Final answer:
Half-life is the period it takes for half of a radioactive element to decay. Over multiple half-lives, the quantity of the radioactive element exponentially decreases, and the amount remaining can be calculated using the half-life formula. For an element with a half-life of 5×108 years, after the 4.6 billion years since the formation of the solar system, only a small fraction of the original element would remain.
Step-by-step explanation:
Understanding Radioactive Decay and Half-Life
The concept of half-life is critical to understanding the decay process of a radioactive element. A half-life is the time required for half of the radioactive nuclei in a sample to undergo radioactive decay. This decay results in a decrease in the amount of the original element, while the number of daughter elements, created from the decay, increases. For instance, if you start with a 1 gram sample of a radioactive nucleus with a half-life of 100 years, after 100 years you will have 0.5 grams of the element. After 200 years, only 0.25 grams remain. This process continues with each half-life period, resulting in an exponential decay of the radioactive nuclei.
When considering a radioactive nucleus with a half-life of 5×108 years, like an element in a rock from the early solar system, we can estimate the fraction of the element remaining today. If the age of the solar system is approximately 4.6 billion years, this would equate to a little over 9 half-lives. Applying the half-life formula, we find a significant reduction in the amount of the original radioactive element, leaving a small fraction of it remaining.