The decay constant (λ) quantifies the rate at which a radioactive isotope undergoes decay. It represents the probability that an individual nucleus will decay per unit of time. A higher decay constant indicates a faster rate of disintegration, providing insight into the isotope’s inherent instability.
The Fundamentals of Radioactive Decay
Radioactive decay is a natural process where unstable atomic nuclei spontaneously transform into more stable configurations by emitting particles or energy. This transformation results in the formation of different elements or isotopes. Each specific radioactive isotope decays at a predictable rate, which is unique to that particular type of atom.
A fundamental concept in understanding decay rates is half-life (t½), defined as the time required for half of the radioactive atoms in a given sample to decay. For instance, if a sample begins with 1000 radioactive nuclei, its half-life is the time it takes for that number to reduce to 500 nuclei. Half-lives can vary enormously, from mere microseconds for some highly unstable isotopes to billions of years for others, such as Uranium-238.
The half-life serves as a direct indicator of an isotope’s decay rate. A shorter half-life signifies that a substance decays more rapidly. This inherent likelihood of decay per unit time is precisely what the decay constant measures, providing a direct numerical link to the observed rate of radioactive transformations.
Calculating the Decay Constant from Half-Life
One common method for determining the decay constant involves using an isotope’s known half-life. The decay constant (λ) and half-life (t½) share a direct mathematical relationship, reflecting the inverse nature of a rapid decay rate corresponding to a short half-life. This relationship is expressed by the formula: λ = ln(2) / t½. Here, ln(2) represents the natural logarithm of 2, which is approximately 0.693.
To calculate the decay constant, one needs the half-life of the specific radioactive isotope. It is important to ensure that the units of time for the half-life are consistent with the desired units for the decay constant, usually expressed as inverse seconds (s⁻¹) or inverse years (yr⁻¹).
For example, Carbon-14, a radioactive isotope widely used in archaeological dating, has a half-life of 5,730 years. To find its decay constant, one would substitute this value into the formula. The calculation becomes λ = 0.693 / 5,730 years, yielding approximately 0.0001209 per year (yr⁻¹).
Determining the Decay Constant from Activity Data
Another approach to finding the decay constant involves measuring a radioactive sample’s activity over time. Activity (A) refers to the rate at which decay events occur in a sample, typically measured in becquerels (Bq), representing one decay per second. Activity is directly proportional to the number of radioactive nuclei present; as nuclei decrease, so does activity.
The relationship between activity, initial activity, time, and the decay constant is described by the radioactive decay law: A = A₀ e^(-λt). Here, A₀ represents the initial activity, A is the activity after time (t), and ‘e’ is Euler’s number, approximately 2.718. This formula shows how activity diminishes exponentially over time.
To determine the decay constant (λ) from activity measurements, one can rearrange this equation. By dividing both sides by A₀, the equation becomes A/A₀ = e^(-λt). Taking the natural logarithm of both sides allows for the isolation of λ: ln(A/A₀) = -λt. Rearranging further to solve for λ yields λ = -ln(A/A₀) / t, which can also be written as λ = ln(A₀/A) / t. This rearranged formula allows for direct calculation of the decay constant using measured activity values and the elapsed time.
As an illustration, imagine a radioactive sample initially exhibits an activity of 1000 becquerels. After 10 days, its activity is measured again and found to be 800 becquerels. To calculate the decay constant, one would use the rearranged formula: λ = ln(1000 Bq / 800 Bq) / 10 days. This simplifies to λ = ln(1.25) / 10 days. Calculating the natural logarithm of 1.25 gives approximately 0.223. Therefore, the decay constant is 0.223 / 10 days, resulting in a value of 0.0223 per day (day⁻¹).