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Half-Life Calculator

Chemistry

Calculate half-life, decay constant, and remaining quantity for radioactive decay or first-order chemical reactions, with time unit conversion.

Reviewed by the thecalcu.com team · Last updated April 26, 2026

100
5,730
11,460

Remaining Amount

25
Percent Remaining (%)
25
Decay Constant λ (per time unit)
0
Number of Half-Lives Elapsed
2

This calculator computes your Remaining Amount, Percent Remaining (%), Decay Constant λ (per time unit), Number of Half-Lives Elapsed from the values you enter.

Inputs
Initial Amount (N₀)Half-LifeElapsed TimeTime Unit
Outputs
Remaining AmountPercent Remaining (%)Decay Constant λ (per time unit)Number of Half-Lives Elapsed

What is a Half-Life?

The Half-Life Calculator determines the remaining quantity of a radioactive isotope or first-order reacting substance after a given time, using the decay formula N(t) = N₀ × (1/2)^(t/t₁/₂). Enter the initial amount, the half-life, and the elapsed time to get the remaining amount, percent remaining, the decay constant, and the number of half-lives elapsed.

Half-life is the most fundamental characterisation of a radioactive isotope: it is the time for exactly half the atoms to decay, independent of how many atoms you started with, what temperature the material is at, or what chemical compound the element is in. This independence from initial conditions is a unique property of first-order decay and stands in stark contrast to most chemical reactions where rate depends on concentration.

The mathematical equivalence between radioactive decay and first-order chemical kinetics, both follow N(t) = N₀ × e^(−λt), means this calculator works equally well for first-order reaction kinetics. The Rate Constant Calculator performs the complementary calculation: finding k from concentration data. The Radioactive Decay Calculator focuses on the λ (decay constant) formulation used in nuclear physics and radiochemistry.

Why Use a Half-Life Calculator?

The two common decay formulas, N(t) = N₀ × (1/2)^(t/t₁/₂) and N(t) = N₀ × e^(−λt), are equivalent but require knowing either t₁/₂ or λ. Converting between them (λ = ln2/t₁/₂) adds a step. This calculator accepts t₁/₂ directly, avoiding the need to compute λ manually.

A common exam error is computing (1/2)^(t/t₁/₂) with t and t₁/₂ in different units, giving answers that are wildly wrong. This calculator shows n = t/t₁/₂ explicitly as the "number of half-lives" output, making the unit-consistency check transparent.

Who Should Use This Calculator?

Class 11 and 12 students covering nuclear chemistry (NCERT Class 12, Chapter 15, Nuclear Chemistry): half-life, radioactive decay, and first-order kinetics are standard examination topics.

JEE and NEET aspirants solving radioactive decay problems, one of the most formula-intensive topics, requiring both the decay law and the ability to extract remaining quantity, percentage, or time from the equation.

Nuclear medicine professionals computing isotope activity at the time of administration from the reference activity and calibration date.

Environmental radiation scientists computing radiological half-lives for contamination assessment, relevant for post-incident response and regulatory compliance under AERB (Atomic Energy Regulatory Board) guidelines.

What Insights Does the Half-Life Calculator Give You?

Remaining Amount is the primary output, the absolute quantity of the substance left after the specified time. The units match whatever units N₀ was entered in.

Percent Remaining directly answers "what fraction of the original activity or mass is left?", useful for checking whether medical isotopes have decayed below usable activity thresholds or whether environmental contamination is below regulatory limits.

Decay Constant λ is the per-unit-time probability of decay, needed for activity calculations (A = λN) and for Bateman equation solutions involving decay chains.

Number of Half-Lives Elapsed is the most intuitive characterisation, "this sample has undergone 3.5 half-lives" immediately tells you the remaining fraction without arithmetic.

How to use this Half-Life calculator

  1. Enter Initial Amount (N₀) in any consistent unit, grams, atoms, Bq (activity), or a relative value.
  2. Enter the Half-Life of the isotope or first-order process. Common values: ¹⁴C = 5730 years; ¹³¹I = 8.02 days; ⁹⁹ᵐTc = 6.01 hours; ²³⁸U = 4.47 × 10⁹ years.
  3. Enter the Elapsed Time in the same units as the half-life, or select a specific time unit from the dropdown.
  4. Read Remaining Amount and Percent Remaining.
  5. Check Number of Half-Lives Elapsed, 1 HL = 50%, 2 HL = 25%, 3 HL = 12.5%, 4 HL = 6.25%, 7 HL ≈ 0.78%.
Show formula & methodology ↓Show less ↑

Formula & Methodology

Half-life decay law:

N(t) = N₀ × (1/2)^(t/t₁/₂) = N₀ × (0.5)^n where n = t / t₁/₂ (number of half-lives)

Decay constant:

λ = ln(2) / t₁/₂ = 0.6931 / t₁/₂

Percent remaining:

% remaining = (N(t) / N₀) × 100 = (0.5)^n × 100

Worked example, iodine-131 medical dosage:

A Iodine-131 therapy dose has an initial activity of 3.7 GBq at calibration. t₁/₂(¹³¹I) = 8.02 days. Time elapsed since calibration: 12 days.

n = 12 / 8.02 = 1.496 half-lives N(t) = N₀ × (0.5)^1.496 = 3.7 × 0.3557 = 1.316 GBq % remaining = 35.6% λ = 0.6931 / 8.02 = 0.0864 per day

The administered activity after 12 days of storage is approximately 1.32 GBq, 36% of the calibration activity. Nuclear medicine physicists calculate this correction routinely when preparing therapeutic doses from calibrated stock.

