Exponential Decay and Half-Life

Half-life is a fundamental concept used to describe any process that follows exponential decay. It is defined as the time required for a quantity to reduce to half of its initial value. While it is most famously associated with the radioactive decay of elements in physics and chemistry, the principle of half-life also applies to many other fields, such as pharmacology (describing how quickly a drug is eliminated from the body), environmental science (the degradation of pollutants), and even finance (the depreciation of certain assets). It provides a predictable and consistent way to measure the rate of decay, regardless of the initial amount of the substance.

The key characteristic of half-life is that it is constant for a given substance or process. For example, the half-life of Carbon-14 is approximately 5,730 years. This means that if you start with 100 grams of Carbon-14, after 5,730 years you will have 50 grams left. After another 5,730 years (a total of 11,460 years), you will have half of that amount remaining, which is 25 grams. This predictable, exponential pattern is what makes half-life such a powerful tool. It allows archaeologists to date ancient organic artifacts through radiocarbon dating, and it enables doctors to determine the correct dosing schedule for medications to maintain a therapeutic level in the bloodstream. This calculator is a versatile tool that allows you to explore this relationship by solving for any of the variables in the half-life equation.

The Half-Life Formula

The mathematical relationship for half-life is described by the following formula:

N(t) = N₀ * (1/2)^(t / T½)

Where:

• N(t) is the final quantity of the substance remaining after a certain amount of time.
• N₀ is the initial quantity of the substance.
• t is the total time elapsed.
• T½ is the half-life of the substance.

This calculator can algebraically rearrange this formula to solve for any one of the four variables, provided the other three are known. For instance, to solve for the time elapsed (t), the formula becomes: t = T½ * log₂(N₀ / N(t)).

Applications of Half-Life

• Radiocarbon Dating (Archaeology): All living organisms absorb Carbon-14 from the atmosphere. When an organism dies, it stops absorbing C-14, and the existing C-14 begins to decay with a half-life of 5,730 years. By measuring the ratio of C-14 to the stable isotope C-12 in an ancient artifact (like bone or wood), scientists can calculate how long it has been since the organism died.
• Medical Imaging and Treatment (Medicine): Radioactive isotopes (radioisotopes) with short half-lives are used as tracers in medical imaging techniques like PET scans. They are chosen to have a half-life long enough to perform the scan but short enough that they decay and are eliminated from the body quickly to minimize radiation exposure.
• Pharmacokinetics (Pharmacology): The biological half-life of a drug is the time it takes for the concentration of the drug in the bloodstream to be reduced by half. This is a critical parameter that helps doctors determine how often a patient needs to take a medication to maintain a steady, effective level in the body.
• Geological Dating (Geology): For dating rocks and the Earth itself, scientists use isotopes with much longer half-lives. For example, the decay of Potassium-40 to Argon-40 has a half-life of 1.25 billion years, and the decay of Uranium-238 to Lead-206 has a half-life of 4.47 billion years. These methods allow geologists to determine the age of ancient rock formations.