If a radioactive element has a half-life of 10,000 years, what fraction of it is left in a rock after 40,000 years? a. \(1 / 2\) b. \(1 / 4\) c. \(1 / 8\) d. \(1 / 16\) e. \(1 / 32\)

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This can happen in the form of alpha, beta, or gamma radiation. The rate at which a radioactive substance decays is constant and unique to each element. This rate is often described in terms of half-life. Understanding radioactive decay is crucial in fields like nuclear medicine, archaeology, and geology. The decay is spontaneous and cannot be influenced by external conditions like temperature or pressure. As the substance decays, it transforms into a different element or a different isotope of the same element.

The term 'half-life' refers to the time it takes for half of the radioactive atoms in a sample to decay. It is denoted by the symbol \(\tau_{1/2}\). For instance, if a substance has a half-life of 10,000 years, this means that after 10,000 years, only half of the original radioactive atoms will remain. After another 10,000 years, half of the remaining atoms will decay, leaving only a quarter (\(\frac{1}{4}\)) of the original number of atoms. This pattern continues with each successive half-life. To calculate the fraction left after a certain number of half-lives, you can use the formula: \[ \text{Remaining fraction} = \frac{1}{2^n} \] where \( n \) is the number of half-lives that have elapsed.

Exponential decay describes any process that decreases at a rate proportional to its current value. In the context of radioactivity, the number of undecayed radioactive atoms decreases exponentially over time. This means that the decay is rapid at first but slows down as time passes. The general formula for exponential decay is given by: \[ N(t) = N_0 e^{-\frac{t}{\tau}} \] where \( N(t) \) is the quantity of substance at time \( t \), \( N_0 \) is the initial quantity, \( e \) is the base of natural logarithms, and \( \tau \) is the mean lifetime of the substance. Although this formula is robust, the half-life formula is often used because it is simpler to understand and apply in real-world scenarios.