Question

A radioactive decay series that begins with \({ }_{90}^{232}\) Th ends with formation of the stable nuclide \({ }^{208} \mathrm{~Pb}\). 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 \({}_{90}^{232}\) Th to \({}^{208}\mathrm{~Pb}\), there are 6 alpha decays and 4 beta decays.

Step-by-step solution

1. Write down the initial and final isotopes in proper notation

We are given a decay series that starts with \({}_{90}^{232}\) Th and ends with \({}^{208}\mathrm{~Pb}\). The initial isotope is represented as \({}_{90}^{232}\text{Th}\), where 90 is the atomic number (Z) and 232 is the mass number (A). The final stable isotope is represented as \({}_{82}^{208}\text{Pb}\), where 82 is the atomic number (Z) and 208 is the mass number (A).

2. Calculate the change in atomic number (ΔZ) and mass number (ΔA)

To determine the number of alpha and beta decays, we first need to calculate the changes in both the atomic number (Z) and mass number (A). ΔZ = Z_final - Z_initial ΔZ = 82 - 90 ΔZ = -8 ΔA = A_final - A_initial ΔA = 208 - 232 ΔA = -24

3. Determine the number of alpha decays (n_alpha)

Since alpha decay results in a loss of 4 mass units (A) and 2 atomic number units (Z), we can determine the number of alpha decays by dividing the change in mass number (ΔA) by -4. n_alpha = ΔA / (-4) n_alpha = -24 / (-4) n_alpha = 6 There are 6 alpha decays in this decay series.

4. Determine the number of beta decays (n_beta)

Next, we will determine the number of beta emissions. Since beta decay results in an increase of 1 atomic number unit (Z) with no change in mass number (A), we can determine the number of beta decays by using the number of alpha decays (n_alpha) and the change in atomic number (ΔZ): n_beta = ΔZ + 2 * n_alpha n_beta = -8 + 2 * 6 n_beta = 4 There are 4 beta decays in this decay series.

5. Present the final results

We have now determined the number of alpha and beta particle emissions during the radioactive decay series: - Number of alpha decays (n_alpha): 6 - Number of beta decays (n_beta): 4

Key concepts

Alpha Particle Emissions

Alpha particle emissions are a type of radioactive decay where an unstable nucleus releases an alpha particle. This alpha particle consists of 2 protons and 2 neutrons, equivalent to a helium-4 nucleus. When an atom undergoes alpha decay, it loses 2 units from its atomic number and 4 units from its mass number. This leads to the formation of a new element that is two places back in the periodic table from the original element.

During the decay series from thorium-232 to lead-208, each alpha decay results in the reduction of the atomic number by 2 and the mass number by 4. Given the significant changes needed to reach lead-208, alpha emissions play a crucial role in decreasing the mass number significantly in the decay series.

To explore further, consider the example of an alpha decay in the sequence: - Start Element: Thorium-232 - End Element: Radium-228 Here, an alpha particle is emitted, resulting in thorium becoming radium.

Beta Particle Emissions

Beta particle emissions are another form of radioactive decay. In beta decay, a neutron in the nucleus is transformed into a proton, emitting a beta particle, which is an electron. This process results in an increase in the atomic number by 1, but the mass number remains unchanged.

This type of emission is essential for adjusting the atomic number in a decay series to achieve a stable state. In the thorium-lead series, beta decay helps increase the atomic number by offsetting the drastic reductions caused by alpha decay.

The useful aspect of beta decay is that it balances out the negative change in atomic number by increasing it slightly without affecting the mass number. This factor makes beta emissions key to understanding the probable end product of a decay series. For example, when undergoing beta decay: - Start Element: Actinium-228 - End Element: Thorium-228 In this case, actinium becomes thorium by converting a neutron to a proton.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are variants of chemical elements that have an unstable nucleus. These isotopes undergo radioactive decay, emitting alpha, beta, or gamma particles to achieve a more stable form.

The decay series discussed involves isotopes such as thorium-232, radium-228, and several others until the stable lead-208 is reached. Each of these isotopes follows a specific path of decay processes, characterized by both alpha and beta emissions.

The timeline of decay for a radioactive isotope is determined by its half-life, which is the time required for half of the isotope's nuclei to decay. This predictable decay allows scientists and students to trace the pathways of decay series like the thorium-lead series, helping in various applications from medicine to dating archaeological findings. By understanding isotopes' behavior, one can predict and manage the outcomes of radioactive decay.

Nuclear Chemistry

Nuclear chemistry is the study of the chemical and physical properties of elements as influenced by changes in the nucleus. This branch of chemistry particularly focuses on radioactivity, nuclear processes, and properties.

A significant aspect of nuclear chemistry is understanding decay series, where unstable isotopes transform to become stable. This process is seen in the thorium to lead decay sequence, where nuclear processes guide the transformation of isotopes through alpha and beta decay.

Since nuclear changes involve large energies, they play crucial roles in applications like nuclear energy and medical therapy. Nuclear chemists explore how these transformations can be effectively utilized while ensuring safety and environmental concerns. The study of nuclear chemistry serves as a bridge between understanding atomic interactions and practical applications in technology and health.

Atomic Number Changes

Atomic number changes are a core concept in nuclear chemistry and radioactive decay. The atomic number, denoted by Z, represents the number of protons in an atom's nucleus. When a decay occurs, the atomic number can increase or decrease, shifting the element's identity in the periodic table.

For instance, alpha decay reduces the atomic number by 2, while beta decay increases it by 1. Considering the decay series from thorium-232 to lead-208, these changes in atomic number are crucial in achieving a stable nucleus.

Understanding atomic number changes helps predict the sequence of elements in a decay series. With thorium-232 having atomic number 90 and lead-208 with 82, a series of six alpha decays (totaling -12) and four beta decays (totaling +4) bring the sequence appropriately from 90 to 82.