Question

Write an equation describing the radioactive decay of each of the following nuclides. (The particle produced is shown in parentheses, except for electron capture, where an electron is a reactant.) a.\(_{1}^{3} \mathrm{H}(\beta)\) b. \(_{3}^{8} \operatorname{Li}(\beta \text { followed by } \alpha)\) c. \(\quad_{4}^{7}\) Be (electron capture) d. \(_{5}^{8} \mathrm{B}(\text { positron })\)

Short answer

The short version of the answer for each nuclide radioactive decay is: a. \[_{1}^{3} \mathrm{H} \to _{2}^{3} \mathrm{He} + \beta^{-} + \bar{\nu}_e\] b. \[_{3}^{8} \operatorname{Li} \to _{4}^{8} \mathrm{Be} + \beta^{-} + \bar{\nu}_e\] _ \[_{4}^{8} \mathrm{Be} \to _{2}^{4} \mathrm{He} + _{2}^{4} \mathrm{He}\] c. \[_{4}^{7} \mathrm{Be} + e^{-} \to _{3}^{7} \mathrm{Li} + \nu_e\] d. \[_{5}^{8} \mathrm{B} \to _{4}^{8} \mathrm{Be} + \beta^{+} + \nu_e\]

Step-by-step solution

Identify the initial nuclide and decay process

For this problem, we have Hydrogen-3 undergoing beta (β) decay. In a beta decay, the initial nuclide emits an electron (β-) and an antineutrino, converting a neutron to a proton.

Write the equation for the decay process

We will write the equation as follows: \[_{1}^{3} \mathrm{H} \to _{2}^{3} \mathrm{He} + \beta^{-} + \bar{\nu}_e\] #b. Beta decay followed by alpha decay of Lithium-8#

Recognize the initial nuclide and decay processes

For this decay, we have Lithium-8 undergoing beta (β) decay followed by alpha (α) decay. We'll handle these two decay processes separately.

Write the equation for the beta decay process

The initial nuclide emits an electron (β-) and an antineutrino, converting a neutron to a proton: \[_{3}^{8} \operatorname{Li} \to _{4}^{8} \mathrm{Be} + \beta^{-} + \bar{\nu}_e\]

Write the equation for the alpha decay process

After the beta decay, the produced Beryllium-8 undergoes an alpha decay, during which an alpha particle (Helium-4 nucleus) is emitted. \[_{4}^{8} \mathrm{Be} \to _{2}^{4} \mathrm{He} + _{2}^{4} \mathrm{He}\] #c. Electron capture of Beryllium-7#

Identify the initial nuclide and decay process

The initial nuclide is Beryllium-7, which undergoes electron capture. In electron capture, an inner-shell electron is captured by the nucleus and combines with a proton to form a neutron. This process also emits a neutrino.

Write the equation for the electron capture process

We will write the equation as follows: \[_{4}^{7} \mathrm{Be} + e^{-} \to _{3}^{7} \mathrm{Li} + \nu_e\] #d. Positron decay of Boron-8#

Identify the initial nuclide and decay process

The initial nuclide is Boron-8, which undergoes positron decay. In positron decay, the nucleus emits a positron (β+), i.e., an anti-electron, which converts a proton into a neutron. This process also emits a neutrino.

Write the equation for the positron decay process

We will write the equation as follows: \[_{5}^{8} \mathrm{B} \to _{4}^{8} \mathrm{Be} + \beta^{+} + \nu_e\]

Key concepts

Beta Decay

Beta decay is a form of radioactive decay where a beta particle, which is an electron or a positron, is emitted from an atomic nucleus. This process transforms a neutron into a proton or vice versa. There are two types of beta decay: beta-minus (β-) and beta-plus (β+). In beta-minus decay, a neutron in an unstable nucleus is converted into a proton, emitting an electron (β-) and an antineutrino. This increases the atomic number by one, but the mass number remains the same. For example, in the decay of Hydrogen-3, the equation is written as:

• \[_{1}^{3} ext{H} ightarrow _{2}^{3} ext{He} + eta^{-} + \bar{u}_e\]

Conversely, in beta-plus decay (positron emission), a proton is transformed into a neutron, releasing a positron and a neutrino, decreasing the atomic number by one. Both processes are essential for the stability of elements with unbalanced proton-to-neutron ratios.

Electron Capture

Electron capture is an intriguing process where an atomic nucleus captures one of its inner atomic electrons. This electron combines with a proton, transforming it into a neutron and emitting a neutrino. Electron capture is generally observed in proton-rich nuclei. This process decreases the atomic number by one while leaving the mass number unchanged. It's the opposite of beta decay. Consider the case of Beryllium-7 undergoing electron capture:

• \[_{4}^{7} ext{Be} + e^{-} \rightarrow _{3}^{7} ext{Li} + u_e\]

This results in the transformation of Beryllium-7 into Lithium-7. Electron capture provides a way for an atom to become more stable when other decay methods are energetically unfavorable.

Positron Emission

Positron emission is another type of beta decay known as beta-plus decay. In this process, a proton within an unstable nucleus is converted into a neutron, releasing a positron (β+) and a neutrino. This decreases the atomic number by one while keeping the mass number unchanged. Positron emission is a way for proton-rich nuclei to achieve greater stability by increasing their neutron-to-proton ratio. Let's take a closer look at Boron-8:

• \[_{5}^{8} ext{B} \rightarrow _{4}^{8} ext{Be} + \beta^{+} + u_e\]

During this decay, Boron-8 is transformed into Beryllium-8. This process is crucial in medical applications, such as positron emission tomography (PET) scans, which help visualize the function of tissues and organs.

Alpha Decay

Alpha decay is a common radioactive decay process where an unstable nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons, equivalent to a Helium-4 nucleus. This emission reduces the atomic number by two and the mass number by four, leading to the formation of a new element. Alpha decay is typically observed in heavy elements.An example of alpha decay can be seen after the beta decay of Lithium-8. Here’s the equation:

• \[_{4}^{8} \text{Be} \rightarrow _{2}^{4} \text{He} + _{2}^{4} \text{He}\]

In this reaction, the Beryllium-8 nucleus emits an alpha particle, leaving behind Helium nuclei. Alpha decay results in a significant change in the nucleus's structure and is often accompanied by the release of energy, making it detectable by various instruments used in radiometric dating and nuclear medicine.