A \(1.40\)-g sample containing radioactive cobalt was kept for \(2.50 \mathrm{a}\), at which time it was found to contain \(0.266 \mathrm{~g}\) of \({ }^{67} \mathrm{Co}\). The half-life of \({ }^{67} \mathrm{Co}\) is \(5.27\) a. What percentage (by mass) of the original sample was \({ }^{67} \mathrm{Co}\) ?

In nuclear chemistry, half-life is a critical concept referring to the time it takes for half of the atoms of a radioactive isotope to decay. For example, if we start with a 1.40g sample of a radioactive isotope, after one half-life has passed, only 0.70g of the original isotope would remain unchanged; the rest would have decayed into other elements or isotopes. This decay process is random but statistically consistent over large numbers of atoms, hence allowing us to predictably measure how long a half-life period is.

In the given exercise, the half-life of cobalt-67 is stated to be 5.27 years. This means that every 5.27 years, the amount of cobalt-67 in the sample is reduced to half its previous amount. Understanding this concept is crucial because it allows us to backtrack the initial quantity of cobalt-67 that must have been present in the sample before observing it 2.50 years later, leading to calculations of what percentage it constituted of the whole sample at the start.

Exponential decay is the process by which a quantity decreases at a rate proportional to its current value. This is represented by the exponential decay formula, which is ubiquitously expressed as \( N(t) = N_0(\frac{1}{2})^{\frac{t}{t_{1/2}}} \) where \( N(t) \) is the amount of substance that remains after time \( t \) has passed, \( N_0 \) is the initial amount of the substance, and \( t_{1/2} \) is the half-life of the substance.

The exercise illustrates this where after a given number of half-lives, the mass of the cobalt-67 present in the sample reduces according to the formula. For instance, if one half-life passes, the substance is reduced to half its original amount. After two half-lives, it’s reduced to a quarter, and so on. Exponential decay is non-linear, which means that the rate of decay of the substance slows down as time progresses. Being able to manipulate this formula is essential for accurately determining the original mass of a decaying sample over time, as was necessary in the solution to the given exercise.

Nuclear chemistry is the subfield of chemistry dealing with radioactivity, nuclear processes, and transformations in the nuclei of atoms. It encompasses the study of the chemical effects resulting from the absorption of radiation within living animals, the environment, and the industrial processes that harness the nuclear fission or fusion reactions.

In the context of the problem provided, nuclear chemistry principles are applied to understand the radioactive decay of cobalt-67. Insights into nuclear chemistry enable us to comprehend how elements change over time and allow us to calculate the duration over which a radioactive element will remain active or potentially hazardous. These calculations are not only pivotal for academic purposes but also significantly important for handling and storing nuclear materials, determining dosages for medical treatments that use radioactive substances, and estimating the age of artifacts through methods like carbon dating.