Half-Life Calculator

What a half-life is

A half-life is the time it takes for half of a sample to decay, transform or clear. It is not the time until the sample is gone. Start with 100 mg of something whose half-life is 6 hours and you have 50 mg after 6 hours, 25 mg after 12, 12.5 mg after 18 and 6.25 mg after 24. The amount keeps halving and never reaches zero, which is why a long half-life means a small amount stays around for a long time.

The shape of that decline is exponential. Equal spans of time remove equal fractions, not equal amounts. Between hour 0 and hour 6 you lose 50 mg, and between hour 18 and hour 24 you lose 6.25 mg, even though both spans are 6 hours. This is the part that trips people up when they read about radiation or drug clearance: the loss is fast at the start and slows down because there is less left to lose.

Half-life applies to more than radioactivity. A medicine has a biological half-life in the body, a pollutant has an environmental half-life in soil or water, and a bank balance has a doubling time that is the same math with a positive sign. The formula does not care what is decaying, only that the rate is proportional to how much remains.

One caution before using any number: a half-life is an average over a large population of atoms or molecules. With a single atom you cannot say when it will decay, only that the chance of surviving one half-life is 50 percent. The smooth curve on a chart is a statement about millions of atoms, not about one.

The formula and the decay constant

Two forms of the same equation cover every question on this page. The first counts half-lives directly, and the second uses the decay constant, which is the decay rate expressed per unit of time.

The decay constant is the half-life written a different way, so a half-life of 6 hours is a decay constant of 0.1155 per hour. The NIST radiation physics program publishes measured half-lives to several significant figures for the isotopes used in calibration, and those are the numbers to use when a calculation has to match a lab certificate.

The mean life is 1.4427 times the half-life, which surprises people who expect the two to match. The reason is that a few atoms survive for many half-lives and pull the average lifetime up: after 10 half-lives 0.0977 percent of the sample is still there, and that tail counts toward the mean.

Worked example: 100 mg with a 6 hour half-life

Take 100 mg of a substance with a 6 hour half-life and ask what is left after 24 hours.

• Half-lives elapsed: 24 / 6 = 4
• Fraction left: (1/2)⁴ = 1/16 = 0.0625
• Amount left: 100 x 0.0625 = 6.25 mg
• Percent left: 6.25 percent
• Decay constant: 0.6931 / 6 = 0.1155 per hour

Six and a quarter milligrams, from a starting point of 100. Four half-lives is all it takes to remove 93.75 percent of the sample, and the arithmetic fits on one line.

The same table answers the question in reverse. If a reading shows 12.5 mg left, then 3 half-lives have passed and the sample is 18 hours old. If it shows 25 percent of the starting amount, the elapsed time equals two half-lives, which is 12 hours here.

Watch the units. A half-life in days with an elapsed time in hours gives a wrong answer by a factor of 24, and the mistake looks plausible on the page because the exponent is still small. Convert both numbers to the same unit before dividing, which is what this calculator does internally.

Half-lives people look up

The range of half-lives runs from fractions of a second to billions of years, and the values below are the ones that come up in medicine, environmental reports and news stories about radiation.

The medical values come from NIST measurements and FDA labeling, and the environmental values from the EPA radionuclide pages. Note how the units are chosen for the use: minutes for a tracer that has to be injected within hours, years for something that stays in soil for a generation.

A half-life is a property of the isotope, not of the sample size, the temperature or the chemical form. Heating a gram of cesium-137 does not change its 30.17 year half-life. Chemistry can change how a substance moves through a body or through soil, which is why environmental reports sometimes quote an effective half-life that combines physical decay with biological clearance.

Carbon dating and the 5,700 year clock

Living plants take in carbon from the air, and a small share of that carbon is carbon-14. When the plant or animal dies, the intake stops and the carbon-14 already inside starts to decay with a half-life of about 5,700 years. Measure how much is left and the ratio gives the time since death.

