Exponential Decay Law
The Exponential Decay Law describes a process in which a quantity decreases over time at a rate proportional to its current value. This phenomenon is characterized by a consistent percentage reduction, leading to a progressively smaller absolute decrease as the quantity diminishes. A common illustration of this principle is found in the evaporation of water, where, for example, 1% of the remaining amount evaporates daily, resulting in a continually decreasing volume that becomes harder to detect over time.
A significant application of exponential decay is observed in the domain of radioactive materials. Radioactive substances, such as uranium, undergo a transformation known as radioactive decay, occurring at a constant exponential rate. This decay is quantitatively described using the concept of "half-life," which indicates the time required for half of the substance to decay. Each radioactive isotope has its unique half-life; for instance, uranium-238 has a half-life of about 4.5 billion years.
Mathematically, exponential decay is represented through a differential equation, allowing for precise calculations of the remaining quantity over time. As time progresses, the quantity approaches zero, with the decrements becoming smaller, making it an important concept in various scientific fields, including physics and chemistry. Understanding exponential decay is crucial for applications ranging from nuclear physics to population dynamics.
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Exponential Decay Law
The exponential decay law describes a change in a quantity over time. More specifically, the word "decay" refers to a decrease in the quantity that is proportional to the quantity itself, during a fixed unit of time. For example, if 1 liter of water were left out in the sun to evaporate under conditions such that 1% of the water evaporated each day, there would be 99 centiliters remaining after the first day because 1 liter equals 100 centiliters and 1% of 100 is 1. During the second day, the amount that would evaporate would again be 1% of the total: 99 centiliters x 0.01 = 0.99 centiliters. Thus, after the second day the amount remaining would be 99 − 0.99 = 98.01 centiliters.
As the quantity reduces in size, so does the amount by which it decays during each period (though the proportion between the two remains constant). Depending on the nature of the quantity, it can become more and more difficult to detect the amount by which it is reduced. In the example used here, it would first be easy to notice a reduction of 1 centiliter in the volume of the water, but after many days had passed it would be much more difficult to detect because the amount evaporating would be only 1/100th of the few drops of water remaining.
Overview
One of the best-known examples of the exponential decay law concerns radioactive materials. Radioactive materials, such as uranium, have somewhat unstable atomic structures, causing them to gradually break down into different elements in a process called radioactive decay. For each radioactive substance, this process of decay happens at a constant rate of exponential decay. To make it easier to understand and discuss radioactive decay, scientists have come to use the term "half life" to describe the amount of time it takes a particular amount of a radioactive substance to decrease in quantity by 50%, as its atoms essentially break apart into different elements. Each radioactive substance has its own half life; that is, it takes different radioactive substances different amounts of time to decay, according to their atomic characteristics.
For example, uranium-238 has a half life of approximately 4.5 billion years. So, taking a 10 gram sample of uranium-238, it would take almost 4.5 billion years for 5 grams of the uranium-238 to decay into uranium-234.Exponential decay is represented mathematically through the use of a differential equation, since it is essentially a process by which a quantity, over time, approaches closer and closer to zero, by increments which become smaller and smaller with each time interval. The general formula for this relationship is
where N is the quantity that is undergoing exponential decay and
is the exponential decay constant (i.e., the rate of decay for each time period). The time period being examined is represented by t, so in the example of water evaporation, determining how much water had evaporated after three days would require t to have a value of 3.
Bibliography
Briet, Philippe, and Claudio Fernández. "Exponential Decay and Resonances in a Driven System." Journal of Mathematical Analysis & Applications 396.2 (2012): 513-22.
Mall, Alison L., and Mike Risinger. "Modeling Exponential Decay." Mathematics Teacher 107.5 (2013): 400.
Reed, Martin B. Core Maths for the Biosciences. New York: Oxford UP, 2011.
Saxena, Pratiksha. Modeling and Simulation. Oxford: Alpha, 2011.
Wheater, Carolyn C. Practice Makes Perfect Algebra. New York: McGraw, 2010.