Use the law of radioactive decay to determine the activity of (a) a \(1.0\)-g sample of \({ }^{235} \mathrm{UO}_{2}\left(t_{1 / 2}=7.1 \times 10^{8} \mathrm{a}\right)\); (b) a \(1.0-\mathrm{g}\) sample of cobalt that is \(1.0 \%{ }^{60} \mathrm{Co}\left(t_{1 / 2}=5.26 \mathrm{a}\right)\); (c) a \(5.0-\mathrm{mg}\) sample of thallium-200 \(\left(t_{1 / 2}=26.1 \mathrm{~h}\right)\). The mass of each nuclide as a multiple of the atomic mass constant \(\left(m_{\mathrm{u}}\right)\) is equal to its mass number, within two significant figures.

Understanding the activity of a radioactive substance is crucial in the field of nuclear chemistry. It tells us the rate at which a radioactive element is undergoing decay. Activity is defined as the number of disintegrations per unit time, and it's measured in units called becquerels (Bq) in the International System of Units (SI), where one becquerel corresponds to one decay per second.

To calculate the activity (\( R \)) of a radioactive substance, we use the formula: \[ R = -\frac{dN}{dt} = \- \-lambda N \. \] Here, \( \-lambda \) represents the decay constant of the nuclide, and \( N \) is the number of undecayed nuclei present in the sample. It's important to remember that as the radioactive substance decays, its activity decreases over time because \( N \) gets smaller.

The decay constant (\( \-lambda \) is a probability factor that helps in determining the likelihood of a nuclear decay event per atom per unit time. It is unique to each radioactive nuclide. To calculate the decay constant from the nuclide's half-life (\( t_{1/2} \) is straightforward with the use of logarithms. The relationship is given by: \[ \-lambda = \-frac{\text{ln}(2)}{t_{1/2}} \. \]

To put it into practice, you will first need to convert the half-life into seconds to keep your units consistent since activity is usually expressed in decays per second. Once you have the decay constant, you can use the formula for activity to determine the rate of decay at any given time.

The half-life of a radioactive nuclide is the time it takes for half of the radioactive atoms in a sample to decay. This is an important concept because it determines how quickly a radioactive substance changes over time. Half-life values can range from fractions of a second to billions of years, depending on the nuclide in question.

Knowing the half-life of a nuclide provides insight into the stability of the nucleus and the potential for radioactive decay. To understand the practical application, consider a medical isotope used in diagnosis—its half-life determines the window of time the substance is effective and safe for the patient. In radioactive dating, half-lives allow scientists to date materials by comparing the ratio of parent nuclides to daughter products.

Nuclear chemistry is the subfield of chemistry dealing with radioactivity, nuclear processes, and transformations in the nuclei of atoms. This includes the study of both naturally occurring and artificial radioisotopes. Key areas of focus are radioactive decay; nuclear fission and fusion reactions; and the synthesis of new elements.

Applications of nuclear chemistry are vast and entail areas like energy production in nuclear reactors, the development of nuclear weapons, medical diagnostic and treatment methods involving radioisotopes, and the use of radioactive tracers in biological and environmental systems. Understanding nuclear chemistry is essential for managing the benefits and risks associated with the use of radioactive materials.