Problem 32 - Chemistry The Science in Context

Question

Bromine has isotopes with mass numbers from 74 through 90\. Two of them, \(^{79} \mathrm{Br}\) and \(^{81} \mathrm{Br}\), are stable. a. How many of the others emit positrons or undergo electron capture? b. How many of the others emit \(\beta\) particles? c. Which one of the other isotopes can emit either positrons or \(\beta\) particles?

Step-by-Step Solution

Answer

Answer: There are 5 unstable isotopes that undergo positron emission or electron capture (mass numbers 74-78), 9 unstable isotopes that undergo beta decay (mass numbers 82-90), and the isotope \(^{80} \mathrm{Br}\) can emit either positrons or beta particles.

Step 1: Identify the Stable and Unstable Isotopes of Bromine

We are given that there are two stable isotopes of bromine: \(^{79} \mathrm{Br}\) and \(^{81} \mathrm{Br}\). All other isotopes with mass numbers from 74 through 90 are unstable. So, we have a total of 15 unstable isotopes to analyze.

Step 2: Determine Decay Processes for Each Isotope

In general, isotopes lighter than the stable isotopes (mass numbers 74 to 78) will tend to undergo either positron emission or electron capture, while heavier isotopes (mass numbers 82 to 90) will undergo beta decay. Let's break this down further: a. Isotopes from 74 through 78 will tend to undergo positron emission or electron capture, so there are 5 isotopes in this category. b. Isotopes from 82 through 90 will tend to undergo beta decay, so there are 9 isotopes in this category.

Step 3: Identify Isotope That Can Emit Either Positrons or Beta Particles

We need to find the one isotope that lies between the positron-emitting/electron-capturing and beta-decaying isotopes, which is \(^{80} \mathrm{Br}\). This isotope has a mass number in between the stable isotopes and can emit either positrons or beta particles.

Step 4: Final Results

a. There are 5 isotopes that emit positrons or undergo electron capture: isotopes with mass numbers 74 through 78. b. There are 9 isotopes that emit beta particles: isotopes with mass numbers 82 through 90. c. The isotope which can emit either positrons or beta particles is \(^{80} \mathrm{Br}\).

Key Concepts

Isotope Stability

When discussing elements such as bromine, it's crucial to understand the concept of isotope stability. Isotopes are variants of a particular chemical element that have the same number of protons but a different number of neutrons within their nuclei. In the world of isotopes, not all are created equal—some are stable, and some are not.

Stable isotopes, like Bromine's ^{79}Br and ^{81}Br, do not change over time. This stability is because the configurations of protons and neutrons within these isotopes result in a balanced nuclear force that does not require the isotope to shed particles to reach a lower energy state. However, other isotopes of bromine are unstable and seek stability through various decay processes, such as emitting positrons, undergoing electron capture, or beta decay.

The stability of an isotope can be influenced by several factors including the neutron-to-proton ratio and the presence of 'magic numbers' of neutrons or protons, which correspond to filled nuclear shells that confer extra stability. In your exercises, understanding which isotopes are stable and why can greatly enhance your grasp of the overall concept of nuclear chemistry.

Electron Capture

Electron capture is a process that plays a crucial role in the quest for stability of certain isotopes. During this process, an inner orbital electron is drawn into the nucleus where it combines with a proton to form a neutron and a neutrino. The neutrino escapes from the nucleus, and the result is a decrease in the atomic number of the element.

This type of decay is common among isotopes that have too many protons to be stable. These protons can attract an electron to balance out their charge surplus, effectively turning into neutrons and leading towards more stable configurations. It's important to note that while electron capture changes the element to another up the periodic table, the mass number remains the same because the atomic mass of a proton roughly equals that of a neutron.

Beta Decay

Beta decay is another mechanism by which unstable isotopes may attain a more stable form. Essentially, this decay process involves the transformation of a neutron into a proton, which results in the emission of an electron (beta particle) and an antineutrino from the nucleus.

There are two types of beta decay: beta-minus and beta-plus. Beta-minus decay increases the atomic number of an element by one, while the mass number stays consistent. This is typical for isotopes with an excess of neutrons. The lighter isotopes of bromine, however, are not prone to this type of decay; it is heavier isotopes, like those with mass numbers 82 to 90, that undergo beta-minus decay to achieve a more favorable neutron-to-proton ratio. As part of understanding this process, recognizing the signs of beta decay can help you predict the stability of isotopes and the probable pathways they might take in their decay.

Positron Emission

Positron emission is the less common counterpart to beta decay, representing the beta-plus decay process. In this process, a proton in an unstable nucleus is converted into a neutron with the release of a positron—the antimatter equivalent of an electron—and a neutrino. This type of decay decreases the atomic number by one but, like its beta decay sibling, leaves the mass number unchanged.

Positron emission occurs in isotopes that have too many protons to maintain stability, which as noted in the textbook solution, includes lighter bromine isotopes with mass numbers from 74 to 78. The emission of a positron effectively reduces the repulsive forces among protons within the nucleus, bringing the isotope closer to the band of stability. It's an intriguing process that highlights the balance forces within the atomic nucleus seek to maintain.