Radioactive decay is a spontaneous process where an unstable atomic nucleus transforms into a more stable configuration by releasing energy and particles, known as radiation. Elements undergoing this transformation are called radionuclides, and the process continues until a stable, non-radioactive state is achieved. This transformation causes the parent nucleus to change into a different element, the decay product. Although the decay of a single atom is random, the overall rate of decay for a massive collection of identical atoms is highly consistent and can be described using precise mathematical rules.
Understanding Half-Life
Half-life, symbolized as \(T_{1/2}\), is a fundamental concept in measuring the rate of radioactive decay, representing the time it takes for exactly half of the radioactive nuclei in any given sample to undergo decay. After one half-life period has passed, 50% of the original radioactive material remains. After a second half-life passes, the remaining 50% is again halved, leaving 25% of the original sample.
This characteristic time interval is unique to every specific radioactive isotope. The half-life is constant and is not influenced by external factors like temperature, pressure, or the initial amount of the substance. Half-lives exhibit an enormous range, from fractions of a second for highly unstable isotopes to billions of years for primordial radionuclides.
The predictability of the half-life stems from the statistical nature of the decay process when dealing with a large population of atoms. While it is impossible to know when any single atom will decay, the sheer number of atoms in a macroscopic sample ensures that the average decay rate is reliable. This reliable rate allows scientists to use half-life in applications such as radioactive dating to determine the age of ancient materials. Half-life provides the simplest method for calculating how much material remains when the time is a clean multiple of the half-life.
Step-by-Step Decay Calculation Using Half-Lives
Calculating the amount of radioactive material remaining is straightforward when the elapsed time corresponds to a whole number of half-lives. This simple method relies on repeatedly dividing the remaining quantity by two for each half-life that passes, focusing only on the initial amount and the number of half-lives that have occurred.
To begin, you must determine how many half-lives have passed by dividing the total elapsed time by the isotope’s known half-life period. For example, if a hypothetical isotope has a half-life of 10 days, and 40 days have passed, then four half-lives have occurred (\(40 \text{ days} / 10 \text{ days per half-life} = 4\) half-lives).
Next, the remaining fraction of the substance is calculated by starting with the initial amount (represented as 1) and dividing it by two for each half-life. After the first half-life, half the substance remains (\(\frac{1}{2}\)); after the second, half of that remains (\(\frac{1}{4}\)); and after the third, half of that remains (\(\frac{1}{8}\)). This process is represented by the fraction \(\frac{1}{2^n}\), where \(n\) is the number of half-lives.
Consider a sample of 80 grams of the hypothetical isotope with a 10-day half-life, where 40 days have passed. After the first 10 days (1 half-life), 40 grams remain; after 20 days (2 half-lives), 20 grams remain; and after 30 days (3 half-lives), 10 grams remain. Finally, after 40 days (4 half-lives), only 5 grams of the original radioactive material are left.
The Exponential Decay Formula
For situations where the elapsed time is not an exact multiple of the half-life, the exponential decay formula provides a more precise mathematical model. This formula describes the continuous decay of the radioactive material over any given time interval. The standard form of this equation is \(N(t) = N_0 e^{-\lambda t}\), which models the amount of radioactive nuclei remaining at time \(t\).
In this formula, \(N(t)\) represents the amount of the radioactive substance remaining after time \(t\) has passed, and \(N_0\) is the initial amount present. The symbol \(e\) is Euler’s number, an irrational mathematical constant which is the base of the natural logarithm. The term \(t\) represents the total elapsed time, which must be expressed in the same unit of time used for the decay constant.
The symbol \(\lambda\) (lambda) is the decay constant, a specific value for each isotope that quantifies the probability of decay per unit of time. The decay constant is directly related to the half-life by the equation \(\lambda = \frac{\ln(2)}{T_{1/2}}\), where \(\ln(2)\) is the natural logarithm of 2. This relationship connects the conceptual simplicity of half-life with the mathematical rigor of the exponential decay model.
To use the full formula, one must first calculate the decay constant \(\lambda\) using the known half-life of the isotope. For example, if an isotope has a half-life of 5 days, \(\lambda\) is \(0.693 / 5 \text{ days}\), resulting in a decay constant of \(0.1386 \text{ days}^{-1}\). This constant is then used in the primary exponential formula, allowing for the calculation of the remaining amount after any time period, such as \(6.5\) days.