Frequently Asked Questions

What is half-life?
Half-life (t₁/₂) is the time required for exactly half of a radioactive isotope (or first-order reactive substance) to decay or react. After one half-life, 50% remains; after two half-lives, 25% remains; after n half-lives, (1/2)ⁿ of the original amount remains. Half-life is a constant for a given isotope, it does not depend on the initial quantity, temperature, or chemical form of the element.
What is the formula for half-life and radioactive decay?
The decay formula is N(t) = N₀ × (1/2)^(t/t₁/₂) = N₀ × e^(−λt), where N₀ is the initial amount, N(t) is the remaining amount at time t, t₁/₂ is the half-life, and λ = ln(2)/t₁/₂ is the decay constant. The two forms are equivalent: (1/2)^(t/t₁/₂) = e^(−λt). The exponential form uses the decay constant λ; the half-life form uses t₁/₂ directly.
What is the decay constant and how is it related to half-life?
The decay constant λ (lambda) is the probability of a single nucleus decaying per unit time. It is related to half-life by λ = ln(2)/t₁/₂ ≈ 0.6931/t₁/₂. A large λ means rapid decay (short half-life); a small λ means slow decay (long half-life). For carbon-14, t₁/₂ = 5,730 years, so λ = 0.6931/5730 = 1.21 × 10⁻⁴ per year. For uranium-238, t₁/₂ = 4.47 × 10⁹ years, so λ = 1.55 × 10⁻¹⁰ per year, extraordinarily slow.
What are the half-lives of common radioactive isotopes?
Polonium-214: 1.6 × 10⁻⁴ seconds (extremely short). Radon-222: 3.82 days. Iodine-131: 8.02 days (used in thyroid cancer treatment). Caesium-137: 30.2 years (Chernobyl and Fukushima contaminant). Carbon-14: 5,730 years (used in radiocarbon dating). Plutonium-239: 24,100 years. Uranium-235: 704 million years. Uranium-238: 4.47 billion years (comparable to Earth's age). Potassium-40: 1.25 billion years (used in geological dating).
How do I use the Half-Life Calculator?
Enter the Initial Amount (N₀) in any unit (grams, atoms, Becquerels, relative units). Enter the Half-Life value. Enter the Elapsed Time in the same units as the half-life, or select a different time unit from the dropdown. The calculator returns the remaining amount, percent remaining, the decay constant λ, and the number of half-lives that have elapsed.
How many half-lives does it take to reduce radioactivity to negligible levels?
After 7 half-lives, approximately 0.78% remains, less than 1%. After 10 half-lives, approximately 0.098% remains. After 20 half-lives, approximately 9.5 × 10⁻⁵% remains. Nuclear waste regulators typically consider 10 half-lives as the practical decay period for low-level radioactive waste (reducing to about 0.1% of initial activity). For iodine-131 (t₁/₂ = 8 days), 10 half-lives = 80 days; for caesium-137 (t₁/₂ = 30.2 years), 10 half-lives = 302 years.
Does half-life apply to first-order chemical reactions as well as radioactive decay?
Yes, half-life is the time for the concentration to fall to half its initial value in any first-order process. For a first-order reaction with rate constant k, t₁/₂ = ln(2)/k = 0.6931/k, independent of initial concentration, just like radioactive decay. This is why first-order reactions are mathematically identical to radioactive decay. For second-order reactions, t₁/₂ = 1/(k[A]₀), it depends on initial concentration. Use the [Rate Constant Calculator](/rate-constant-calculator/) to determine k from concentration data.
What is activity and how is it related to half-life?
Radioactive activity A is the rate of disintegrations per unit time: A = λN = N × ln(2)/t₁/₂. The SI unit is the Becquerel (Bq) = 1 disintegration/second. The older unit is the Curie (Ci) = 3.7 × 10¹⁰ Bq. Activity decreases exponentially with time at the same rate as the number of atoms: A(t) = A₀ × (1/2)^(t/t₁/₂). Medical isotopes are specified by their activity at a reference date; knowing the half-life allows calculation of activity at any later date.
How is half-life used in radiocarbon dating?
Radiocarbon dating uses the half-life of carbon-14 (5,730 years) to determine the age of organic materials. Living organisms maintain a constant ¹⁴C/¹²C ratio (from atmospheric carbon fixation). After death, ¹⁴C decays without replenishment. Measuring the remaining ¹⁴C ratio allows calculation of age: t = (t₁/₂ / ln2) × ln(N₀/N) = (5730/0.6931) × ln(initial ratio/current ratio). This technique is effective for materials up to approximately 50,000 years old (about 8–9 half-lives). The [Radiocarbon Dating Calculator](/radiocarbon-dating-calculator/) specialises in this application.
Is half-life relevant to nuclear medicine in India?
Critically so. India's Nuclear Medicine departments at AIIMS, Tata Memorial Hospital, and major government hospitals use radioactive isotopes for diagnosis and therapy. Technetium-99m (t₁/₂ = 6.01 hours) is the most widely used diagnostic isotope, administered in bone scans, cardiac perfusion imaging, and renal function tests. Because of the short half-life, generators must be shipped daily from the reactor (BARC in Mumbai supplies most of India's Tc-99m generators). Iodine-131 (t₁/₂ = 8.02 days) is used for thyroid cancer treatment at major cancer centres.