curve that falls steeply at first and then flattens, approaching the axis without touching it. Each equal horizontal step halves the height, so the drop from 100 to 50 takes the same time as the drop from 50 to 25. That constant proportional decrease is the visual signature of first-order decay.

Plotted on a logarithmic vertical axis, the same data becomes a straight line. The slope is −λ, or −ln 2 divided by the half-life, and the intercept is the initial amount. This is why semi-log plots are standard in decay work: a straight line confirms a single exponential, and a bend reveals a second component with a different half-life or a change in the system.

Real samples often contain a mixture. A curve that drops quickly and then settles into a slower straight line may reflect a short-lived parent in equilibrium with a long-lived daughter, or a drug with a rapid distribution phase followed by slow elimination. Fitting two exponentials is not a failure of the model; it is the model telling you that two processes are present.

Half-Life and Doubling Time

Growth and decay are the same equation with the sign flipped. For a quantity growing at a constant rate r, the doubling time is ln 2 / r; for a quantity shrinking at the same rate, the half-life is ln 2 / r. The constants are identical, which is why a process cannot have a half-life without also having a doubling time if the rate were reversed.

The rule of 72 is the familiar approximation: at r percent per year, a balance doubles in roughly 72/r years. At 6%, that is 12 years against an exact 11.9; at 2%, it is 36 years against 35.0. The approximation is good in the middle of the range and drifts at very low or very high rates, where 69.3/r or a continuous-compounding formula is more accurate.

The symmetry is useful in public health. An epidemic's doubling time describes how fast cases accumulate, while the half-life of a drug or an antibody describes how fast protection fades. Neither number is a policy by itself, but together they set the tempo for testing, treatment, and booster schedules.

Environmental Persistence

After a release, the isotopes present determine the timeline of the hazard. Iodine-131, with an eight-day half-life, dominates the first weeks and poses a thyroid risk that can be reduced by early iodine prophylaxis. Cesium-137, at about 30 years, and strontium-90, at about 29 years, define the decades-long contamination zones because they persist in soil and enter the food chain. The Chernobyl and Fukushima accidents both followed this pattern.

Ecological half-life is not the same as physical half-life. A radionuclide can decay quickly but be re-released from soil, resuspended in dust, or concentrated by organisms, which lengthens the time it remains a practical problem. Cesium binds to clay minerals and moves slowly through some soils, while strontium behaves chemically like calcium and accumulates in bone and milk. The environmental number is a combination of physics, chemistry, and biology.

Organic pollutants have their own persistence measures. DDT and PCBs break down slowly, accumulate in fatty tissue, and biomagnify up food chains, so their environmental half-lives can be measured in years to decades even though they are not radioactive. Monitoring programs use the same exponential math to project how long a contaminated site will take to fall below a threshold, and the same caveats about mixtures apply.

Units and Conversions

Half-life can be measured in nanoseconds, minutes, days, or billions of years. The only rule that matters for the arithmetic is consistency: elapsed time and half-life must use the same unit. The calculator's ratio is dimensionless, so it cannot detect a mismatch; entering 5,730 years and 12,000 days produces a number that is wrong by a factor of 365.25 and looks entirely reasonable.

Convert deliberately. A day is 86,400 seconds, a year is about 365.25 days, and a year is roughly 31.56 million seconds. For short-lived isotopes, seconds or minutes are conventional; for geological dating, millions or billions of years. Mixing them is common when a paper reports a half-life in years and a lab measurement in days, which is exactly the situation where an explicit conversion step prevents a silent error.

Activity is a separate unit from quantity. The becquerel is one decay per second, the curie is 3.7 × 10¹⁰ becquerels, and the gray and sievert measure absorbed and equivalent dose rather than decay rate. A sample can have a short half-life and a low activity if only a few atoms are present, so do not substitute one for the other when reading a safety data sheet.

Practical Workflow

Start by writing down the three known values with their units, then decide which variable is unknown. Convert time units to a single base before entering anything. If you are using the conversion feature, enter the constant you trust and copy the other two into your notes rather than re-deriving them later from memory.

A quick verification trick is to set the initial amount to 100 and the half-life to 1. After 1, 2, 3, and 4 units of elapsed time, the remaining amounts should be 50, 25, 12.5, and 6.25. If the calculator returns those values, the formula is being applied correctly and any remaining problem is in the inputs.

Check the order of magnitude before you interpret the result. A remaining fraction of 0.001 after a stated interval implies about ten half-lives have passed; if your input implied two, one of the numbers is wrong. Similarly, a computed age of 80,000 years for a radiocarbon sample should raise a flag, because the method's practical limit is around 50,000 years and contamination becomes dominant well before that.

Five Mistakes That Produce Confident Wrong Answers

Mismatched time units. The most common error by a wide margin. Convert before calculating, and write the unit next to every number in your notes so the mismatch is visible.

Treating (1/2)ⁿ as n/2. After three half-lives, 12.5% remains, not 50% divided by three. The fraction falls geometrically, and linear intuition underestimates the decay badly after the first step.

Asking when the substance will be completely gone. The exponential never reaches zero. In practice, a material is treated as decayed after about ten half-lives, when less than 0.1% remains, and the threshold for a specific regulation may be different again.

Solving for time without logarithms. Rearranging t = t_1/2 × log₂(N₀/N_t) is straightforward, but substituting a natural logarithm without dividing by ln 2 gives an answer that is off by a factor of 1.44. Check one known case, such as a 50% remainder, which must return exactly one half-life.

Confusing activity with mass or concentration. A sample can lose activity through decay while the total mass of the element stays almost constant, because the daughter product remains. If you are measuring a drug or a pollutant rather than a radionuclide, the exponential describes the concentration of the original substance, not the total material in the system.

For related arithmetic, the log calculator handles the rearrangements, the exponent calculator checks powers of two and e, and the scientific calculator covers the rest of the working.