Those numbers explain the method's ceiling. At 50,000 years only about 0.23 percent of the original carbon-14 remains, and the leftover signal is close to what contamination and instrument background produce. The USGS puts the practical limit near 50,000 years for that reason, even though the math keeps returning smaller fractions forever.

Ages are reported in "years before present", where present means 1950, because atmospheric nuclear testing after that date loaded the air with extra carbon-14. Labs also use a defined half-life of 5,568 years and apply a calibration curve, so a hand calculation with 5,700 can differ from a published date by a century or two. Treat your own figure as a check on the arithmetic, not as a replacement for a lab result.

Medical isotopes: why six hours changes a schedule

Nuclear medicine works on a clock. A tracer with a 6 hour half-life cannot be prepared on Monday for a Wednesday scan, and a department that ignores the decay curve either wastes doses or images with too little signal.

• Tc-99m, 6.0067 hours: after 12 hours 25.0 percent is left, and after 24 hours 6.27 percent.
• F-18, 109.77 minutes: after 5.5 hours (330 minutes) 12.45 percent is left, which is 3.006 half-lives.
• I-131, 8.02 days: after 30 days 7.48 percent is left, which is 3.74 half-lives.

Those percentages come from the same equation as every other example on this page. A dose calibrated to 1,000 MBq at 8 am holds 501 MBq at 2 pm for Tc-99m, because 6 hours is one half-life. The label on a radiopharmaceutical vial lists a calibration time for exactly this reason, and the technologist corrects the volume to the activity the protocol asks for.

For F-18 the practical window is tighter. A 110 minute half-life means 5.5 hours costs 87.55 percent of the activity, so a late patient gets a longer scan or a weaker image. None of this is medical advice: dose selection and timing belong to the clinician and the radiation safety officer.

Finding the half-life from two measurements

Sometimes the half-life is the unknown. Two readings of the same sample, taken a known time apart, are enough: t_1/2 = t x ln(2) / ln(N₀ / N_t).

A source reads 80 MBq at the start of a shift and 60 MBq five hours later. The ratio is 80 / 60 = 1.333, and ln(1.333) = 0.2877. Then 5 x 0.6931 / 0.2877 = 12.05 hours, so the half-life is about 12 hours.

The same method works outside physics. A medicine that peaks at 200 mg in the blood and falls to 125 mg four hours later has a half-life of 5.90 hours, from 4 x 0.6931 / ln(1.6). That figure is a biological half-life, which depends on the liver and kidneys rather than on the nucleus, and it is why two people can clear the same drug at different rates.

Three conditions make the two-point method trustworthy: nothing new enters the sample during the interval, both readings come from the same instrument and geometry, and the drop is large enough to sit above the measurement noise. A change from 100 to 96 over an hour is mostly noise, and dividing by it produces a wild half-life.

Where the simple model stops

The equation assumes a single substance decaying at a constant rate. Real samples break that assumption in five common ways.

Mixtures. A sample with two isotopes decays on two clocks, and a plot of the log of activity against time bends instead of forming a straight line. Reading the first day of such a curve as if it were one isotope overestimates how fast the whole sample fades.

Parent and daughter. Tc-99m comes from a Mo-99 generator with a 66 hour parent half-life. Early on the daughter is replenished as fast as it decays, so the activity you measure does not follow the 6 hour curve until the parent has faded. This is the transient equilibrium case, and it is why generator elution schedules exist.

Biological clearance. In a living body, physical decay and biological removal act at once, and the effective half-life is shorter than either one alone. A substance with a 12 hour physical half-life and a 12 hour biological half-life has an effective half-life of 6 hours.

Detection limits. At 10 half-lives 0.098 percent remains, and at 20 half-lives 0.0001 percent. Long before zero, the signal disappears into background, so "gone" in a report means "below what the instrument can see", not "zero atoms".

Rounding. Published half-lives are averages with uncertainty. Carbon-14 is quoted as 5,700 plus or minus 30 years, and a calculation that carries more digits than the input supports suggests a precision that does not exist.