Understanding Radioactive Half-Life
Isotopes are variations of a chemical element, distinguished by their differing numbers of neutrons. Some are unstable, transforming into a more stable configuration by releasing energy and particles in a process known as radioactive decay. This phenomenon occurs at a predictable rate for each unstable isotope.
The half-life of a radioactive isotope is the time it takes for half of a given sample to undergo radioactive decay. This value is constant for a particular isotope, regardless of the initial quantity of the substance present.
Radioactive decay proceeds exponentially. Over each successive half-life period, a fixed fraction (half) of the remaining radioactive atoms will decay, rather than a fixed amount. For example, if an isotope has a half-life of one hour, after one hour, half will remain; after another hour, half of that remaining amount (or one-quarter of the original) will be left.
Half-life measures an isotope’s stability. Isotopes with shorter half-lives are less stable, decaying more rapidly. Conversely, isotopes with longer half-lives are more stable, decaying at a slower pace over extended periods.
The Half-Life Calculation Formula and Its Variables
Calculating half-life relies on understanding the mathematical relationships governing radioactive decay. The fundamental formula describing the quantity of a radioactive substance remaining after a certain time is N(t) = N₀ e^(-λt).
In this formula, N(t) represents the quantity remaining at time t, while N₀ signifies the initial quantity at time zero. The variable ‘e’ is Euler’s number, approximately 2.71828. The variable ‘t’ denotes the elapsed time.
Central to these calculations is λ (lambda), the decay constant. The decay constant quantifies the probability per unit time that a nucleus will undergo decay. A larger decay constant signifies a faster decay rate and a shorter half-life.
The half-life (t₁/₂) can be directly calculated from the decay constant using the formula: t₁/₂ = ln(2) / λ. Here, ln(2) is the natural logarithm of 2, approximately 0.693. Ensure all units are consistent throughout any calculation.
How to Calculate Half-Life: A Step-by-Step Example
To illustrate the calculation, consider a sample of a radioactive isotope. Suppose you start with an initial quantity (N₀) of 100 grams. After 20 days, the quantity remaining (N(t)) is 25 grams. The goal is to determine the half-life.
First, use the radioactive decay formula: N(t) = N₀ e^(-λt). Substitute the known values: 25 g = 100 g e^(-λ 20 days). Divide both sides by 100 g, resulting in 0.25 = e^(-λ 20 days).
Next, to solve for the decay constant (λ), take the natural logarithm (ln) of both sides. This leads to ln(0.25) = -λ 20 days. The natural logarithm of 0.25 is approximately -1.386. So, -1.386 = -λ 20 days.
Now, calculate λ by dividing -1.386 by -20 days, which yields λ ≈ 0.0693 day⁻¹. With λ determined, use the half-life formula: t₁/₂ = ln(2) / λ. Substitute λ: t₁/₂ = 0.693 / 0.0693 day⁻¹.
Performing this division gives a half-life (t₁/₂) of 10 days.
Applications of Half-Life Calculations
Half-life calculations are fundamental across various scientific and practical domains. One application is in radiometric dating, a technique used to determine the age of ancient materials.
By measuring the ratio of a radioactive isotope to its stable decay product, scientists can ascertain the age of geological formations, archaeological artifacts, and ancient organic matter, such as through carbon-14 dating.
In medicine, half-life is important for diagnostic and therapeutic procedures involving radioactive isotopes. For instance, in medical imaging like PET scans, isotopes with short half-lives are chosen to ensure they decay quickly, minimizing patient radiation exposure. Similarly, in radiation therapy, isotopes with specific half-lives deliver targeted radiation doses over a controlled period.
Half-life calculations also play a role in nuclear waste management. Understanding the half-lives of radioactive waste products is important for designing safe, long-term storage solutions. Isotopes with extremely long half-lives require containment for thousands to millions of years, posing challenges for secure disposal.