Question

Write balanced nuclear equations for these reactions and identify \(\mathrm{X}:\) (a) \(\mathrm{X}(\mathrm{p}, \alpha){ }_{6}^{12} \mathrm{C},\) (b) \({ }_{13}^{27} \mathrm{Al}(\mathrm{d}, \alpha) \mathrm{X}\) (c) \({ }_{25}^{55} \mathrm{Mn}(\mathrm{n}, \gamma) \mathrm{X}\)

Short answer

The balanced nuclear equations for the given reactions are: (a) \(_6^{12}C(p, \alpha)_4^{8}Be\), (b) \(_{13}^{27}Al(d, \alpha)_{12}^{25}Mg\), (c) \(_{25}^{55}Mn(n, γ)_{25}^{56}Mn\)

Step-by-step solution

Balance equation (a)

We'll start with the equation \(_6^{12}C(p, \alpha)X\). Here a proton(p) interacts with a Carbon-12 nucleus producing alpha particle(α) and X. An alpha particle is essentially a helium-4 nucleus, denoted by \(_2^{4}He\). To balance the atomic numbers (on the bottom), the atomic number of X would be 6 - 2, which equals 4. Similarly, for mass numbers (on the top) the mass number of X would be 12 - 4, which equals 8. Therefore, X in this reaction is \(_4^{8}Be\). So, the balance equation is \(_6^{12}C(p, \alpha)_4^{8}Be\).

Balance equation (b)

Next, we'll balance the equation \(_{13}^{27}Al(d, \alpha)X\). In this reaction, a deuteron(d) interacts with an Aluminum-27 nucleus producing an alpha particle(α) and product X. An alpha particle is known as a helium-4 nucleus, represented by \(_2^{4}He\), and a deutron, the nucleus of a deuterium atom (hydrogen-2), is represented by \(_1^{2}H\). To balance the atomic numbers, the atomic number of X would be 13 + 1 - 2, which equals 12. Similarly, the mass number of X would be 27 + 2 - 4, which equals 25. Thus, X in this reaction is \(_{12}^{25}Mg\), and the balanced equation is \(_{13}^{27}Al(d, \alpha)_{12}^{25}Mg\)

Balance equation (c)

Finally, we'll balance the equation \(_{25}^{55}Mn(n, γ)X\). A neutron(n) interacts with a Manganese-55 nucleus producing a gamma ray(γ) and X. The gamma ray doesn't have any atomic or mass number so it won't affect the balance. In this context, X is simply the product of a neutron being absorbed by a Manganese-55 nucleus, meaning it's just Manganese with one more nucleon (the absorbing neutron). So, X in this reaction is \(_{25}^{56}Mn\), and the balanced equation is \(_{25}^{55}Mn(n, γ)_{25}^{56}Mn\)

Key concepts

Balancing Nuclear Reactions

Balancing nuclear reactions involves making sure that both sides of the equation have the same total number of protons (atomic number) and the same mass number (sum of protons and neutrons). This is quite similar to balancing chemical equations in chemistry, but here we focus on the composition of atomic nuclei.

To write a balanced nuclear reaction, follow these steps:

• Identify the known particles involved, like protons, neutrons, alpha particles, etc.
• Assign the correct atomic number and mass number to these particles.
• Ensure that the sum of atomic numbers on the reactant side equals the sum on the product side. Do the same for mass numbers.

For example, if you have a reaction in which a proton (a hydrogen nucleus) interacts with another nucleus to produce an alpha particle and an unknown "X," you would calculate the missing atomic and mass numbers based on these conservation rules.

In practice, this step-by-step approach assists in keeping the nuclear reactions balanced and understanding the transformation of elements during nuclear reactions.

Alpha Particle

An alpha particle is a common product in nuclear reactions and carries significant importance. It is simply the nucleus of a helium-4 atom, consisting of 2 protons and 2 neutrons. Thus, it has an atomic number of 2 and a mass number of 4.

Alpha particles are often emitted during the radioactive decay processes. They are represented as \(_{2}^{4}\mathrm{He}\). In nuclear equations, knowing the properties of an alpha particle helps in balancing the equations as it contributes both mass and atomic numbers.

For instance, when a nucleus emits an alpha particle, it loses 2 protons and 2 neutrons. This means:

• The atomic number decreases by 2.
• The mass number decreases by 4.

Recognizing these significant changes is crucial in accurately balancing nuclear reactions where alpha particles appear, assisting in identifying unknown products within the reaction.

Neutron Reactions

Neutron reactions are an integral component of nuclear chemistry, particularly due to the role of neutrons in initiating nuclear transformations. Neutrons, unlike protons and other charged particles, carry no charge, allowing them to penetrate atomic nuclei more easily without being repelled.

In nuclear reactions, a neutron is depicted simply by \(_{0}^{1}\mathrm{n}\), with a mass number of 1 and an atomic number of 0. Neutron reactions can lead to different types of reactions, including the absorption of neutrons by a nucleus. For example:

• When a neutron is absorbed, the target nucleus will gain one additional nucleon, increasing its mass number by 1 but not affecting its atomic number.
• This absorption may lead to a subsequent release of energy, commonly in the form of gamma radiation (\(\gamma\)).

Such neutron-induced reactions are not uncommon and play a significant role in processes like nuclear fission or the synthesis of heavier elements in reactors or stars. Understanding the behavior and role of neutrons helps unravel the complexity of nuclear transformations.