Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This pattern results in a curve that drops steeply at first and then levels out as the remaining quantity shrinks. Calculating this rate of reduction is fundamental across many scientific disciplines, providing a reliable method for determining the age of ancient materials or monitoring substances in a controlled environment. The calculation of exponential decay relies on the concept of half-life.
Understanding Half-Life
Half-life, symbolized as \(T_{1/2}\), is the specific time interval required for exactly half of a substance to undergo decay. This concept is most commonly applied to the radioactive decay of unstable atomic nuclei, where the half-life is a fixed, inherent property of that particular isotope. It is entirely unaffected by external conditions like temperature or pressure.
The predictability of this time constant makes it the foundation for all decay calculations in physics and chemistry. While it is impossible to predict when any single atom will decay, the behavior of a large collection of atoms is highly reliable. After one half-life, 50% of the original sample remains. After a second half-life, 50% of the remaining amount (25% of the original) is left. This consistent reduction by half forms a geometric progression, which defines exponential decay.
Calculating Decay Over Whole Half-Lives
For a simple understanding of decay, the easiest calculation involves determining the remaining amount after a number of whole half-lives have passed. This method avoids complex mathematics and relies only on repeated multiplication by one-half. To use this method, one must know the initial amount of the substance and its specific half-life duration.
Imagine a radioisotope sample starts with 100 grams and has a half-life of 5 hours. After the first 5 hours (one half-life), the remaining amount is \(100 text{ grams} times 1/2\), which equals 50 grams.
After 10 total hours (two half-lives), the remaining 50 grams is reduced by half again, leaving 25 grams. Extending this calculation to a third half-life (15 total hours), the 25 grams is halved, resulting in 12.5 grams. The pattern can be expressed as a fraction of the original amount: \(1/2\) after one half-life, \(1/4\) after two, \(1/8\) after three, and so on.
This calculation can be generalized by the formula \(N = N_0 times (1/2)^n\), where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the total number of half-lives that have elapsed. For example, after four half-lives, the remaining fraction is \((1/2)^4\), or \(1/16\) of the initial quantity. This method is effective for estimating decay when the total time is a perfect multiple of the half-life.
The Exponential Decay Formula
When the elapsed time is not a perfect multiple of the half-life, a more generalized mathematical model is necessary to calculate the remaining quantity precisely. This is the exponential decay formula, commonly written as \(N(t) = N_0 e^{-lambda t}\). This equation accounts for continuous decay over any time period.
In this formula, \(N(t)\) represents the quantity remaining after time \(t\), and \(N_0\) is the initial quantity. The constant \(e\) is Euler’s number, approximately \(2.718\), which serves as the base for the natural logarithm and is standard in continuous exponential modeling. The term \(lambda\) is the decay constant, which quantifies the probability of decay per unit of time.
The decay constant (\(lambda\)) and the half-life (\(T_{1/2}\)) are mathematically linked, providing a way to convert the half-life into the constant required for the exponential function. The relationship is defined by the equation \(T_{1/2} = ln(2)/lambda\). Since \(ln(2)\) is approximately \(0.693\), the decay constant is calculated as \(lambda approx 0.693/T_{1/2}\). Once \(lambda\) is determined, it can be substituted into the primary decay equation to calculate the remaining material at any point in time \(t\).
Why Calculating Decay Matters
Applying these decay calculations is a foundational tool in several specialized fields. One recognized application is radiometric dating, used by archaeologists and geologists to determine the age of samples. Carbon-14 dating relies on the known half-life of 5,730 years to measure the remaining Carbon-14 in formerly living organic matter, allowing scientists to date artifacts up to approximately 50,000 years old.
The medical field also depends heavily on precise half-life calculations, particularly in nuclear medicine and therapy. Radioactive isotopes, such as Technetium-99m, are used as tracers for diagnostic imaging of organs like the brain, heart, and bones. Technetium-99m has a short half-life of about six hours, which is long enough for the imaging procedure but short enough to decay quickly, minimizing the patient’s radiation exposure.
Similarly, isotopes like Iodine-131 are used in targeted treatments, such as for thyroid cancer. Calculating its half-life of eight days is necessary to determine the proper dosage and to predict the rate at which the radioisotope will be eliminated from the body. These calculations ensure the administered dose delivers the intended therapeutic effect while maintaining patient safety.