Uranium-238 decays through a series of \(\alpha\) and \(\beta\) emissions to lead-206, with an overall half-life for the entire process of \(4.5 \mathrm{Ga}\). How old is a uranium-bearing ore that is found to have a \({ }^{238} \mathrm{U} /{ }^{206} \mathrm{~Pb}\) ratio of (a) \(1.00\) and (b) \(1.25\) ?

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation, such as an alpha particle or a beta particle, which are represented by \( \alpha \) and \( \beta \) respectively in scientific notation. This process transmutes the element into another at a rate determined by the substance's half-life.

For example, Uranium-238 decays to lead-206 via a chain of these \( \alpha \) and \( \beta \) emitters. Each element in this decay chain has its own unique half-life, but the overall transformation from Uranium-238 to lead-206 occurs with a half-life of approximately 4.5 billion years. That's how long it takes for half of a sample of Uranium-238 to convert into lead-206.

Understanding this process is crucial for estimating the age of uranium-bearing ores and thus has significant applications in fields such as geology and archeology for dating rocks and fossils.

The half-life of a radioactive substance is a measure of the time it takes for half of the radioactive atoms in a sample to decay. In simpler terms, after one half-life, the amount of the substance is reduced to half of its initial quantity.

To calculate the age of a substance using its half-life, you need to know the starting amount of the substance and the ratio of the undecayed atoms to the decay products. For instance, in the exercise involving Uranium-238, knowing the ratio of Uranium-238 to lead-206 enables scientists to use the half-life of Uranium-238 to estimate the age of the sample.

The half-life is crucial when dealing with radioactive materials, as it helps determine the longevity of their radioactivity and can be used to assess potential risks or to date ancient objects, aiding researchers in piecing together the history of our planet.

The exponential decay formula is a mathematical representation that describes the decrease of a quantity at a rate proportional to its current value. For radioactive decay, the formula is often expressed as:

\[ N(t) = N_0 (1/2)^{t/T_{1/2}} \]

where \( N(t) \) represents the number of undecayed atoms at time \( t \), \( N_0 \) is the initial number of atoms, \( t \) is the time that has elapsed, and \( T_{1/2} \) is the half-life of the decaying substance.

When working with decay problems, we often want to find out the elapsed time since the formation of the material. To do this, we can rearrange the above formula to isolate \( t \), usually involving logarithms due to the exponential nature of the equation. This is the method used to determine the age of the uranium-bearing ore in the given exercise, highlighting the power and utility of the exponential decay formula in understanding radioactive decay processes.