What is half-life and how does radioactive decay work?

Half-life is the time required for half of a radioactive substance to decay. Radioactive decay follows an exponential pattern described by the formula N(t) = N₀ × (1/2)^(t/T), where N(t) is the remaining amount, N₀ is the initial amount, t is time elapsed, and T is the half-life period. This means that after one half-life, 50% remains; after two half-lives, 25% remains; after three, 12.5% remains, and so on. The decay is a random process at the atomic level, but becomes predictable for large numbers of atoms. Each radioactive isotope has a unique half-life ranging from fractions of a second to billions of years.

How do I use the half-life calculator?

Using the half-life calculator is simple: First, select your calculation type (remaining amount, time required, or half-life determination). Enter the initial amount of substance, choose your time unit (seconds, minutes, hours, days, or years), and input either the half-life and elapsed time, or the remaining amount depending on what you want to calculate. You can also select from common isotopes like Carbon-14 or Uranium-235 to auto-fill half-life values. Click Calculate to see results including decay curves, milestone tables, and percentage breakdowns. The calculator provides both numerical results and visual representations of the exponential decay process.

What are common isotopes and their half-lives?

Common radioactive isotopes have vastly different half-lives for different applications. Carbon-14 (t½ = 5,730 years) is used for archaeological dating of organic materials up to 50,000 years old. Uranium-235 (t½ = 703.8 million years) is used in nuclear reactors and dating ancient rocks. Uranium-238 (t½ = 4.47 billion years) helps date the Earth itself. For medical applications, Iodine-131 (t½ = 8.02 days) treats thyroid conditions, Technetium-99m (t½ = 6 hours) is used in medical imaging, and Cobalt-60 (t½ = 5.27 years) is used in radiation therapy. Each half-life makes the isotope suitable for specific practical applications.

What is the decay constant and how does it relate to half-life?

The decay constant (λ) is the probability that a single atom will decay per unit time, related to half-life by the formula λ = ln(2)/T, where T is the half-life. The decay constant appears in the exponential decay equation N(t) = N₀ × e^(-λt). A larger decay constant means faster decay and a shorter half-life. For example, if λ = 0.693 per year, the half-life is 1 year. The decay constant is fundamental in nuclear physics because it represents an intrinsic property of the isotope, independent of the amount present. Understanding this relationship helps in converting between different representations of radioactive decay rates used in various scientific fields.

How is radioactive decay used in nuclear medicine?

Nuclear medicine uses radioactive isotopes with specific half-lives for diagnosis and treatment. Short half-life isotopes like Technetium-99m (6 hours) are ideal for imaging because they provide strong signals while clearing from the body quickly, minimizing radiation exposure. Iodine-131 (8 days) treats thyroid cancer by concentrating in thyroid tissue and delivering targeted radiation. Cobalt-60 (5.3 years) and other isotopes treat various cancers through external beam radiation. The half-life must be long enough for medical procedures but short enough to avoid long-term radiation exposure. Calculating decay helps doctors determine proper dosing, timing, and when radioactive materials can be safely disposed of or have decayed to safe levels.

Half-Life Calculator - Calculate Radioactive Decay & Remaining Amount | AICalculator

Understanding Half-Life and Radioactive Decay

Half-life is one of the most important concepts in nuclear physics, describing how quickly radioactive materials decay over time. This fundamental principle has applications ranging from carbon dating ancient artifacts to treating cancer with radiation therapy.

What is Half-Life?

Half-life (t½) is defined as the time required for exactly half of a radioactive substance to undergo decay. This is a constant property for each radioactive isotope, independent of the initial amount or external conditions like temperature or pressure. For example, if you start with 100 grams of a substance with a half-life of 10 years, after 10 years you'll have 50 grams remaining, after 20 years you'll have 25 grams, and after 30 years you'll have 12.5 grams.

