Half-Life Calculator

Free Online Half-Life Calculator

This Half-Life Calculator works out how much of a radioactive substance remains after a given time, how long decay has been happening, or the half-life of a sample based on measured quantities. You can enter any consistent quantity unit, such as number of atoms, grams, activity, or percentage, together with matching time units. The calculator applies the standard half-life equation and shows a full, copyable step-by-step solution alongside the final answer.

The decay diagram below the calculator shows the defining pattern of radioactive decay: after every completed half-life, exactly half of whatever remained at the start of that interval is left. This free tool is meant for modern physics homework, chemistry revision, radiometric dating examples, nuclear science coursework, and general practice with exponential-decay problems that appear again and again in science exams.

What Is Half-Life?

Half-life is the time required for half of the unstable nuclei in a radioactive sample to decay into a more stable form. It is usually written as T½ or t½. Unlike a countdown timer for a single atom, half-life describes the statistical behaviour of an entire population of atoms. It is impossible to say when any one particular nucleus will decay, but the fraction of a large sample that remains after a fixed amount of time is remarkably predictable and repeatable in the laboratory.

After one half-life, fifty percent of the original quantity remains; after two half-lives, twenty-five percent remains; after three, 12.5 percent, and so on. In the ideal mathematical model the substance never reaches exactly zero, since each halving still leaves a smaller positive amount, but in practice a sample can become far too small to detect or to matter for any real purpose. Different radioactive isotopes have extremely different half-lives, ranging from tiny fractions of a second to many billions of years.

Half-Life Formula

The standard half-life equation used by this calculator is N = N₀(1/2)^(t/T½). Here N is the remaining quantity, N₀ is the initial quantity, t is the elapsed time, and T½ is the half-life. The exponent t/T½ represents the number of half-lives that have passed, and it does not need to be a whole number; fractional numbers of half-lives are handled correctly by the exponential form.

This formula applies equally well to atom number, mass, radioactive activity, or count rate, as long as the same unit is used consistently for both N and N₀. For example, a sample that starts at 100 grams with an eight-year half-life will have 12.5 grams remaining after 24 years, because 24 divided by 8 gives exactly three half-lives, and 100 multiplied by one-half three times equals 12.5. The bar diagram on this page shows that repeated halving visually, making the pattern easy to see at a glance.

Calculating Remaining Radioactive Quantity

To find what remains after a certain time, divide the total elapsed time by the half-life to get the number of half-lives that have passed. Raise one-half to that power, then multiply the result by the initial quantity. Whole numbers of half-lives can be worked out by simple repeated halving, but the exponential formula also handles fractional numbers of half-lives just as accurately, which is important when the elapsed time does not divide evenly into the half-life.

For example, if 10 grams of a substance has a half-life of 5 days, then after 15 days three complete half-lives have passed, leaving 1.25 grams. After only 2.5 days, half of one half-life has passed, so the exponent used is 2.5 divided by 5, which equals 0.5. It is important not to round a fractional number of half-lives to the nearest whole number before calculating, since that shortcut introduces a noticeable error into the final answer.

Finding Elapsed Time or Half-Life From Measurements

When the initial and remaining quantities are already known from measurement, logarithms are used to find either the elapsed time or the half-life itself. Elapsed time is found using t = T½ × log₂(N₀/N). This expression first calculates the number of halvings needed to move from the initial quantity down to the remaining quantity, then multiplies that count by the length of one half-life. For a drop from 100 down to 12.5, the logarithm gives exactly 3 half-lives.

To find the half-life instead, when time and both quantities are known, rearrange the equation to T½ = t / log₂(N₀/N). Accurate laboratory measurement matters here: scientists typically repeat readings several times and correct for background radiation, detector efficiency, and statistical uncertainty rather than trusting a single measurement, since real data is always noisier than a textbook example.

Radioactive Decay and Activity

Radioactive activity is the number of nuclear decay events happening per second, measured in a unit called the becquerel, abbreviated Bq. Because activity is directly proportional to how many undecayed nuclei remain in a sample, it follows exactly the same half-life pattern as mass or atom count. If a sample starts at an activity of 800 Bq, it falls to 400 Bq after one half-life and 200 Bq after two half-lives, following the same repeated halving seen with any other quantity.

Activity should never be confused with radiation dose or personal risk. The actual danger from radiation depends on the type of radiation emitted, its energy, how long a person is exposed, distance from the source, the shielding in place, and whether the material has entered the body. A half-life calculation on its own cannot determine whether something is safe to be near. Always follow trained professionals and official guidance for any real radioactive material.

