A current \(1.00\)-g sample of carbon shows 921 disintegrations per hour. If \(1.00 \mathrm{~g}\) of charcoal from an archaeological dig in a limestone cave in Slovenia shows \(5.50 \times 10^{3}\) disintegrations in \(24.0 \mathrm{~h}\), what is the age of the charcoal sample?

Understanding radioactive decay is essential when analyzing materials with radioisotopes, like carbon-14. Imagine each unstable carbon-14 nucleus as an unpredictable timepiece that can 'tick' at random intervals. Once it does, it transforms into a stable nitrogen-14 atom. This 'ticking' is a disintegration and is part of a natural process called radioactive decay.

Radioactive decay is spontaneous and occurs without any external force. It follows first-order kinetics, meaning the rate of decay is proportional to the number of unstable atoms present. Over time, as more atoms decay, fewer remain to decay in the future, which is why the rate decreases over time. Scientists exploit this predictable decay pattern by measuring how many 'ticks' or disintegrations occur over a specific period. By comparing this rate to a known standard, we can deduce much about the sample's age and initial conditions.

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. For carbon-14, the half-life is 5730 years. This means that if you start with 1 gram of pure carbon-14, after 5730 years, only 0.5 grams will remain; the other 0.5 grams will have decayed into nitrogen-14.

To compute the half-life, you can use the formula
\[ t_{1/2} = \frac{\ln 2}{k} \]
where \( t_{1/2} \) is the half-life, \( \ln 2 \) is the natural logarithm of 2 (approximately 0.693), and \( k \) is the decay constant. The decay constant represents the probability that a single atom will decay per unit of time. Once we find the decay constant, we can easily translate this into a powerful tool for dating archaeological samples, as it gives us a clock that runs at a predictable speed.

When dating an archaeological sample, like charcoal, scientists measure its current level of radioactivity and compare it to fresh carbon to comprehend its age. In the exercise, disintegration rates are compared between the current sample and one recovered from a Slovenian cave.

The process involves first determining the decay rate per hour for both samples. By dividing the archaeological sample's disintegration events by 24 hours, we find its rate and express the relative activity by comparing this rate to the modern standard. Finally, we apply a logarithmic function to solve for the sample's age, taking into account carbon-14's half-life.
\[ t = \frac{-1}{k} \ln \frac{A}{A_0} \]
where \( A \) is the activity of the archaeological sample, \( A_0 \) is the activity of the current sample, and \( k \) is the decay constant. This method enables us to wind back the clock, providing a window into the past to determine when the sample stopped accumulating carbon-14 and effectively started the decay stopwatch—providing invaluable information on the history and chronology of human and environmental events.