Problem 15 - Chemistry

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. \(_{5}^{8} \mathrm{B}(\text { positron })\)

Step-by-Step Solution

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 1: 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.

Step 2: 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#

Step 1: 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.

Step 2: Write the equation for the beta decay process

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

Step 3: 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#

Step 1: 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.

Step 2: 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#

Step 1: 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.

Step 2: 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.