Question

A radioactive decay series that begins with \({ }_{9}^{23} \mathrm{Th}\) ends with formation of the stable nuclide \({ }_{82}^{208} \mathrm{~Pb}\). How many alpha-particle emissions and how many beta-particle emissions are involved in the sequence of radioactive decays?

Short answer

In the radioactive decay series from Th-233 to Pb-208, there are 4 alpha-particle emissions and 17 beta-particle emissions involved. This result is obtained by setting up and solving proton and neutron balance equations, using the facts that alpha decay involves the loss of 2 protons and 2 neutrons while beta decay involves a neutron becoming a proton or vice versa.

Step-by-step solution

Identify initial and final nuclides and decay information

We are given that the decay series starts with Th-233 (\(^{233}_{\ 90}\text{Th}\)) and ends with Pb-208 (\(^{208}_{\ 82}\text{Pb}\)). We also know that in an alpha decay, an atom emits 2 protons and 2 neutrons, and in a beta decay, a neutron is converted into a proton (or vice versa).

Write the balance equations for proton and neutron counts

Let \(a\) be the number of alpha decays and \(b\) be the number of beta decays. We can write two equations to represent the changes in proton numbers and neutron numbers: - Proton balance: \(90 - 2a = 82\) - Neutron balance: \(233 - 90 - 2a = 208 - 82 - b\)

Solve the equations for the number of alpha and beta decays

Now we will solve for the variables \(a\) (number of alpha decays) and \(b\) (number of beta decays). From the proton balance equation, we see that \(90 - 82 = 8 = 2a\) \(\Rightarrow a = 4\) Now, we plug the value of \(a\) in the neutron balance equation: \(233 - 90 - 2(4) = 208 - 82 - b\) \(143 - 8 = 126 - b\) \(\Rightarrow b = 17\)

Interpret the results

So, there are 4 alpha-particle emissions and 17 beta-particle emissions involved in the sequence of radioactive decays from Th-233 to Pb-208.

Key concepts

Alpha-Particle Emissions

Alpha-particle emissions are a form of radioactive decay in which an unstable atom releases an alpha particle, consisting of two protons and two neutrons—the same as a helium-4 nucleus. This process decreases the mass number of the original nucleus by four and the atomic number by two.

Let's consider our original problem where thorium-233 decays into lead-208. During an alpha-particle emission from this series, the thorium atom loses four units of mass and two units of atomic number, essentially becoming a different element. These alpha emissions are significant in the decay series since they can drastically change the identity of the radioactive nuclide, leading towards a stable end product.

Understanding the impact of alpha emissions on radioactive decay is crucial in fields like nuclear physics, medicine, and environmental science, as these emissions can be both beneficial and potentially harmful to life.

Beta-Particle Emissions

In beta-particle emissions, a beta particle, which can be either an electron or positron, is released from an atomic nucleus. A neutron gets converted into a proton (or vice versa), which does not change the mass number but does increase (or decrease) the atomic number by one.

In our exercise, where we observe the decay from thorium-233 to lead-208, beta-particle emissions are responsible for changing the number of protons within the nucleus without altering the mass number. Since mass remains unchanged, beta emissions fine-tune the stability of the nuclide by altering the neutron-proton ratio. Specifically, 17 beta emissions were necessary to reach the stable state of lead-208.

Students studying nuclear chemistry must grasp the concept of beta-particle emissions, as they play a pivotal role in both natural and induced radioactive decay processes.

Balance Equations in Nuclear Chemistry

Balancing equations in nuclear chemistry is vital for understanding the stoichiometry of nuclear reactions, just as it is with chemical reactions. However, instead of balancing the count of atoms, we balance the mass number and atomic number to preserve the laws of conservation of mass and charge.

In the provided exercise, the balance equations allowed us to find the exact count of alpha and beta emissions by comparing the change in protons and neutrons from thorium-233 to lead-208. Balancing equations in such decay series involves setting up algebraic equations where the sum of mass and atomic numbers before the decay equals the sum after decay, with the inclusion of the mass and charge of emitted particles.

This concept is instrumental for scientists and engineers working in sectors dealing with radioactive materials or studying nuclear reactions, as it ensures a quantitative understanding of nuclear changes.

Nuclear Decay Pathways

Nuclear decay pathways represent the sequence of transformations an unstable nuclide undergoes to reach a stable state. This involves a series of radioactive decays, each with specific characteristics and rules, such as alpha-particle and beta-particle emissions.

In our exercise example, thorium-233 undergoes multiple decays, involving 4 alpha and 17 beta emissions, to eventually become stable lead-208. The detailed pathway would show each intermediate nuclide and the type of decay it undergoes. By studying nuclear decay pathways, we gain insights into the behavior of radioactive materials, the rates of decay, and the potential hazards or applications of different nuclides.

Navigating through nuclear decay pathways empowers students to predict and understand the mechanics behind the stability of elements, an essential part of nuclear chemistry and physics education.