Half-Life Calculator

Online calculator and formulas to calculate half-life for a given decay rate per period

Half-Life Calculator

Half-Life Calculation

This function calculates either the time period needed for a value to halve at a given decay rate, or the decay rate required to halve in a specified time.

Select Calculation Mode
Input Values
Example: 5 for 5% decay per period
Result precision
Half-Life Calculation Result
Half-Life:
About Half-Life

Half-life is fundamental in nuclear physics and radioactive dating. It's the time needed for any quantity to decay to half its original amount. Unlike exponential decay (any factor), half-life is specifically for a factor of 0.5.

Half-Life Visualization

Half-life shows exponential decay where the amount is cut in half each period.
Higher decay rates result in shorter half-lives.

Half-Life Decay Chart 0% 50% 100% 1st 2nd 3rd 100% Decay Curve Half-Life Points 100%


What is Half-Life?

Half-life is the time required for any quantity to decay to exactly half its initial value:

  • Definition: The time for decay to 50% (factor of 0.5)
  • Special Case: Half-life is exponential decay with factor = 0.5
  • Fixed Ratio: Always reduces to exactly half, regardless of starting amount
  • Exponential Process: Each period reduces by constant percentage
  • Fundamental: Key concept in nuclear physics and radiocarbon dating
  • Predictable: Can model decay chains (1st, 2nd, 3rd half-life)

Mathematical Foundations

The half-life calculation is a special case of exponential decay mathematics:

Calculate Half-Life
t = log(0.5) / log(1 - d/100)

Calculates periods needed to reach 50% at decay rate d%.

Calculate Decay Rate
d = -(e^(ln(0.5)/t) - 1) × 100

Calculates the required decay rate % to halve in t periods.

Half-Life Series and Decay Chains

Half-life has a unique property: after each half-life period passes, the remaining amount is halved again:

Periods Elapsed Half-Lives Passed Remaining Percentage Decay Formula
1 1st 50% 0.5^1 = 0.5
2 2nd 25% 0.5^2 = 0.25
3 3rd 12.5% 0.5^3 = 0.125
5 5th 3.125% 0.5^5 = 0.03125
10 10th 0.098% 0.5^10 ≈ 0.00098
Key Insight

After n half-lives, the remaining amount is 0.5^n (or (1/2)^n) of the original. This predictable relationship makes half-life incredibly useful for radiocarbon dating and decay prediction.

Applications of Half-Life

Half-life is essential in many fields:

Nuclear Physics & Radiochemistry
  • Radioactive element decay
  • Nuclear waste disposal planning
  • Radiation safety calculations
  • Reactor management
Archaeology & Dating
  • Radiocarbon (C-14) dating of artifacts
  • Determining age of fossils
  • Dating ancient materials
  • Archaeological age verification
Medicine & Pharmacology
  • Drug elimination from body
  • Medical imaging isotopes (PET, SPECT)
  • Medication dosing schedules
  • Radiation therapy dosing
Environmental Science
  • Pollutant degradation in environment
  • Soil and water contamination tracking
  • Atmospheric compound breakdown
  • Bioaccumulation modeling

Calculation Examples

Example 1: Carbon-14 Half-Life
Given

Carbon-14 decays with a half-life of 5,730 years. What is the annual decay rate?

Question: What percentage of Carbon-14 decays per year?

Solution
\[d = -(e^{\ln(0.5)/5730} - 1) \times 100\] \[d = -(e^{-0.000121} - 1) \times 100 = 0.0121\%\]
Result

Carbon-14 decays at approximately 0.0121% per year. After 5,730 years, half remains.

Example 2: Medicinal Half-Life
Given

A medication has a half-life of 6 hours. What is the hourly elimination rate from the body?

Question: What percentage of drug is eliminated per hour?

Solution
\[d = -(e^{\ln(0.5)/6} - 1) \times 100\] \[d = -(e^{-0.1155} - 1) \times 100 = 10.92\%\]
Result

The drug is eliminated at 10.92% per hour. After 6 hours, half the original dose remains in the body.

Example 3: Dating Ancient Artifact
Given

An archaeological sample has only 12.5% of its original Carbon-14 remaining (C-14 half-life = 5,730 years).

Question: How old is the artifact?

Solution
\[\text{12.5\% = 0.125 = 0.5}^3\] \[\text{Therefore: 3 half-lives have passed}\] \[\text{Age} = 3 \times 5,730 = 17,190 \text{ years}\]
Interpretation

The artifact is approximately 17,190 years old. The 12.5% remaining matches exactly 3 half-lives (0.5^3 = 0.125).

Key Points About Half-Life

Core Concepts
  • Definition: Time to decay to 50% (factor = 0.5)
  • Special Case: Exponential decay with fixed reduction ratio
  • Predictable: Always halves in the same time period
  • Series: After n half-lives, (0.5)^n remains
Practical Applications
  • Radiocarbon dating (archaeology)
  • Medical isotope imaging
  • Drug metabolism calculations
  • Radiation safety planning
Summary

Half-life is a fundamental concept describing exponential decay where material reduces to half its amount in a fixed time period. From radioactive isotopes used in archaeology and medicine to medications eliminated from the body, half-life provides an intuitive way to understand decay processes. The predictable nature of half-life—particularly the series property where after n half-lives, (0.5)^n remains—makes it invaluable for dating ancient artifacts, medical dosing, and nuclear physics. Whether calculating decay rates or estimating ages using radiocarbon dating, understanding half-life is essential for accurate predictions.