Half-Life Calculator
Calculate the remaining amount of a substance after radioactive decay or any first-order process. Enter the initial amount, half-life period, and elapsed time to determine how much remains. This calculator also provides the decay constant, fraction remaining, and number of half-lives elapsed. See also our Radioactive Decay Calculator and Exponential Growth Calculator for related computations.
How to Calculate Half-Life Decay
Half-life is the time required for a quantity to reduce to half its initial value. The concept is most commonly associated with radioactive decay, but it applies to any process that follows first-order kinetics, including drug metabolism, chemical reactions, and even the decay of internet memes. Understanding half-life calculations is essential in nuclear physics, pharmacology, archaeology (carbon dating), and environmental science.
- Identify the initial amount (N₀) of the substance.
- Determine the half-life (t½) of the substance from reference data.
- Determine the elapsed time (t) since the process began.
- Calculate the number of half-lives: n = t / t½.
- Calculate remaining amount: N(t) = N₀ × (1/2)^n.
- Optionally calculate the decay constant: λ = ln(2) / t½ ≈ 0.693 / t½.
After one half-life, 50% remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains. The pattern continues exponentially — the substance never completely disappears mathematically, but after about 10 half-lives (less than 0.1% remaining), it is generally considered negligible for practical purposes.
Half-Life Formula
N(t) = N₀ × (1/2)^(t/t½)
Equivalent exponential form:
N(t) = N₀ × e^(-λt)
Decay constant:
λ = ln(2) / t½ = 0.693147 / t½
Finding half-life from decay constant:
t½ = ln(2) / λ = 0.693147 / λ
Number of half-lives:
n = t / t½
Where:
N(t) = amount remaining at time t
N₀ = initial amount
t½ = half-life period
λ = decay constant
t = elapsed time
The half-life formula is derived from the first-order rate law. The exponential decay function N(t) = N₀e^(-λt) describes how the quantity decreases continuously over time. Setting N(t) = N₀/2 and solving for t gives t½ = ln(2)/λ. The two forms of the equation (using 1/2 raised to a power, or using e raised to a negative power) are mathematically equivalent and can be used interchangeably.
Example Calculation
Problem: A sample contains 100 g of Carbon-14 (t½ = 5,730 years). How much remains after 11,460 years?
Given:
• N₀ = 100 g
• t½ = 5,730 years
• t = 11,460 years
Solution:
Number of half-lives: n = 11,460 / 5,730 = 2
N(t) = 100 × (1/2)² = 100 × 0.25 = 25 g
Decay constant: λ = 0.693147 / 5,730 = 1.2097 × 10⁻⁴ per year
Answer: After 11,460 years (2 half-lives), 25 g remains. 75% has decayed.
Verification: Using exponential form: N = 100 × e^(-1.2097×10⁻⁴ × 11460) = 100 × e^(-1.3863) = 100 × 0.25 = 25 g ✓
Half-Life Reference Table
| Isotope | Half-Life | Decay Type | Application |
|---|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | β⁻ | Archaeological dating |
| Uranium-238 (²³⁸U) | 4.47 billion years | α | Geological dating |
| Potassium-40 (⁴⁰K) | 1.25 billion years | β⁻/EC | Rock dating |
| Iodine-131 (¹³¹I) | 8.02 days | β⁻ | Thyroid treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | γ | Medical imaging |
| Cobalt-60 (⁶⁰Co) | 5.27 years | β⁻/γ | Radiation therapy |
| Radon-222 (²²²Rn) | 3.82 days | α | Indoor air quality |
| Plutonium-239 (²³⁹Pu) | 24,100 years | α | Nuclear fuel |
| Tritium (³H) | 12.32 years | β⁻ | Luminous paint, dating |
| Strontium-90 (⁹⁰Sr) | 28.8 years | β⁻ | Nuclear fallout |
Frequently Asked Questions
What is half-life?
