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In the realm of nuclear physics, isotopes play a fascinating role due to their ability to undergo radioactive decay. An isotope is a variant of a particular chemical element, which, while sharing the same number of protons, has different numbers of neutrons, leading to differing atomic masses. Over time, some isotopes are unstable and will "decay" or transform into other isotopes or elements. This process is referred to as isotope decay.
When the isotope \({ }^{40}K\) decays, it can turn into either \({ }^{40}Ca\) or \({ }^{40}Ar\). Such transformation is essential in the calculations for determining the age of geological samples. As decay progresses, the number of original isotopes decreases, while the number of decay products increases. This transformation proceeds in a predictable manner, allowing scientists to use it as a time gauge.
The consistent rate of decay follows an exponential decay model. Understanding this model is crucial when working with radioactive materials, as it helps predict how long a sample has been undergoing decay. Scientists use this knowledge for various purposes, such as understanding the age of rocks and meteorites.

The concept of half-life is central to understanding how radioactive decay helps us calculate the age of materials. The half-life is the time taken for half the amount of a radioactive isotope in a sample to decay. It indicates the stability and speed of decay of an isotope.
For \({ }^{40}K\), the half-life is \(1.26 \times 10^{9} \, \text{years}\). This means that after approximately 1.26 billion years, half of the original \({ }^{40}K\) isotopes will have decayed into \({ }^{40}Ca\) and \({ }^{40}Ar\).
In this exercise, with equal amounts of \({ }^{40}K\) and \({ }^{40}Ar\), it implies that one half-life has passed. Scientists can use the half-life to calculate the time since the sample had only pure \({ }^{40}K\), which in this case, is indeed \( ext{1.26 billion years}\). This calculation assumes a constant decay rate which is a property of the isotope.

Radioactive dating, also called radiometric dating, is a technique used to date materials such as rocks or carbon, based on knowledge of their radioactive isotopes. This method is possible because these isotope decays occur at a consistent rate over time, making them a "clock" of sorts for scientists to interpret.
By measuring the ratio of the original radioactive isotope to its decay products, such as \({ }^{40}K\) to \({ }^{40}Ar\), geologists can ascertain the age of the sample. In this case, the method helped determine that our sample is \( ext{1.26 billion years}\) old.
The precision of radioactive dating makes it indispensable in studying Earth's history. It provides a way to quantify the age of geological formations and can be used to trace the evolutionary history of Earth, as well as for archaeological purposes. Understanding the principles behind radiometric dating allows us to piece together significant events in the history of our planet.