An ancient skull has a carbon-14 decay rate of 0.85 disintegrations per minute per gram of carbon \((0.85\) dis/ min \(\cdot \mathrm{g}\) C). How old is the skull? (Assume that living organisms have a carbon-14 decay rate of 15.3 dis/ min \(\cdot \mathrm{g} \mathrm{C}\) and that carbon- 14 has a half- life of 5730 yr.)

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting particles or radiation. This process transforms the unstable nucleus into a more stable form, often resulting in a different element. For example, carbon-14, a radioactive isotope of carbon, decays over time into nitrogen-14.

The rate at which this decay occurs is measurable and characterized by a specific half-life, which is the amount of time it takes for half of the radioactive atoms in a sample to decay. In the context of carbon dating, scientists use the decay of carbon-14 to estimate the age of organic material. When an organism dies, it stops absorbing carbon-14, and the existing carbon-14 starts to decay at a predictable rate. By measuring the decay rate, or disintegrations per minute per gram, researchers can estimate the time elapsed since the death of the organism.

The half-life of a radioactive element is essential in understanding its rate of decay. For carbon-14, the half-life is approximately 5730 years, which means that every 5730 years, half of the carbon-14 in a given sample will have decayed into nitrogen-14.

To determine the number of half-lives that have passed since an organism died, scientists use the current decay rate in comparison to the decay rate when the organism was alive. This involves calculating a ratio and then using logarithms to solve for the number of half-lives, as shown in the solution steps for the skull's age. The use of logarithms is fundamental because it allows us to invert the exponential decay equation and isolate the number of half-lives that have occurred.

The exponential decay formula is integral to understanding the process of radioactive decay. It is represented by the equation: \[\begin{equation}N(t) = N_0 \times \bigg(\frac{1}{2}\bigg)^\frac{t}{T}\text{where:}\begin{itemize} \text{•} N(t) \text{ is the amount of the substance that still remains after time } t \text{•} N_0 \text{ is the original amount of the substance} \text{•} t \text{ is the time elapsed} \text{•} T \text{ is the half-life of the substance}\text{\end{itemize}}\text{\end{equation}\]}

The formula illustrates how the quantity of a substance decreases over time. Applying this formula to the age of the skull, scientists use the known half-life of carbon-14 and the ratio of remaining carbon-14 to solve for the elapsed time since the organism's death. By rearranging the formula to isolate the time variable, one can determine the age of an archaeological find with precision. This method is an excellent tool for understanding the history and chronology of ancient remains and artifacts.