Radioactive Isotope Half-Life Computer

Select a preset isotope or enter custom values. Solve for final amount or time elapsed. The exponential decay curve renders live.

☢ Isotope Selection
🔍
Time unit for elapsed:
N0 Empty
Initial Amount
t Empty
Time Elapsed
Unit: Days
Nt Empty
Final Amount
Same unit as N0
Enter a half-life and initial amount, then fill either Time Elapsed or Final Amount to solve for the other.
📚 Key Terms Explained
Half-Life (t1/2)
The time required for exactly half of the atoms in a radioactive sample to decay. It is a fixed, characteristic property of each isotope and is independent of the sample size or external conditions such as temperature and pressure.
Radioactive Decay
The spontaneous disintegration of an unstable atomic nucleus, releasing energy in the form of alpha particles, beta particles, or gamma rays. Decay transforms the parent isotope into a different element or a lower-energy state.
Isotope
Atoms of the same element (same number of protons) that have different numbers of neutrons. Carbon-12, Carbon-13, and Carbon-14 are all isotopes of carbon. Isotopes with unstable nuclei are radioactive; those with stable nuclei are not.
Decay Constant (lambda)
A probability measure indicating how quickly an isotope decays per unit of time. Calculated as lambda = ln(2) divided by the half-life. A large lambda means rapid decay; a small lambda means the isotope is long-lived.
Exponential Decay
A mathematical pattern in which a quantity decreases at a rate proportional to its current value. The formula Nt = N0 times e raised to the power of negative lambda times t describes this curve. The characteristic shape falls steeply at first and then flattens asymptotically toward zero.
Radiocarbon Dating
A technique that uses the known half-life of Carbon-14 (5,730 years) to estimate the age of once-living organic material. Scientists measure the ratio of C-14 to stable C-12 remaining in a sample and solve the decay equation for time elapsed since the organism died.
Activity
The rate at which a radioactive sample produces decay events, expressed in becquerels (Bq) or curies (Ci). Activity equals the decay constant multiplied by the number of atoms currently present: A = lambda times N. As atoms decay, activity decreases over time along the same exponential curve.
Parent and Daughter Isotopes
The parent isotope is the unstable atom that decays. The daughter isotope is the product created after decay. Some daughters are themselves radioactive, forming a chain (for example, Uranium-238 decays through 14 steps before reaching stable Lead-206).

The Complete Guide to Radioactive Half-Life and Exponential Decay

Whether you are a student calculating how much Carbon-14 remains in an archaeological sample, a nuclear medicine professional estimating when a patient's dose will clear, or simply someone curious about how scientists date ancient objects and predict isotope behavior, this tool handles the math. Below is everything you need to understand what is happening under the hood.

How to Use This Calculator

Start by selecting an isotope from the searchable list or switch to Custom mode to enter any half-life value you like. Choose the time unit (Seconds, Minutes, Hours, Days, or Years) that fits your scenario - the same unit will apply to both the half-life field and the time elapsed field. Then fill in the Initial Amount and one of the two remaining variables. The tool will instantly solve for the missing variable, display the Decay Constant, and draw the exponential curve scaled to exactly five half-life intervals.

The unit selector for the Initial Amount lets you work in Grams, Milligrams, Micrograms, or Percentage. All calculations are internally standardized so that the half-life and time elapsed are always compared in the same unit before any formula is applied. This prevents scale-mismatch errors that can occur when, for example, a half-life is entered in days but the elapsed time is mentally imagined in hours.

The Mathematics Behind the Curve

The exponential decay formula is Nt = N0 multiplied by e raised to the power of negative lambda multiplied by t, where N0 is the initial amount, Nt is the amount remaining after time t, and lambda is the decay constant. Lambda is derived from the half-life using lambda = ln(2) divided by t1/2. To solve for time when you know the final amount, rearrange the formula to t = negative ln(Nt divided by N0) divided by lambda.

