Can it handle growth?

Half-life specifically describes decay. For growth processes (such as population doubling), you would use doubling time instead and replace the (1/2) factor with 2 in the exponent. This calculator is set up for decay scenarios.

What is the decay constant λ used for?

The decay constant λ is used in the continuous exponential decay equation N(t) = N₀ × e^(−λt). It’s especially helpful in differential-equation models, kinetic analyses, and when combining decay with other processes in more advanced math.

Can I use this for multi-phase or multi-compartment decays?

This calculator assumes a single half-life and a simple exponential model. Many real systems (e.g., some drugs, environmental contaminants) have multi-phase decays with different half-lives in different compartments. Those require more detailed modeling than this tool provides.

Does this calculator replace official tables or regulatory tools?

No. It is intended for education and quick calculations. For regulatory work, safety-critical decisions, or precise medical dosing, always rely on official data, detailed models, and professional guidance.

science calculator

Half-Life Calculator

Compute remaining quantity after radioactive or exponential decay over time.

Results

Remaining amount: 18.95
Percent remaining: 18.95%
Half-lives elapsed: 2.40
Decay constant (λ): 0.14

Overview

Half-life is a core concept in nuclear physics, chemistry, pharmacology, and many other fields. It describes how long it takes for half of a substance to decay or be eliminated. Because the process is exponential, the math can feel unintuitive: the same half-life applies over and over, no matter how much remains.

This half-life calculator handles the exponential decay for you. By entering an initial amount, the substance’s half-life, and the elapsed time, it computes the remaining quantity, percent remaining, the number of half-lives that have passed, and the decay constant λ used in more advanced kinetic equations. It’s useful for quick homework checks, lab prep, and back-of-the-envelope calculations in research or clinical contexts.

How to use this calculator

Enter the Initial amount of the substance—this might be mass (grams, milligrams), activity (curies, becquerels), concentration, or even a simple count, as long as you use the same unit throughout.
Enter the substance’s Half-life in any time unit you like (seconds, hours, days, years, etc.).
Enter the Elapsed time using the same units you used for half-life. For example, if half-life is in days, elapsed time should also be in days.
The calculator divides elapsed time by half-life to find how many half-lives have passed, raises 1/2 to that power, and multiplies by the initial amount to find the Remaining amount.
It then reports Percent remaining, Half-lives elapsed, and the Decay constant λ so you can interpret the decay both in half-life terms and in terms of continuous exponential decay.
Adjust half-life or elapsed time to explore different scenarios or to compare substances with different decay rates.

Inputs explained

Initial amount: The starting quantity of the substance before decay begins. This can be in any unit—grams, milligrams, curies, becquerels, counts, concentration, etc.—as long as you use the same unit when interpreting the remaining amount.
Half-life (time units): The time it takes for half of the substance to decay or be eliminated. Enter it in any convenient time unit (seconds, minutes, hours, days, years), but be sure to use the same units for elapsed time.
Elapsed time: How much time has passed since the initial amount was measured. To keep the math correct, use the same time units you used for the half-life input.

How it works

We model exponential decay using the classic half-life relationship: after one half-life, half of the original amount remains; after two half-lives, one quarter remains; after three, one eighth, and so on.

First, we compute how many half-lives have elapsed: half-lives elapsed = elapsed time ÷ half-life. This can be a whole number or a fraction.

We then apply the exponential decay formula using powers of 1/2: Remaining amount = Initial amount × (1/2)^(half-lives elapsed).

Percent remaining is simply Remaining amount ÷ Initial amount, expressed as a percentage.

For users who need the decay constant used in differential equations, we compute λ = ln(2) ÷ half-life. This relates to the continuous decay equation N(t) = N₀ × e^(−λt).

Because half-life and elapsed time can be given in any consistent time units (seconds, minutes, hours, days, years), the calculator is time-unit agnostic—just make sure both values use the same units.

Formula

Half-lives elapsed = t ÷ t½\nN(t) = N₀ × (1/2)^(t ÷ t½)\nPercent remaining = N(t) ÷ N₀\nλ = ln(2) ÷ t½\nAlternative continuous form: N(t) = N₀ × e^(−λt)

When to use it

Radioactive decay calculations in nuclear physics, medical imaging, or radiation safety to determine how much activity remains after a certain period.
Drug elimination and pharmacokinetics problems, where half-life describes how quickly a medication leaves the body and how much remains at a given time after dosing.
Environmental science scenarios, such as pollutant breakdown or isotope tracing in ecosystems.
Any exponential decay process that can be approximated by a single half-life, including some capacitor discharge problems, population decay models, and simple first-order reactions.
Educational demonstrations that show how exponential decay works and why it never quite reaches zero, even as the amount becomes very small.

