Question

Consider the following process of decay, \({ }_{92} \mathrm{U}^{234} \rightarrow{ }_{90} \mathrm{Th}^{230}+{ }_{2} \mathrm{He}^{4} ; t_{1 / 2}=2,50,000\) years \({ }_{90} \mathrm{Th}^{230} \rightarrow{ }_{88} \mathrm{Ra}^{226}+{ }_{2} \mathrm{He}^{4} ; t_{1 / 2}=80,000\) years \({ }_{88} \mathrm{Ra}^{226} \rightarrow{ }_{86} \mathrm{Rn}^{222}+{ }_{2} \mathrm{He}^{4} ; t_{1 / 2}=1600\) years After the above process has occurred for a long time, a state is reached where for every two thorium atoms formed from \({ }_{92} \mathrm{U}^{234}\), one decomposes to form \({ }_{88} \mathrm{Ra}^{226}\) and for every two \({ }_{88} \mathrm{Ra}^{226}\) formed, one decomposes. The ratio of \({ }_{90} \mathrm{Th}^{230}\) to \({ }_{88} \mathrm{Ra}^{226}\) will be (a) \(250000 / 80000\) (b) \(80000 / 1600\) (c) \(250000 / 1600\) (d) \(251600 / 8\)

Short answer

(a) 250000 / 80000

Step-by-step solution

Understanding Secular Equilibrium

Secular equilibrium occurs when the rate of production of a daughter nuclide is equal to its rate of decay. Given the ratios of formation and decay for thorium and radium, we can infer they are in a state of secular equilibrium.

Setting Up the Ratio

For every two atoms of thorium-230 formed, one atom decays to radium-226, and for every two atoms of radium-226 formed, one atom decays. Thus, the decay rates must be proportional to their half-lives.

Calculate The Ratio

Since the decay rates are inversely proportional to their half-lives, the ratio of thorium-230 to radium-226 is the inverse ratio of their half-lives. Therefore, the correct ratio is the half-life of Th-230 over the half-life of Ra-226 or (250,000 years / 80,000 years).

Key concepts

Nuclear Decay Processes

Nuclear decay processes are fundamental to understanding how radioactive isotopes transform over time. It involves an unstable atomic nucleus losing energy by emitting radiation. This radiation can take the form of alpha particles (as seen in our exercise), beta particles, or gamma rays.

Each decay process follows unique rules and leads to a different element or isotope. For our example, Uranium-234 ( Uranium-234 (_92U^{234}) decays into Thorium-230 (_90Th^{230}) through alpha decay, where the Uranium nucleus ejects an alpha particle (_2_2He^{4}) to become a new element, Thorium. Alpha decay typically occurs in the heaviest elements, and the ejected alpha particles consist of two protons and two neutrons, hence reducing the atomic number by two and the mass number by four.

Understanding these processes is essential because they shape the nature of radioactive materials, their applications, and the precautions necessary when handling them.

Half-life Calculation

The half-life of a radioactive isotope is the period it takes for half of its atoms to decay. This concept is critical in the fields of geology, archaeology, and nuclear physics because it can help determine the age of objects or materials (radiometric dating) and inform the handling of nuclear materials.

To calculate the half-life, one must understand the decay rate, which is inversely proportional to the half-life. This means the longer the half-life, the slower the rate of decay. In our exercise example, the half-lives of different decaying isotopes, like Thorium-230 and Radium-226, give us insights into how quickly these atoms will disappear over time. You'll often encounter the half-life expressed in various units of time, depending on the isotope's stability – from fractions of a second to billions of years.

Radioactive Decay Series

A radioactive decay series occurs when a radioactive isotope decays into another radioactive isotope, which then decays into yet another isotope, and this process continues until a stable isotope is reached. This sequence of transformations is critical for understanding how different elements are formed naturally.

In our textbook exercise, we examine a part of the decay series starting with Uranium-234 and ending with Radon-222, each undergoing alpha decay. Keeping track of such a series is crucial because it can tell us about the different types of isotopes produced and the timeline of their existence. Additionally, it helps in understanding secular equilibrium, where the rate of production of a specific isotope is balanced by the rate of its decay, resulting in a constant proportion of isotopes after a sufficiently long period, as indicated by the ratios provided in the exercise solution.