Calculate the age of a uranium mineral, if the ratio by weight of \(\mathrm{Pb}^{206}\) to \(\mathrm{U}^{238}\) is \(0.5 .\) (Half life of \(\mathrm{U}^{238}\) \(=4.5 \times 10^{9}\) vears).

Understanding the half-life of Uranium-238 is crucial for determining the age of geological samples. The half-life is the period it takes for half of the radioactive atoms in a given sample to decay. For Uranium-238, this half-life is approximately 4.5 billion years. This remarkably long half-life makes it suitable as a natural clock to date ancient rocks and minerals.

When scientists mention the half-life in a practical context, such as the exercise above, they use it to track how much time has passed since the sample's formation. If a mineral shows a half-to-half ratio by weight of lead-206 to uranium-238, this indicates that one half-life has occurred since the mineral was formed.

The decay constant, represented by the symbol \( k \), is essential for calculating the rate at which a radioactive substance decays. It is expressed mathematically by the formula \( k = \frac{\ln(2)}{T_{1/2}} \), where \( \ln \) represents the natural logarithm, and \( T_{1/2} \) is the half-life of the radioactive substance.

This formula establishes a direct relationship between the decay constant and the half-life. The longer the half-life, the smaller the decay constant, indicating that the substance decays more slowly. In the context of our Uranium-238 example, we would divide the natural logarithm of 2 by 4.5 billion years to find its decay constant. This value helps us predict how quickly the uranium will decay to lead-206 over time.

Radioactive decay is a statistical process, and the decay formula \( N = N_0 e^{(-kt)} \) models the number of undecayed nuclei remaining over time. In this equation, \( N_0 \) is the original number of nuclei, \( N \) is the number remaining after time \( t \), and \( k \) is the decay constant. The \( e \) represents the base of natural logarithms, which is approximately equal to 2.71828.

In solving our decay problem, we set up the decay formula to reflect the given conditions, such as the initial and remaining quantities of Uranium-238. For instance, a 0.5 ratio of lead-206 to uranium-238 implies that \( N_0 = 2N \). By substituting this and the decay constant into our formula, we can solve for the time \( t \), giving us the age of the uranium. It's a critical tool used to understand the behavior of unstable isotopes and to find the age of various samples.