Tips & cautions

Always keep half-life and elapsed time in the same units. If a nuclide’s half-life is quoted in days but your problem uses hours, convert one to match the other before entering values.
If your initial amount is extremely large or extremely small, consider using scientific notation or consistent prefixes (mg vs g) so your numbers are easy to read and interpret.
Use the decay constant λ when you need to plug the process into differential equations or continuous models (for example, in more advanced kinetics or simulation work).
If you know the fraction remaining and elapsed time but not the half-life, you can rearrange the continuous decay equation with λ to solve for half-life; this calculator focuses on the forward direction but the relationships are invertible.
Remember that half-life describes an average statistical behavior in large populations of particles or molecules; individual atoms or molecules decay randomly, not on a fixed schedule.
Assumes ideal first-order exponential decay with a single half-life. Real-world systems can involve multiple phases, non‑exponential behavior, or changing decay rates.
Does not account for production terms (sources adding new material) or multi-compartment pharmacokinetics, where different tissues or phases have different effective half-lives.
Relies on user-supplied half-life; it does not look up nuclide data, drug half-lives, or environmental decay constants automatically.
Focuses on nominal amounts and does not model measurement uncertainty, background noise, or detection limits in experimental setups.
Not intended for regulatory calculations or safety-critical decisions without validation against official data and methods.

Worked examples

Example 1: Iodine-131 (half-life 8 days) after 24 days

Half-life t½ = 8 days; elapsed time t = 24 days.
Half-lives elapsed = 24 ÷ 8 = 3.
Remaining amount = N₀ × (1/2)^3 = N₀ × 1/8 = 0.125 N₀.
Percent remaining ≈ 12.5% of the initial activity or mass.

Example 2: Drug with t½ = 6 hours after 18 hours

Half-life t½ = 6 hours; elapsed time t = 18 hours.
Half-lives elapsed = 18 ÷ 6 = 3.
Remaining concentration ≈ (1/2)^3 = 1/8 ≈ 12.5% of the initial level.
Interpretation: about 87.5% of the dose has been eliminated under a simple single-compartment model.

Example 3: General decay over fractional half-lives

Suppose t½ = 10 years and t = 15 years.
Half-lives elapsed = 15 ÷ 10 = 1.5.
Remaining amount = N₀ × (1/2)^1.5 ≈ N₀ × 0.3536 → about 35.4% remaining.
This shows how partial half-lives still follow the same exponential pattern, not just integer multiples.

Deep dive

Use this half-life calculator to compute remaining quantity, percent remaining, half-lives elapsed, and the decay constant λ from an initial amount, known half-life, and elapsed time. It’s ideal for physics, chemistry, pharmacology, and environmental science problems that involve exponential decay.

Enter a starting amount, the half-life in your preferred time units, and the time that has passed to get fast, accurate exponential decay answers without manually working through logarithms and exponents.

FAQs

Is time unit-agnostic?: Yes. You can use any time unit you like (seconds, minutes, hours, days, years) as long as the half-life and elapsed time use the same unit. The math depends only on their ratio, not on the specific units.
Can it handle growth?: Half-life specifically describes decay. For growth processes (such as population doubling), you would use doubling time instead and replace the (1/2) factor with 2 in the exponent. This calculator is set up for decay scenarios.
What is the decay constant λ used for?: The decay constant λ is used in the continuous exponential decay equation N(t) = N₀ × e^(−λt). It’s especially helpful in differential-equation models, kinetic analyses, and when combining decay with other processes in more advanced math.
Can I use this for multi-phase or multi-compartment decays?: This calculator assumes a single half-life and a simple exponential model. Many real systems (e.g., some drugs, environmental contaminants) have multi-phase decays with different half-lives in different compartments. Those require more detailed modeling than this tool provides.
Does this calculator replace official tables or regulatory tools?: No. It is intended for education and quick calculations. For regulatory work, safety-critical decisions, or precise medical dosing, always rely on official data, detailed models, and professional guidance.

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This half-life calculator illustrates ideal exponential decay based on user-entered initial amounts, half-lives, and elapsed times. It does not account for complex, multi-phase, or non-exponential behavior, nor does it incorporate safety factors, regulatory rules, or experimental uncertainty. It is intended for educational and general planning purposes only. Radioactive materials, medical dosing, and other safety-critical applications must be handled under the guidance of qualified professionals and appropriate regulations.