The isotope \(^{208} \mathrm{Tl}\) undergoes \(\beta\) decay with a half-life of \(3.1 \mathrm{min}\). (a) What isotope is produced by the decay? (b) How long will it take for \(99.0 \%\) of a sample of pure \(^{208} \mathrm{Tl}\) to decay? (c) What percentage of a sample of pure \(^{208} \mathrm{Tl}\) remains un- decayed after \(1.0 \mathrm{h}\) ?

Understanding the concept of half-life is essential when studying isotopes and their radioactive decay. The half-life is defined as the time required for half the atoms in a sample of a radioactive substance to undergo decay. This decay results in the transformation of the isotopes into different elements or isotopes.

In the given exercise, for instance, the isotope thallium-208 (^{208}Tl) has a half-life of 3.1 minutes. This information is crucial for predicting how long the sample will take to decay to a certain level. The use of half-life allows us to calculate, for example, how long it will take for 99.0% of a thallium-208 sample to decay, which is illustrated in step 2 and step 3 of the solution.

An important aspect to consider is that half-life is not influenced by the initial amount of the substance or by external factors such as temperature; it is a constant property for a given isotope. This predictability makes half-life a powerful tool in fields such as archeology for carbon-dating, in medicine for radioactive tracers, and in nuclear power generation for waste management.

Radioactive decay is a fundamental concept within nuclear chemistry where an unstable atomic nucleus loses energy by emitting radiation. There are several different types of radioactive decay, including alpha decay, beta decay, and gamma decay.

In beta decay, which is the focus of the provided exercise, a neutron in the nucleus is converted into a proton, and an electron called a 'beta particle' is emitted. This transformation results in the increase of the atomic number by one, producing a new element. For example, when ^{208}Tl undergoes beta decay, it transforms into lead-208 (^{208} Pb) as a neutron converts into a proton.

Understanding these changes allows students to predict products of nuclear reactions. Being able to identify the new element formed after a beta decay is a critical skill in deciphering the complexity of nuclear reactions and the transformation of elements.

Nuclear chemistry is the study of the changes that occur in atomic nuclei. It encompasses various aspects such as the behavior of radioisotopes, nuclear reactions, and the laws governing radioactive decay.

Within nuclear chemistry, being familiar with concepts like half-life and types of radioactive decay helps us understand processes such as fission, fusion, and transmutation of elements. For example, in the context of the exercise, understanding beta decay as a nuclear process informs us not only about the identity of the decay product but also about the stability of nuclei and their potential applications or risks.

Additionally, nuclear chemistry is important in many practical applications, including energy production in nuclear reactors, the creation of new elements, medical diagnostics and treatments, and the study of environmental radioactive contaminants. The depth of learning in nuclear chemistry enables us to harness the power of the nucleus responsibly and innovatively.