Half-Life Calculator

Half-life is the time required for exactly half of a substance to decay or transform into another form. In radioactive decay, it's the period it takes for half of the radioactive atoms in a sample to undergo nuclear decay. The process follows an exponential decay pattern, meaning that after each half-life period, 50% of the remaining substance decays. For example, if you start with 100g of a radioactive material with a 10-year half-life, after 10 years you'll have 50g, after 20 years 25g, after 30 years 12.5g, and so on. The decay rate remains constant regardless of external conditions like temperature or pressure. Use our time calculator to help with half-life time period calculations.

The decay constant (λ, lambda) is a probability rate that describes how quickly a radioactive substance decays. It represents the probability per unit time that a given atom will decay. The decay constant and half-life are inversely related through the formula: λ = ln(2)/t½ ≈ 0.693/t½. A larger decay constant means faster decay and a shorter half-life. The decay constant is essential in the exponential decay equation N(t) = N₀ × e^(-λt), where N₀ is the initial amount and t is time. This relationship allows scientists to convert between half-life (which is easier to understand) and decay constant (which is more useful in mathematical calculations). Our calculator automatically computes the decay constant when you perform half-life calculations.

Carbon-14 dating is an archaeological technique that uses the half-life of Carbon-14 (5,730 years) to determine the age of organic materials up to about 50,000 years old. Living organisms constantly absorb Carbon-14 from the atmosphere, maintaining a steady ratio with stable carbon isotopes. When an organism dies, it stops absorbing Carbon-14, and the existing Carbon-14 begins to decay. By measuring the remaining Carbon-14 in a sample and comparing it to the expected initial amount, scientists can calculate how many half-lives have passed and thus determine the age of the specimen. For instance, if a sample has 25% of the expected Carbon-14, approximately 2 half-lives (11,460 years) have elapsed. This method has been crucial for dating archaeological artifacts, fossils, and historical events. Use our percentage calculator to determine what percentage of the original substance remains.

In pharmacology, the biological half-life of a medication is the time it takes for the body to eliminate half of the drug through metabolism and excretion. This concept is crucial for determining proper dosing schedules. For example, caffeine has a half-life of 5-6 hours in most adults, meaning if you consume 200mg of caffeine, approximately 100mg remains after 5-6 hours, 50mg after 10-12 hours, and so on. Medications are typically considered eliminated from the body after 5-7 half-lives, when less than 3% remains. Understanding half-life helps doctors determine how often patients should take medications to maintain therapeutic levels, avoid toxic accumulation, and predict when a drug will be cleared from the system. Factors like age, liver function, kidney function, and genetics can affect medication half-life in different individuals.

The exponential decay formula describes how a quantity decreases over time at a rate proportional to its current value. There are two equivalent forms: N(t) = N₀ × (1/2)^(t/t½) and N(t) = N₀ × e^(-λt), where N(t) is the amount at time t, N₀ is the initial amount, t½ is the half-life, λ is the decay constant, and e is Euler's number (approximately 2.71828). The first form is more intuitive as it directly shows the halving process, while the second form is preferred in scientific calculations. Both formulas produce identical results because they're mathematically equivalent through the relationship λ = ln(2)/t½. This exponential nature means the decay rate is always proportional to the current amount, which is why radioactive materials never completely disappear but approach zero asymptotically. Our calculator uses these formulas to perform all half-life computations accurately.

While radioactive substances theoretically never completely disappear due to exponential decay, they become practically negligible after a certain number of half-lives. After 10 half-lives, less than 0.1% (1/1024) of the original substance remains. In medicine, drugs are considered eliminated after 5-7 half-lives (when 3% or less remains). For radiation safety, 10 half-lives is often used as the threshold for considering a radioactive source effectively gone. The percentage remaining after each half-life follows this pattern: 1 half-life = 50%, 2 = 25%, 3 = 12.5%, 4 = 6.25%, 5 = 3.125%, 6 = 1.56%, 7 = 0.78%, 8 = 0.39%, 9 = 0.20%, 10 = 0.098%. The specific threshold depends on the application and acceptable risk level. Use our log calculator for advanced exponential decay calculations.

Radioactive isotopes have vastly different half-lives ranging from fractions of seconds to billions of years. Carbon-14 (5,730 years) is used for archaeological dating. Uranium-238 (4.5 billion years) is used to date the Earth and rocks. Iodine-131 (8.02 days) is used in medical imaging and thyroid treatments. Cobalt-60 (5.27 years) is employed in cancer radiation therapy. Radon-222 (3.82 days) is a concern in indoor air quality. Technetium-99m (6.01 hours) is the most common isotope in medical diagnostic imaging. Plutonium-239 (24,110 years) is relevant in nuclear energy and weapons. Strontium-90 (28.8 years) is a nuclear fission product of concern. Tritium (12.3 years) is used in self-illuminating devices. Polonium-210 (138 days) is highly radioactive and toxic. Our reference table provides half-lives for these and other common isotopes.

For radioactive decay, the half-life is an intrinsic nuclear property that cannot be changed by external conditions like temperature, pressure, chemical bonding, or physical state. This is because radioactive decay occurs in the atomic nucleus, which is unaffected by these external factors. This constancy makes radioactive half-lives extremely reliable for dating and other applications. However, there are rare exceptions: electron capture decay can be very slightly affected by extreme changes in electron density (chemical environment), though these effects are negligible in practical applications. In contrast, biological half-lives of medications can be significantly affected by factors like metabolism rate, kidney function, liver function, age, and interactions with other substances. It's important to distinguish between nuclear half-life (constant) and biological half-life (variable).

Radiometric dating of rocks uses the known half-lives of radioactive isotopes to determine when rocks formed. The most common methods include Uranium-Lead dating (for rocks billions of years old), Potassium-Argon dating (for volcanic rocks 100,000 to billions of years old), and Rubidium-Strontium dating (for igneous and metamorphic rocks). When molten rock crystallizes, it incorporates radioactive parent isotopes but excludes decay products. Over time, the parent isotope decays into the daughter isotope at a known rate. By measuring the ratio of parent to daughter isotopes and knowing the half-life, scientists can calculate how many half-lives have passed since the rock crystallized. For example, if a rock has equal amounts of Uranium-238 and its decay product Lead-206, one half-life (4.5 billion years) has passed. Multiple dating methods are often used together to cross-verify ages and ensure accuracy.

Half-life (t½) and mean lifetime (τ, tau) are two different but related ways to characterize exponential decay. Half-life is the time for half of the substance to decay, while mean lifetime is the average time an individual atom exists before decaying (mathematically, the time for the quantity to reduce to 1/e or about 36.8% of its initial value). They're related by the formula: τ = t½/ln(2) ≈ 1.443 × t½, meaning the mean lifetime is always about 44% longer than the half-life. For example, if a substance has a half-life of 10 years, its mean lifetime is about 14.4 years. Half-life is more commonly used because it's more intuitive (50% is easier to visualize than 36.8%), while mean lifetime is sometimes preferred in theoretical physics because it's the reciprocal of the decay constant (τ = 1/λ). Both describe the same decay process from different perspectives.