The decay process follows an exponential pattern, which means it never truly reaches zero—theoretically, there are always infinitesimally small amounts remaining. However, after about 10 half-lives, less than 0.1% of the original substance remains, which is often considered negligible for practical purposes.

The Exponential Decay Formula

Radioactive decay is described by the exponential decay equation:

N(t) = N₀ × (1/2)^(t/T)

or equivalently

N(t) = N₀ × e^(-λt)

Where:

N(t) = remaining amount at time t
N₀ = initial amount at time 0
t = elapsed time
T = half-life period
λ = decay constant = ln(2)/T ≈ 0.693/T
e = Euler's number (approximately 2.71828)

Both formulas are equivalent, with the first being more intuitive for understanding half-lives, and the second being more common in advanced physics. The decay constant λ represents the probability that a single atom will decay per unit time.

Common Radioactive Isotopes and Their Half-Lives

Different radioactive isotopes have vastly different half-lives, making them suitable for different applications:

Isotope	Half-Life	Primary Use
Carbon-14	5,730 years	Archaeological dating
Uranium-235	703.8 million years	Nuclear fuel, geological dating
Uranium-238	4.47 billion years	Dating Earth's age
Plutonium-239	24,110 years	Nuclear weapons, research
Cobalt-60	5.27 years	Cancer radiation therapy
Iodine-131	8.02 days	Thyroid treatment
Technetium-99m	6 hours	Medical diagnostic imaging
Radon-222	3.82 days	Geological research, home safety

Carbon-14 Dating: Archaeological Applications

One of the most well-known applications of half-life is radiocarbon dating using Carbon-14. Living organisms continuously absorb carbon from the atmosphere, including a small amount of radioactive C-14 alongside stable C-12. While alive, the ratio of C-14 to C-12 remains constant. When an organism dies, it stops taking in new carbon, and the C-14 begins to decay with a half-life of 5,730 years.

By measuring the remaining C-14 in organic materials (wood, bone, textiles, etc.) and comparing it to the expected amount, scientists can calculate when the organism died. For example:

50% remaining = 1 half-life = ~5,730 years old
25% remaining = 2 half-lives = ~11,460 years old
12.5% remaining = 3 half-lives = ~17,190 years old
6.25% remaining = 4 half-lives = ~22,920 years old

Carbon-14 dating is effective for materials between 500 and 50,000 years old. Beyond this range, too little C-14 remains for accurate measurement. For more information on carbon dating methodology, visit the American Physical Society's article on radiocarbon dating.

Medical Applications of Radioactive Decay

Nuclear medicine relies heavily on understanding half-lives to safely use radioactive materials for diagnosis and treatment:

Diagnostic Imaging

Technetium-99m (t½ = 6 hours) is the most widely used medical radioisotope. Its short half-life means it produces strong signals for imaging while clearing from the body quickly, minimizing radiation exposure. Patients receive an injection of Tc-99m attached to specific molecules that concentrate in organs or tissues being examined. After 24 hours (4 half-lives), less than 10% of the radioactivity remains, and after 48 hours it's essentially gone.

Cancer Treatment

Iodine-131 (t½ = 8.02 days) treats thyroid cancer because the thyroid naturally concentrates iodine. The radiation from I-131 decay kills cancer cells while having minimal effect on surrounding tissues. Cobalt-60 (t½ = 5.27 years) provides external beam radiation therapy. Its intermediate half-life makes it practical for medical facilities—long enough that sources don't need frequent replacement, but short enough that they can be safely disposed of after use.

For authoritative information on medical uses of radioactive isotopes, see the National Cancer Institute's guide to radiation therapy.

Geological Dating and Earth Science

Isotopes with extremely long half-lives help scientists date rocks and determine Earth's age:

Uranium-238 (t½ = 4.47 billion years) decays into lead-206 through a series of intermediate steps. By measuring the ratio of U-238 to Pb-206 in rocks, geologists can determine their age. This method has shown that the oldest Earth rocks are about 4.0 billion years old, and meteorites are around 4.5 billion years old, providing our best estimate of when the solar system formed.

