Question

In a cancer treatment called boron neutron-capture therapy, a. compound containing boron- 10 is injected into a patient where it selectively binds to cancer cells. Irradiating the affected area with neutrons then induces the following reaction: $$ { }^{10} \mathrm{~B}+{ }^{1} \mathrm{n} \longrightarrow{ }^{4} \mathrm{He}+{ }^{7} \mathrm{Li}+\gamma $$ The \(\alpha\) radiation kills the cancer cells, leaving the surrounding tissue unharmed. The reactants in this nuclear process have essentially no kinetic energy, but the products have a total kinetic energy of \(2.31 \mathrm{MeV}\). What is the energy of the \(\gamma\) photon released? Relevant masses are: \({ }^{4} \mathrm{He}(4.002603 \mathrm{u}),{ }^{7} \mathrm{Li}(7.016004 \mathrm{u})\), \({ }^{10} \mathrm{~B}(10.012937 \mathrm{u}), \mathrm{e}^{-}(0.0005486 \mathrm{u}),{ }_{0}^{1} \mathrm{n}(1.008665 \mathrm{u})\)

Short answer

The energy of the gamma photon released is 0.479 MeV.

Step-by-step solution

Calculate Total Mass of Reactants

First, determine the total mass of the reactants, which are boron-10 \( \left(^{10} \mathrm{B}\right) \) and a neutron \( \left(^{1} \mathrm{n}\right) \). The mass of \( ^{10}\mathrm{B} \) is 10.012937 u and the mass of a neutron is 1.008665 u. Therefore, the total mass of reactants is:\[10.012937 \text{ u} + 1.008665 \text{ u} = 11.021602 \text{ u}\]

Calculate Total Mass of Products

Next, calculate the total mass of the products which are helium-4 \( \left(^{4} \mathrm{He}\right) \), lithium-7 \( \left(^{7} \mathrm{Li}\right) \), and a gamma photon \( \gamma \). The mass of \( ^{4} \mathrm{He} \) is 4.002603 u and the mass of \( ^{7} \mathrm{Li} \) is 7.016004 u. Summing these gives:\[4.002603 \text{ u} + 7.016004 \text{ u} = 11.018607 \text{ u}\]

Find Mass Defect

Calculate the mass defect by subtracting the total mass of the products from the total mass of the reactants:\[11.021602 \text{ u} - 11.018607 \text{ u} = 0.002995 \text{ u}\]

Convert Mass Defect to Energy

Convert the mass defect obtained in Step 3 to energy using the equation \( E=mc^2 \), where \( c \) is the speed of light. Using the conversion factor 1 u = 931.5 MeV/c²:\[E = 0.002995 \times 931.5 \text{ MeV} = 2.789 \text{ MeV}\]

Calculate Energy of Gamma Photon

The total kinetic energy of the products is 2.31 MeV. The energy of the gamma photon \( \gamma \) is the total energy minus this kinetic energy, using the energy calculated in Step 4:\[E_{\gamma} = 2.789 \text{ MeV} - 2.31 \text{ MeV} = 0.479 \text{ MeV}\]

Key concepts

Nuclear Reaction

In a nuclear reaction, changes occur in the nucleus of atoms, resulting in the transformation of elements and the release or absorption of energy. These reactions are distinct from chemical reactions, which involve electrons and do not alter the nucleus. In boron neutron-capture therapy ( extit{BNCT}), a specific nuclear reaction occurs where a neutron is captured by a boron-10 \(^{10} \mathrm{~B}\) atom, leading to the production of helium-4 \(^{4} \mathrm{He}\), lithium-7 \(^{7} \mathrm{Li}\), and a gamma photon \(\gamma\). This particular nuclear reaction is beneficial in medical treatments due to its targeted nature.

These reactions often release a significant amount of energy, which can be harnessed for various applications, including power generation and, as seen in BNCT, medical therapies. Understanding the specifics of nuclear reactions, including the particles involved and the energy implications, is essential for safely leveraging their capabilities.

Cancer Treatment

Boron Neutron-Capture Therapy (BNCT) is a special cancer treatment that uses nuclear reactions to target and destroy cancer cells without harming healthy tissues. In BNCT, a compound containing boron-10 is administered to the patient, where it selectively accumulates in cancer cells. When the affected area is then bombarded with neutrons, an energetic reaction occurs.

The result of this reaction is the production of alpha particles, which are very effective at killing cancer cells due to their high energy and short range. This confined impact means that only cancer cells in proximity to boron are destroyed, while surrounding healthy cells remain largely unaffected. Its targeted approach makes BNCT a promising option for treating difficult-to-reach tumors and minimizing side effects.

Kinetic Energy

Kinetic energy refers to the energy possessed by an object due to its motion. In the context of nuclear reactions, when reactants undergo a nuclear transformation, the products often emerge with high kinetic energy. For BNCT, the nuclear reaction products include helium-4, lithium-7, and gamma photons. These products collectively hold a kinetic energy of 2.31 MeV, indicating they are moving rapidly post-reaction.

The high kinetic energy of these particles is crucial in the effectiveness of BNCT. The energy released is sufficient to cause cellular damage, effectively targeting cancer cells. Understanding kinetic energy's role in nuclear reactions helps delineate how energy is partitioned among reaction products, influencing the reaction's practical applications.

Mass Defect

Mass defect is a fundamental concept in nuclear physics describing the difference between the mass of reactants and the mass of products in a nuclear reaction. It arises because some mass is converted into energy during the reaction, as per Einstein's famous equation \(E = mc^2\). In the context of the BNCT reaction, the mass defect comes to about 0.002995 u.

This lost mass is converted into energy, which is released as gamma photons and contributes to the kinetic energy of the reaction products. By calculating the mass defect, scientists can predict how much energy will be available from a given nuclear reaction. This understanding is crucial, especially in medical applications, where precise energy control is needed to ensure effective treatment without harming the patient.

Photon Energy

Photon energy is the energy carried by a photon, a type of elementary particle in electromagnetic radiation. In the BNCT reaction, a gamma photon is one of the products, and its energy can be calculated based on the difference in energy before and after the reaction.

The energy of the gamma photon in this particular therapy is 0.479 MeV. This energy value is obtained by subtracting the total kinetic energy of the reaction products from the total energy calculated from the mass defect. Photon energy plays a significant role in medical treatments like BNCT, as it contributes to accurately targeting cancer cells without interfering with adjacent healthy tissues.

The calculation of photon energy using mass-energy equivalence is pivotal in ensuring the effectiveness of treatments using nuclear reactions, allowing for the precise calibration of energy release needed for therapeutic applications.