Bromine-82 has a half-life of 36 hours. A sample containing Br-82 was found to have an activity of \(1.2 \times 10^{5}\) disintegrations \(/ \mathrm{min}\). How many grams of Br-82 were present in the sample? Assume that there were no other radioactive nuclides in the sample.

The half-life of an isotope is the time taken for half of the radioactive nuclei in a sample to decay. It's an intrinsic property of radioactive substances and is symbolically denoted by the letter 'T'. During each half-life period, the amount of the radioactive isotope decreases by 50%. Knowing the half-life is essential in calculating how long it will take for a given isotope to reduce to a certain amount.

For instance, in the exercise with Bromine-82 ('Br-82'), we're told that its half-life is 36 hours. That means in every 36-hour period, the number of active Br-82 atoms will halve due to radioactive decay. Importantly, the half-life can help us determine the decay constant and, from there, the initial quantity of the radioactive material.

The decay constant, represented by the Greek letter λ, is a probability factor that describes the likelihood of a single atom decaying per unit time. In simpler terms, it's a measure of how quickly radioactive atoms are expected to decay. The decay constant and the half-life of an isotope are inversely related; as shown in the exercise solution, you can find one if you know the other using the formula: \[ \lambda = \frac{0.693}{T} \].

This relation is critical since the decay constant directly feeds into our calculations for the original activity or number of atoms present in a sample. In nuclear chemistry, understanding the decay constant is vital for making accurate predictions about a substance's behavior over time, which has practical applications in fields like medicine, archaeology, and environmental science.

Nuclear chemistry is the branch of chemistry that deals with nuclear reactions and their implications. It encompasses various phenomena, including fission, fusion, and radioactive decay. Among these, radioactive decay is particularly important for understanding how unstable isotopes, like Br-82, transform into stable ones over time by emitting radiation in the form of particles or photons.

This type of chemistry is not only about comprehending the processes that go on within a nucleus but also involves calculations like those performed in our exercise. By using principles of nuclear chemistry, we can predict quantities, dates, and even the potential energy released in nuclear reactions. The subject is integral to energy generation, medical diagnostic and treatment techniques, and it is also key to ensuring safety from radiation hazards.