Problem 29 - Chemistry Structure and Dynamics

Question

Calculate the energy released in the following reaction. $$ { }^{10} \mathrm{~B}(n, \alpha)^{7} \mathrm{Li} $$ Use the following data for the masses of the particles involved in the reaction: \({ }^{10} \mathrm{~B}=10.0129 \mathrm{amu} ;{ }^{7} \mathrm{Li}=\) \(7.01600 \mathrm{amu} ;{ }^{4} \mathrm{He}=4.00260 \mathrm{amu}\). Explain why the abil- ity of \({ }^{10} \mathrm{~B}\) to release a high-energy \(\alpha\) particle after it absorbs a thermal neutron generated considerable interest in getting boron compounds to absorb preferentially into the fastgrowing tumor in patients who suffer from brain tumors.

Step-by-Step Solution

Answer

The energy released is approximately 2.76 MeV. Boron compounds target brain tumors due to their selective absorption and energy release.

Step 1: Identify Reaction Components

The reaction is \(^{10}B(n, \alpha)^{7}Li\). It involves a neutron \((n)\), boron \(^{10}B\), an alpha particle \((\alpha = ^{4}He)\), and lithium \(^7Li\).

Step 2: Write Total Mass of Reactants

The mass of the reactants includes the boron and neutron. Given: \(^{10}B = 10.0129\ amu\). Mass of neutron \(n = 1.00866\ amu\). Total: \(10.0129\ amu + 1.00866\ amu = 11.02156\ amu\).

Step 3: Write Total Mass of Products

The mass of the products includes lithium and the alpha particle. Given: \(^7Li = 7.01600\ amu\) and \(\alpha = ^4He = 4.00260\ amu\). Total: \(7.01600\ amu + 4.00260\ amu = 11.01860\ amu\).

Step 4: Calculate Mass Defect

Mass defect is the difference in mass between reactants and products. Mass defect \(\Delta m = 11.02156\ amu - 11.01860\ amu = 0.00296\ amu\).

Step 5: Convert Mass Defect to Energy

Use Einstein’s equation \(E = \Delta m \cdot c^2\) where \(c = 931.5\ MeV/amu\). Energy \(E = 0.00296\ amu \times 931.5\ MeV/amu = 2.75724\ MeV\).

Step 6: Discuss Medical Significance

The ability of \(^{10}B\) to release high-energy \(\alpha\) particles upon absorbing neutrons allows it to target and destroy cancer cells while minimizing damage to surrounding healthy tissues. This property makes boron compounds particularly attractive for treating brain tumors through boron neutron capture therapy.

Key Concepts

Mass Defect

In nuclear reactions, the concept of mass defect plays a key role. Mass defect is the difference between the total mass of the reactants and the total mass of the products in a nuclear reaction. When you calculate the mass defect, you start by adding up the atomic masses of all reactants and then do the same for the products.
In a nuclear reaction like \[^{10} ext{B}(n, \alpha)^{7} ext{Li}\], the sum of the masses of the boron atom and neutron will be slightly greater than the sum of the masses of the lithium atom and alpha particle produced. This is due to the conversion of some mass into energy, according to Einstein's famous equation, \(E = \Delta m \cdot c^2\).
This "missing" mass or mass defect is crucial in calculating the energy released during a nuclear reaction. In the equation, \(c\) represents the speed of light, and its square highlights how even a small mass defect can produce a significant amount of energy.

Medical Applications of Nuclear Reactions

Nuclear reactions have significant applications in the medical field, aiding in the diagnosis and treatment of various conditions. One of the benefits of medical nuclear reactions is their ability to target specific cells, such as cancer cells, without affecting surrounding healthy tissue.
For example:

• **Cancer Treatment**: Some nuclear reactions can release particles that destroy cancer cells directly.
• **Medical Imaging**: Radioactive isotopes are often used to provide detailed images of the inside of the body.
• **Therapeutic Agents**: Radioisotopes can be employed as therapeutic agents, releasing energy to treat diseases.

The capability to harness nuclear reactions in these ways allows for innovations such as Boron Neutron Capture Therapy, which specifically targets and destroys tumor cells. This precision reduces the risk of damaging healthy tissues.

Boron Neutron Capture Therapy

Boron Neutron Capture Therapy (BNCT) is an innovative cancer treatment that uses nuclear reactions to destroy cancer cells. How does it work? It relies on the reaction of boron-10 with neutrons to produce high-energy alpha particles and lithium nuclei. This reaction is symbolized as:\[^{10} ext{B} + n \rightarrow \alpha + ^{7} ext{Li}\]The reaction produces alpha particles that are highly energetic and can destroy cancer cells.
The key to BNCT's success is the selective uptake of boron compounds by tumor cells. When boron-10 in the body captures a neutron, it results in a localized reaction that releases destructive energy. The surrounding healthy tissues remain largely unaffected, because:

• Alpha particles have a limited range, thus they only affect cells near the reaction site.
• The specific uptake of boron by tumor cells ensures that most reactions occur within or near the tumor.

BNCT is particularly useful in treating brain tumors, as it minimizes damage to critical surrounding brain tissues.

Energy Conversion in Nuclear Reactions

Energy conversion in nuclear reactions involves turning small amounts of mass into significant amounts of energy. Using Einstein's equation \(E = \Delta m \cdot c^2\), one can calculate the energy emitted during such reactions.
During the reaction between boron-10 and a neutron, a small mass defect occurs. Here’s how this is quantified:

• The mass difference, or mass defect, between reactants and products is measured.
• This mass defect is then multiplied by the speed of light squared (as in Einstein's equation) to find the energy produced.

For example, in the reaction \[^{10} ext{B}(n, \alpha)^{7} ext{Li}\], the calculated mass defect translates to approximately 2.75724 MeV of energy. This energy is sufficient to produce high-energy alpha particles that are used in medical applications like BNCT.
The process of energy conversion in nuclear reactions is powerful and sustains the potential for both industrial power generation and medical applications, showcasing the vast capabilities of nuclear science.