Half-life (t½) is the time required for a quantity to decrease to half its initial value. In radioactive decay, it is the time for half the atoms in a sample to undergo nuclear transformation. The concept applies to any exponential decay process, including drug elimination from the body, chemical reaction kinetics, and the degradation of pollutants in the environment.
Does the amount of substance affect the half-life?
No, half-life is independent of the initial amount for first-order processes like radioactive decay. Whether you start with 1 gram or 1 kilogram, the half-life remains the same. This is because radioactive decay is a statistical process — each atom has the same probability of decaying per unit time, regardless of how many other atoms are present.
How is carbon-14 dating used?
Carbon-14 dating works because living organisms continuously exchange carbon with the atmosphere, maintaining a constant ¹⁴C/¹²C ratio. When an organism dies, it stops absorbing ¹⁴C, and the existing ¹⁴C decays with a half-life of 5,730 years. By measuring the remaining ¹⁴C fraction, scientists can determine when the organism died. The method is effective for samples up to about 50,000 years old.
What is the decay constant?
The decay constant (λ) represents the probability of decay per unit time for a single atom. It is related to half-life by λ = ln(2)/t½ ≈ 0.693/t½. A larger decay constant means faster decay (shorter half-life). The activity (rate of decay) of a sample is A = λN, where N is the number of atoms present. Activity decreases exponentially with the same half-life as the amount.
How many half-lives until a substance is gone?
Mathematically, exponential decay never reaches exactly zero. However, after 10 half-lives, only 0.098% remains (about 1/1024 of the original). In practice, after 7 half-lives (less than 1% remaining), a substance is often considered negligible. For radioactive waste disposal, the rule of thumb is that material is safe after 10 half-lives, though regulations vary by isotope and application.
What is biological half-life vs physical half-life?
Physical half-life is the time for radioactive decay alone. Biological half-life is the time for the body to eliminate half of a substance through metabolism and excretion. The effective half-life combines both: 1/t_eff = 1/t_physical + 1/t_biological. For example, Iodine-131 has a physical half-life of 8 days and a biological half-life of about 80 days in the thyroid, giving an effective half-life of about 7.3 days.
Half-Life in Science and Medicine
The concept of half-life permeates many areas of science and has profound practical implications. In nuclear medicine, the choice of radioisotope for diagnostic imaging or therapy depends critically on its half-life. Technetium-99m, with a half-life of just 6 hours, is ideal for imaging because it provides enough time for the procedure while minimizing patient radiation exposure. Iodine-131, with an 8-day half-life, is used for thyroid cancer treatment because it delivers a therapeutic radiation dose over several days.
In pharmacology, the half-life of a drug determines dosing frequency. A drug with a 4-hour half-life might need to be taken every 4-6 hours to maintain therapeutic levels, while a drug with a 24-hour half-life can be taken once daily. The concept of steady state — reached after approximately 5 half-lives of regular dosing — is fundamental to pharmacokinetics and drug therapy optimization.
Environmental science uses half-life to predict how long pollutants persist in ecosystems. DDT has an environmental half-life of 2-15 years, explaining why it persists decades after being banned. Radioactive contamination from nuclear accidents (like Chernobyl or Fukushima) is assessed using the half-lives of released isotopes — Cesium-137 (30 years) and Strontium-90 (29 years) are the primary long-term concerns.
In geology, radioactive dating using long-lived isotopes allows scientists to determine the age of rocks and the Earth itself. Uranium-lead dating (using U-238 with a 4.47 billion year half-life) has established the age of the Earth at approximately 4.54 billion years. Potassium-argon dating is used for volcanic rocks, while carbon-14 dating is limited to organic materials less than about 50,000 years old.
Nuclear power and waste management are dominated by half-life considerations. Spent nuclear fuel contains isotopes with half-lives ranging from seconds to millions of years. The challenge of nuclear waste disposal is that some components (like Plutonium-239 with a 24,100-year half-life) remain hazardous for hundreds of thousands of years, requiring geological repositories designed to contain waste for timescales far exceeding human civilization.