Potassium-40 (t½ = 1.25 billion years) decays to argon-40 and is used for potassium-argon dating of volcanic rocks. This technique has been crucial in dating important archaeological sites, particularly in Africa where early human fossils are found in volcanic layers.

For more information on geological dating methods, visit the U.S. Geological Survey's explanation of rock dating techniques.

Nuclear Power and Waste Management

Understanding half-lives is critical for nuclear power generation and managing radioactive waste:

Uranium-235 (t½ = 703.8 million years) and Plutonium-239 (t½ = 24,110 years) are fissile materials used in nuclear reactors. Their long half-lives mean they remain radioactive for geological timescales, creating significant challenges for waste disposal. Nuclear waste containing these isotopes must be isolated from the environment for tens of thousands of years.

Some nuclear waste products have much shorter half-lives. Iodine-131 (t½ = 8 days) and Xenon-133 (t½ = 5 days) from nuclear fission become non-hazardous relatively quickly. After 10 half-lives (about 80 days for I-131), they've decayed to less than 0.1% of their original radioactivity. However, waste also contains long-lived isotopes like Plutonium-239, requiring long-term storage solutions.

Calculation Methods: Forward and Inverse Problems

There are three main types of half-life calculations, each useful for different scenarios:

1. Forward Calculation (Finding Remaining Amount)

Given initial amount, half-life, and elapsed time, find remaining amount:

N(t) = N₀ × (1/2)^(t/T)

Example: Starting with 100 grams of C-14 (t½ = 5,730 years), how much remains after 17,190 years?

Solution: N(17,190) = 100 × (1/2)^(17,190/5,730) = 100 × (1/2)^3 = 12.5 grams

2. Finding Time Required

Given initial amount, half-life, and remaining amount, find elapsed time:

t = T × log(N/N₀) / log(0.5)

or equivalently: t = T × log₂(N₀/N)

Example: An artifact has 25% of its original C-14. How old is it?

Solution: t = 5,730 × log(0.25) / log(0.5) = 5,730 × 2 = 11,460 years

3. Determining Half-Life

Given initial amount, remaining amount, and elapsed time, find half-life:

T = t × log(0.5) / log(N/N₀)

Example: A sample decreases from 1000 to 250 units in 20 days. What's the half-life?

Solution: T = 20 × log(0.5) / log(0.25) = 20 × 0.5 = 10 days

Decay Rate Classification

Radioactive isotopes can be classified by their decay rates:

Fast Decay: Half-life less than 1 day (Tc-99m, I-131) - Used for medical procedures
Medium Decay: Half-life between 1 day and 1 year (Radon-222) - Environmental concerns
Slow Decay: Half-life greater than 1 year (C-14, U-238) - Dating applications

Practical Tips for Using the Calculator

Match your time units consistently (don't mix years and days)
Use scientific notation for very large or small numbers
For archaeological dating, consider that practical measurements become difficult below 1% remaining
In medical applications, assume radioactivity is negligible after 10 half-lives
Save results using the export functions for documentation
Select common isotopes from the database for quick calculations

Safety Considerations

While this calculator helps understand radioactive decay mathematically, working with actual radioactive materials requires:

Proper licensing and training
Radiation monitoring equipment
Appropriate shielding and containment
Following local and national regulations
Understanding ALARA (As Low As Reasonably Achievable) principles

For information about radiation safety, consult the U.S. Nuclear Regulatory Commission or your country's nuclear safety authority.

Related Calculations

For related mathematical and scientific calculations, you might find these calculators helpful:

Exponential Growth Calculator - For growth processes (opposite of decay)
Logarithm Calculator - Essential for inverse decay calculations
Scientific Calculator - For complex mathematical operations
Percentage Calculator - Calculate decay percentages