Radioactivity is the spontaneous process where an unstable atomic nucleus releases energy and particles to achieve a more stable state. This radioactive decay causes the amount of the original radioactive material, or isotope, to decrease over time. The rate of this decline is measured by the half-life (\(T_{1/2}\)), which is the time required for exactly half of the radioactive material to transform. This characteristic time is unique to every radioactive isotope and measures its stability and longevity.
Understanding the Decay Constant
The decay constant, symbolized by lambda (\(\lambda\)), quantifies the probability of a single atomic nucleus decaying within a specific unit of time. A higher value for the decay constant signifies a greater likelihood of decay, meaning the isotope is less stable and decays more rapidly. Since it represents the rate of transformation, it is typically expressed in reciprocal time units, such as inverse seconds (\(s^{-1}\)) or inverse years (\(yr^{-1}\)).
This constant is intrinsically linked to the half-life of an isotope through a fundamental mathematical relationship. The half-life is inversely proportional to the decay constant: an isotope with a large decay constant will have a short half-life, and vice versa. This connection is formally expressed as \(T_{1/2} = \ln(2) / \lambda\).
The term \(\ln(2)\) represents the natural logarithm of two, which is numerically equal to approximately 0.693. This constant value arises directly from the definition of half-life, as the time when the remaining amount is precisely one-half of the starting amount.
Therefore, the equation \(T_{1/2} = 0.693 / \lambda\) provides a direct path to calculate the half-life once the decay constant is known. This relationship confirms that the half-life is an inherent property of the isotope, independent of the initial quantity of material present.
Calculating Half-Life from Experimental Observations
To determine an unknown half-life, scientists must first experimentally determine the decay constant (\(\lambda\)) by observing how quickly a sample decays over time. This calculation relies on the integrated first-order rate law, which mathematically describes all radioactive decay processes.
The fundamental equation used is \(\ln(N_t/N_0) = -\lambda t\). Here, \(N_0\) represents the initial quantity of the radioactive material (measured by mass, number of atoms, or activity) at the start of the observation period. \(N_t\) is the quantity of the original isotope that remains undecayed after the elapsed time, \(t\).
Experimental data are collected by measuring the sample’s quantity at the start (\(N_0\)) and again at a later time (\(N_t\)), recording the elapsed time (\(t\)). To solve for the decay constant, the integrated rate law is rearranged to isolate \(\lambda\): \(\lambda = – \ln(N_t/N_0) / t\). Since \(N_t\) is less than \(N_0\), the negative sign in the formula ensures that the calculated decay constant is positive.
Once the decay constant (\(\lambda\)) is calculated from the experimental data, the final step uses the known relationship between \(\lambda\) and half-life. Substituting the calculated \(\lambda\) into the formula \(T_{1/2} = 0.693 / \lambda\) provides the numerical value for the isotope’s half-life. The units of the calculated half-life will correspond to the time unit used for the elapsed time (\(t\)) during the initial observation.
Predicting Isotope Decay Over Time
The established mathematical relationships are used not only to calculate the half-life but also to predict the outcome of decay over specified periods. A primary application is determining the remaining quantity of an isotope after a given time has passed. If the half-life is known, the decay constant (\(\lambda\)) is easily calculated using the \(0.693 / T_{1/2}\) relationship.
This calculated decay constant is then used in the integrated rate law, \(N_t = N_0 e^{-\lambda t}\). By inputting the initial quantity (\(N_0\)) and the time elapsed (\(t\)), the equation directly solves for \(N_t\), the remaining amount of the original isotope. This process is useful for predicting safety levels for nuclear waste or the depletion of medical radioisotopes.
A second significant application involves solving for the time elapsed (\(t\)), a technique fundamental to radiometric dating. This requires rearranging the integrated rate law to \(t = – \ln(N_t/N_0) / \lambda\). For example, in carbon dating, the ratio \(N_t/N_0\) is determined by comparing the Carbon-14 remaining in an artifact (\(N_t\)) to the amount initially present (\(N_0\)). Using the known half-life of Carbon-14 (approximately 5,730 years) to find \(\lambda\), the formula calculates the time (\(t\)) that has passed since the organism died.
The calculations emphasize the exponential nature of decay, where the quantity halves precisely during every half-life period. For instance, after three half-lives, only one-eighth of the original material will remain, illustrating the predictable rate of transformation.