The calculator uses JavaScript's Math.log (natural logarithm) and Math.exp (e raised to a power) to apply these formulas with full floating-point precision. When the result becomes very small - for instance, after more than 20 half-lives have passed - the output switches to scientific notation such as 1.24e-6 rather than displaying an inaccurate rounded zero.

Reading the Decay Curve

The SVG graph scales its X-axis to show exactly five half-life intervals based on your current half-life input. Dot markers appear at each interval, labeling the percentage of the original amount remaining at that point: 50% after one half-life, 25% after two, 12.5% after three, 6.25% after four, and 3.125% after five. A subtle grid overlay helps you read intermediate values. If you have entered a Time Elapsed value, a vertical marker shows where that specific moment falls on the curve.

Common Applications by Isotope Type

Medical isotopes such as Technetium-99m (half-life 6.01 hours) and Iodine-131 (8.02 days) are used in diagnostic imaging and cancer treatment. Their short half-lives mean the patient's radiation exposure decreases quickly. Industrial isotopes such as Cobalt-60 (5.27 years) are used in radiography and sterilization, while Cesium-137 (30.17 years) appears in gauges and some historical nuclear accidents. Geological and dating isotopes such as Carbon-14 (5,730 years), Potassium-40 (1.25 billion years), and Uranium-238 (4.47 billion years) span timescales from human history to the age of the solar system.

Frequently Asked Questions

Does a radioactive substance ever truly disappear to zero? ▼
Mathematically, no. The exponential decay formula Nt = N0 times e raised to the power of negative lambda times t approaches zero asymptotically. Each half-life cuts the remaining amount in half, so there is always half of something left, no matter how small. In practice, atoms are discrete particles: you eventually reach one atom, and it either decays or it does not at a random moment. But the continuous exponential model - accurate for large populations of atoms - never produces an exact zero. After about 20 half-lives, the remaining fraction is roughly one-millionth of the original, which is why the calculator switches to scientific notation at that scale.
How is Carbon-14 used for dating ancient artifacts? ▼
Living organisms continuously absorb Carbon-14 from the atmosphere, maintaining a known ratio of C-14 to stable C-12. When an organism dies, absorption stops and the existing C-14 decays with a half-life of 5,730 years. Scientists measure the current C-14 fraction in a sample and compare it to the known initial ratio. By solving the decay formula for time (t = negative ln(Nt divided by N0) divided by lambda), they calculate how many years ago the organism died. The technique is reliable for objects up to about 50,000 years old. Beyond that, too little C-14 remains to measure accurately, and longer-lived isotopes such as Uranium-238 or Potassium-40 are used instead.
What is the difference between biological half-life and physical half-life? ▼
Physical (or radiological) half-life is the time for half of a radioactive isotope's atoms to decay, governed purely by nuclear forces. Biological half-life is the time for the body to eliminate half of a substance through metabolism, excretion, and other physiological processes - it applies to any drug or chemical, not just radioactive ones. In nuclear medicine, the effective half-life combines both: it is the time for the radioactive dose in the body to fall by half due to both physical decay and biological elimination. For Iodine-131, the physical half-life is about 8 days, but the effective half-life in the thyroid is shorter because the body also metabolises iodine. This calculator models physical half-life only.
Why does the decay curve flatten out over time but never immediately hit the bottom? ▼
The shape of the exponential decay curve follows the formula Nt = N0 times e raised to the power of negative lambda times t. Because the exponent is a negative multiple of time, the rate of decrease is proportional to the current amount remaining - not a fixed quantity per unit time. When a large amount is present, the absolute decrease per unit time is large, so the curve drops steeply. As the amount shrinks, the absolute decrease per unit time shrinks in proportion, and the curve flattens. This self-similar property means the curve can never reach zero in finite time: the smaller the amount, the slower the absolute rate of loss, producing the characteristic asymptotic approach to the x-axis.