Decay describes how a quantity or substance diminishes over time. This predictable reduction is important across various scientific disciplines, and understanding how to calculate it is key.
The Science Behind Decay
Many natural processes involve exponential decay, which describes a quantity decreasing at a rate proportional to its current amount. A key concept in exponential decay is “half-life” (t½), the time it takes for a substance to reduce to half of its initial value. This principle is most commonly associated with radioactive decay, where unstable atomic nuclei spontaneously transform into more stable forms by emitting particles and energy.
The half-life of a radioactive isotope is constant, independent of the initial amount. For instance, if a material has a half-life of one hour, half of the original amount decays in the first hour, and half of the remaining material decays in the next. While radioactive decay provides a clear example, the concept of half-life and exponential decay applies to other phenomena, such as the elimination of drugs from the body or the depreciation of assets.
Key Formulas for Decay
The primary mathematical model for exponential decay is N(t) = N₀ e^(-λt). Here, N(t) is the amount remaining after time (t), and N₀ is the initial amount. The constant ‘e’ is Euler’s number, approximately 2.71828.
The term ‘λ’ (lambda) is the decay constant, which quantifies the decay probability per unit time. A larger decay constant signifies a faster decay rate.
The relationship between the decay constant (λ) and half-life (t½) is λ = ln(2) / t½. Here, ln(2) is approximately 0.693. This formula shows that substances with shorter half-lives have larger decay constants, indicating they decay more rapidly.
Calculating Decay Step-by-Step
These formulas allow for direct calculation of decay. Consider a scenario to determine the amount of a radioactive isotope remaining after a certain period.
For example, if you start with 100 grams of Technetium-99m (Tc-99m), which has a half-life of 6 hours, you can calculate how much remains after 24 hours. First, calculate the decay constant (λ) using the half-life: λ = ln(2) / t½ = 0.693 / 6 hours = 0.1155 per hour.
Next, use the exponential decay formula N(t) = N₀ e^(-λt). Plugging in the values, N(24) = 100 g e^(-0.1155 24). This calculation simplifies to N(24) = 100 g e^(-2.772), which is approximately N(24) = 100 g 0.0625. After 24 hours, about 6.25 grams of Technetium-99m would remain.
Carbon-14 dating is another application for determining the age of ancient artifacts. Carbon-14 has a half-life of approximately 5,730 years. If an archaeological sample initially contained 10 micrograms of carbon-14 (N₀ = 10 µg) and now contains 1 microgram (N(t) = 1 µg), you can find its age.
First, calculate the decay constant for carbon-14: λ = ln(2) / 5730 years ≈ 0.00012097 per year. Then, rearrange the decay formula N(t) = N₀ e^(-λt) to solve for time (t): t = -ln(N(t)/N₀) / λ. Substituting the values, t = -ln(1 µg / 10 µg) / 0.00012097. This becomes t = -ln(0.1) / 0.00012097. Since ln(0.1) is approximately -2.302585, the calculation yields t ≈ -(-2.302585) / 0.00012097 ≈ 19,034 years.
Where Decay Calculations Matter
Decay calculations are used in several real-world applications. Carbon dating relies on the predictable decay of carbon-14 to determine the age of organic materials like ancient artifacts and fossils. This technique provides timelines for historical events up to about 60,000 years old.
In medicine, understanding decay is important for diagnostic imaging and cancer treatment. Radioactive isotopes, such as Technetium-99m, are used in PET scans and other medical procedures. Knowledge of their half-lives helps ensure correct dosage and minimizes patient exposure, as these isotopes decay rapidly.
Decay calculations are also essential in nuclear energy and waste management. Radioactive waste, generated during nuclear power production, contains isotopes with varying half-lives. Calculations of their decay rates inform safe storage and disposal strategies, as the radioactivity of these materials decreases over time.