Question

The naturally occurring radioactive decay series that begins with \(_{92}^{235} \mathrm{U}\) stops with formation of the stable \(_{82}^{207} \mathrm{Pb}\) nucleus. The decays proceed through a series of alpha-particle and beta- particle emissions. How many of each type of emission are involved in this series?

Short answer

In the radioactive decay series starting with \(_{92}^{235}\mathrm{U}\) and ending with \(_{82}^{207}\mathrm{Pb}\), there are 7 alpha-particle emissions (A) and 4 beta-particle emissions (B).

Step-by-step solution

Identify the initial and final nuclei and their atomic and mass numbers.

The initial nucleus is \(_{92}^{235}\mathrm{U}\), with an atomic number of 92 and a mass number of 235. The final nucleus is \(_{82}^{207}\mathrm{Pb}\), with an atomic number of 82 and a mass number of 207.

Understand the changes in atomic and mass numbers during alpha-particle and beta-particle emissions.

In an alpha-particle emission, the atomic number of the nucleus decreases by 2, and the mass number decreases by 4. In a beta-particle emission, the atomic number increases by 1, while the mass number remains unchanged.

Determine the change in atomic and mass numbers in the decay series.

The change in atomic number is the final atomic number minus the initial atomic number, so it is 82 - 92 = -10. The change in mass number is the final mass number minus the initial mass number, so it is 207 - 235 = -28.

Set up equations for the number of alpha-particle and beta-particle emissions.

Let A be the number of alpha-particle emissions, and B be the number of beta-particle emissions. From Step 2, we know that alpha-particle emissions cause a decrease of 2 in the atomic number and 4 in the mass number, while beta-particle emissions cause an increase of 1 in the atomic number. Therefore, we can set up the following equations: -2A + B = -10 (change in atomic number) -4A = -28 (change in mass number)

Solve the two equations for A and B.

From the second equation, we can directly find the number of alpha-particle emissions: -4A = -28 A = 7 Now we can substitute the value of A in the first equation: -2(7) + B = -10 -14 + B = -10 B = 4

Interpret the results.

In the radioactive decay series starting with \(_{92}^{235}\mathrm{U}\) and ending with \(_{82}^{207}\mathrm{Pb}\), there are 7 alpha-particle emissions and 4 beta-particle emissions.

Key concepts

Alpha-Particle Emission

Alpha-particle emission is a type of radioactive decay where an unstable nucleus releases an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, equivalent to a helium nucleus. When an alpha particle is emitted from a radioactive element, two major changes occur:

• The atomic number of the original nucleus decreases by 2, as it loses two protons.
• The mass number decreases by 4, due to the loss of the two protons and two neutrons.

These changes explain why the element transforms into a new, lighter element in the decay series. In the problem given, the process starts with \(_{92}^{235}\mathrm{U}\) and ends with \(_{82}^{207}\mathrm{Pb}\). With each alpha emission, the atomic and mass numbers change accordingly, contributing to the overall transformation in the decay series.

Beta-Particle Emission

Beta-particle emission is another form of radioactive decay. It occurs when a neutron in an atom's nucleus transforms into a proton while emitting a beta particle, which is an electron or positron. This transformation leads to specific changes:

• The atomic number increases by 1, as the neutron becomes an additional proton.
• The mass number remains unchanged because the total number of nucleons (protons and neutrons) stays the same.

Beta emissions allow the radioactive decay series to adjust without affecting the mass number. Notably, while the unstable nuclei convert into more stable forms, they increase their atomic number and progress through the decay series. In this decay series from \(_{92}^{235}\mathrm{U}\) to \(_{82}^{207}\mathrm{Pb}\), beta emissions contribute to the changes necessary to balance the transformations initiated by alpha emissions.

Atomic Number Change

The atomic number change is a key aspect of understanding radioactive decay series. It represents the difference in the number of protons within the nucleus from the start to the end of a decay sequence. In alpha-particle and beta-particle emissions:

• Alpha emissions reduce the atomic number by 2 for each decay event.
• Beta emissions increase the atomic number by 1 per event.

By focusing on the problem, the initial atomic number is 92 (uranium), and the final is 82 (lead), resulting in a reduction of 10. This change is achieved through a combination of alpha and beta emissions, allowing us to follow the element's journey from a heavier to a more stable form. The sequence involves calculating the contributions from both alpha and beta decays to achieve this net change of -10.

Mass Number Change

Mass number change is another crucial concept in radioactive decay. The mass number reflects the total count of protons and neutrons in the nucleus. In this decay series:

• Alpha-particle emissions cause the mass number to decrease by 4 for each event since two protons and two neutrons are lost.
• Beta-particle emissions do not alter the mass number, as they convert neutrons into protons without changing total nucleon count.

For the case of \(\mathrm{U}\) to \(\mathrm{Pb}\), the initial mass number, 235, decreases to 207, indicating a net change of -28. This results solely from alpha emissions, as beta emissions do not influence the mass number. The calculation of required decay events, specifically alpha emissions, uses these changes to determine the pathway from \(\mathrm{U}\) to \(\mathrm{Pb}\).