Half-Life Calculator
Calculate radioactive decay, remaining quantity, half-life, or elapsed time. Includes exponential decay charts and common isotope reference.
Formula:N(t) = N₀ × (1/2)^(t/t½)
Half-Life Results
Remaining
250.0000
units
Half-Lives Elapsed2.0000
% Remaining25.0000%
Decay Constant (λ)1.2097e-4
Half-Life Results
Remaining
250.0000
units
Half-Lives Elapsed2.0000
% Remaining25.0000%
Decay Constant (λ)1.2097e-4
Try These Isotope Examples
Quick-start with common scenarios
Practice Half-Life Calculations
Test your skills with practice problems
Practice with 4 problems to test your understanding.
What do you want to calculate?
Input Values
Result
Initial (N₀)
1,000
Remaining (N)
250
Half-Life (t½)
5,730 years
Time (t)
11,460 years
After 2.00 half-lives
25.0000% remaining
Formula & Calculation
N(t) = N₀ × (½)^(t/t½)
N(t) = 1,000 × (0.5)^(11460/5730)
N(t) = 1,000 × (0.5)^2.0000
N(t) = 1,000 × 0.25000000
N(t) = 250.0000
Decay Constant
λ = ln(2) / t½ = 1.2097e-4
Alternative Formula
N(t) = N₀ × e^(-λt)
Exponential Decay Chart
Shows percentage remaining vs number of half-lives elapsed
Decay Progress Table
| Half-Lives | Time | Remaining | % Remaining | Decayed |
|---|---|---|---|---|
| 0 | 0 | 1000.00 | 100.0000% | 0.0000% |
| 1 | 5,730 | 500.00 | 50.0000% | 50.0000% |
| 2 | 11,460 | 250.00 | 25.0000% | 75.0000% |
| 3 | 17,190 | 125.00 | 12.5000% | 87.5000% |
| 4 | 22,920 | 62.50 | 6.2500% | 93.7500% |
| 5 | 28,650 | 31.25 | 3.1250% | 96.8750% |
| 6 | 34,380 | 15.63 | 1.5625% | 98.4375% |
| 7 | 40,110 | 7.81 | 0.7813% | 99.2188% |
| 8 | 45,840 | 3.91 | 0.3906% | 99.6094% |
| 9 | 51,570 | 1.95 | 0.1953% | 99.8047% |
| 10 | 57,300 | 0.98 | 0.0977% | 99.9023% |
Common Radioactive Isotopes
Click an isotope to use its half-life
?How Do You Calculate Half-Life?
Half-life is the time for half of a substance to decay. Formula: N(t) = N0 * (1/2)^(t/t1/2), where N0 is initial amount, t is elapsed time, t1/2 is half-life. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. The decay constant k = ln(2)/t1/2 approximately equals 0.693/t1/2.
What is Half-Life?
Half-life is the time required for exactly half of a substance to decay or transform. In radioactive decay, it describes how long until half the atoms have undergone nuclear decay. The concept applies to any exponential decay process including drug metabolism, chemical reactions, and population dynamics.
Key Facts About Half-Life
- Half-life: time for half the substance to decay
- Formula: N(t) = N0 * (1/2)^(t/t1/2)
- Decay constant: k = ln(2)/t1/2 approximately equals 0.693/t1/2
- After n half-lives: (1/2)^n of original remains
- Exponential decay: quantity never reaches zero
- Carbon-14 half-life: ~5,730 years (used in dating)
- Medical isotopes have short half-lives (hours to days)
- Also applies to drug elimination, population decline
Quick Answer
Half-life is the time for half of a substance to decay. Formula: N(t) = N0 * (1/2)^(t/t1/2), where N0 is initial amount, t is elapsed time, t1/2 is half-life. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. The decay constant k = ln(2)/t1/2 approximately equals 0.693/t1/2.
Frequently Asked Questions
Half-life (t½) is the time required for a quantity to reduce to half its initial value. It's constant regardless of the amount present. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. Used in radioactive decay, pharmacology, and chemistry.
N(t) = N₀ × (1/2)^(t/t½), where N(t) = remaining amount, N₀ = initial amount, t = elapsed time, t½ = half-life. Alternative: N(t) = N₀ × e^(-λt), where λ = ln(2)/t½ is the decay constant.
Carbon-14 has a half-life of 5,730 years. Living organisms maintain constant C-14 levels. After death, C-14 decays. By measuring remaining C-14 compared to living organisms, we calculate time since death. Effective up to ~50,000 years.
Biological half-life is the time for half of a substance to be eliminated from the body through metabolism and excretion. Different from radioactive half-life. Used in pharmacology to determine drug dosing schedules.
Mathematically, exponential decay approaches but never reaches zero. In practice, after ~7 half-lives, less than 1% remains. After 10 half-lives, ~0.1% remains. Eventually, individual atom behavior becomes relevant (quantum effects).
Last updated: 2025-01-15
Related Math Tools
Explore similar calculators
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Half-Life Results
Remaining
250.0000
units
Half-Lives Elapsed2.0000
% Remaining25.0000%
Decay Constant (λ)1.2097e-4