Half-Life and Decay Constant

An equivalent way to describe the same exponential process uses the decay constant, λ, in the form N = N₀e^(−λt). Half-life and decay constant are connected by the relationship T½ = ln(2)/λ. A larger decay constant corresponds to faster decay and a shorter half-life, while a smaller decay constant corresponds to slower decay and a longer half-life. Both descriptions model exactly the same underlying random nuclear process, just written with different mathematical constants.

Some exam questions give a decay constant instead of a half-life directly, so it helps to be comfortable converting between the two forms. The half-life version tends to be easier for quick mental checks, since every equal interval simply halves the previous value, while the exponential version built around e is generally more convenient for continuous-rate calculations found in more advanced physics courses.

Half-Lives of Familiar Isotopes

Different radioactive isotopes cover an enormous range of half-lives, which is part of why half-life is such a useful property for identifying materials and choosing them for specific jobs. Carbon-14, widely used in archaeological dating, has a half-life of roughly 5,730 years, which makes it well suited to dating organic remains that are thousands of years old. Iodine-131, used in some medical procedures, has a much shorter half-life of about eight days, meaning it becomes far less active within a few weeks.

At the other extreme, isotopes such as uranium-238 have half-lives measured in billions of years, which is why uranium is still present in the Earth's crust today despite the planet being about 4.5 billion years old. Comparing these very different timescales helps explain why some radioactive sources are chosen for long-term geological dating while others are chosen specifically because they decay away quickly and safely after serving their purpose.

Half-Life Applications

Half-life calculations are used in radiocarbon dating, medical imaging, cancer treatment planning, nuclear medicine, geological dating, environmental monitoring, and nuclear waste management. Radiocarbon dating estimates the age of once-living material by comparing how much carbon-14 remains against how much would have been present when the organism was alive. Other isotope systems, based on much longer half-lives, are used to date rocks and minerals that are millions or billions of years old.

Medical isotopes are chosen partly for their half-life: they need to remain active long enough to produce a useful image or deliver treatment, but decay away quickly enough afterwards to limit unnecessary radiation exposure to the patient. In every real application, a half-life calculation is only one part of a much larger, carefully controlled process that also involves isotope-specific data, precise measurement equipment, safety regulations, and trained specialist oversight.

Reading the Half-Life Diagram

The bar diagram above the results panel plots the remaining quantity after zero, one, two, and three half-lives, so the falling pattern can be seen directly rather than only calculated. Each bar is exactly half the height of the one before it, which visually reinforces why the decay curve flattens out over time instead of falling in a straight line. The highlighted box on the diagram always reflects the current result from the calculator, so changing the inputs updates the picture immediately.

This kind of visual check is a useful habit for any exponential decay problem, not just half-life questions. If an answer looks like it should sit between two of the marked half-life points on the diagram, but the calculated number does not, that is usually a sign that a unit was entered incorrectly or the wrong mode was selected. Building this habit of a quick visual sanity check catches many simple mistakes before they end up in a final answer.

Units and Problem-Solving Tips

Time units must always match. If half-life is given in years, elapsed time must also be entered in years; if one value is in hours and the other in minutes, convert everything to the same unit before calculating anything. Quantity units can be grams, atoms, becquerels, or a plain percentage, as long as the initial and remaining values use exactly the same unit as each other, since the units cancel out in the ratio N₀/N used inside the logarithm.

Always use positive quantities, and keep the remaining quantity less than or equal to the initial quantity for ordinary decay. Entering a remaining amount larger than the initial amount will produce a negative elapsed time, which is a clear sign of growth or inconsistent data rather than normal radioactive decay. As a quick check, confirm that each completed half-life in your working cuts the remaining percentage exactly in half before moving on to the next step.

How to Use the Half-Life Calculator

Select whether you want to find Remaining Quantity, Elapsed Time, or Half-Life using the mode selector above the input fields. Enter the three known values with consistent units. The results panel then shows the answer, the number of half-lives elapsed, the percentage remaining, and the formula used for that particular mode. The step-by-step area below explains each stage of the substitution in order, and a copy button lets the full working be saved for homework or lab reports.

This calculator is designed for education, revision, and general planning calculations. It does not identify unknown isotopes and it does not determine radiation hazards or safe exposure limits. Any real decision involving actual radioactive materials, medical procedures, disposal, or regulatory compliance should always be guided by qualified specialists and official safety documentation, not by a general-purpose online calculator.

Half-Life Calculator FAQ Summary

Half-life is the time needed for a radioactive sample to fall to half of its current amount. Use N = N₀(1/2)^(t/T½) to find the remaining quantity, and use logarithms in the form T½ = t / log₂(N₀/N) or t = T½ × log₂(N₀/N) to find half-life or elapsed time from measured data. The decay follows a smooth exponential curve rather than a straight line, since it halves repeatedly instead of decreasing by a fixed amount each interval. Keep all units consistent throughout a calculation, and remember that a half-life value alone does not determine whether a radioactive material is safe.