Question

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

Short answer

(a) \({}^{15}_{7} \mathrm{N}\), (b) \({}^{25}_{12} \mathrm{Mg}\), (c) \({}^{56}_{25} \mathrm{Mn}\)

Step-by-step solution

Understanding the Reactions

We need to write balanced nuclear equations for each reaction, determining the unknown nuclear species \( \mathrm{X} \). Each equation must balance in terms of mass number and atomic number. We identify the reactants and products from the notation \((a, b)\) where \(a\) and \(b\) are particles exchanged during the reaction.

Equation (a): Finding X in \(\mathrm{X} (p, \alpha) \) Reaction

In reaction \(\mathrm{X} (p, \alpha) \rightarrow_{6}^{12} \mathrm{C}\), a proton \((p)\) is bombarding an unknown nucleus \(\mathrm{X}\), and an alpha particle \((\alpha)\) is emitted with \({}^{12}_{6} \mathrm{C}\) as a product. The alpha particle is represented as \({}^{4}_{2}\mathrm{He}\) and the proton as \({}^{1}_{1}\mathrm{p}\). We balance mass and atomic numbers:- Mass number: \( A_X + 1 = 4 + 12 \rightarrow A_X = 15 \).- Atomic number: \( Z_X + 1 = 2 + 6 \rightarrow Z_X = 7 \).Thus, the unknown nucleus \(\mathrm{X}\) is \({}^{15}_{7} \mathrm{N}\).

Equation (b): Finding X in \({}^{27}_{13}\mathrm{Al} (d, \alpha) \) Reaction

In this reaction pattern, a deuteron \((d)\), represented as \({}^{2}_{1}\mathrm{H}\), strikes \({}^{27}_{13}\mathrm{Al}\) releasing an alpha particle \(\alpha\) and forming \(\mathrm{X}\). Balance the numbers:- Mass number: \( 2 + 27 = 4 + A_X \rightarrow A_X = 25 \).- Atomic number: \( 1 + 13 = 2 + Z_X \rightarrow Z_X = 12 \).Therefore, \(\mathrm{X}\) is \({}^{25}_{12} \mathrm{Mg}\).

Equation (c): Finding X in \({}^{55}_{25}\mathrm{Mn} (n, \gamma) \) Reaction

In this reaction, a neutron \((n)\) bombards \({}^{55}_{25}\mathrm{Mn}\) and a gamma photon \((\gamma)\) is emitted hand-in-hand with the formation of \(\mathrm{X}\). \(\gamma\) has zero mass and charge, so:- Mass number: \( 55 + 1 = A_X \rightarrow A_X = 56 \).- Atomic number: \( 25 + 0 = Z_X \rightarrow Z_X = 25 \).The species \(\mathrm{X}\) formed is \({}^{56}_{25} \mathrm{Mn}\), the subsequent isotope of manganese.

Key concepts

Mass Number

In nuclear chemistry, understanding the mass number is crucial for balancing equations. The mass number, represented by the symbol \( A \), is the total count of protons and neutrons in an atomic nucleus. It's like the "weight" of an atom's nucleus since neutrons and protons are the heavy parts of an atom.

• Protons have a mass of approximately 1 amu (atomic mass unit).
• Neutrons also have a mass of about 1 amu.

The mass number is not to be confused with the atomic number, as these are two different characteristics of an element. When writing nuclear equations, we always ensure the sum of the mass numbers on the reactant side is equal to the sum on the product side, maintaining the law of conservation of mass.

Atomic Number

The atomic number, symbolized as \( Z \), refers to the number of protons within the nucleus of an atom. This number is fundamental because it determines the element's identity. In a periodic table, elements are arranged by increasing atomic number.

• Each element has a unique atomic number, no two elements share the same number of protons.
• The number of protons directly influences the atom's behavior and chemical properties.

In nuclear reactions, like those in the given problem, the equation must balance in terms of atomic numbers, ensuring the sum of atomic numbers for the reactants equals that of the products. This reflects the conservation of charge.

Nuclear Reactions

Nuclear reactions involve changes in an atom's nucleus and can result in the transformation of elements. These reactions differ from chemical reactions, which involve only the electrons surrounding the nucleus.Nuclear reactions can be described using various notation methods, often shown as \( (a, b) \) reactions, where \( a \) is the projectile and \( b \) is the emitted particle. Key points include:

• Conservation of mass number and atomic number is fundamental.
• Nuclear reactions can produce new elements or isotopes.

Unlike chemical reactions, nuclear reactions have the potential to release or absorb immense quantities of energy. This energy is due to the conversion of mass into energy as described by Einstein's equation \( E = mc^2 \).

Alpha Particle

An alpha particle is a type of nuclear radiation described as \( {}^{4}_{2} \mathrm{He} \), composed of two protons and two neutrons. Alpha particles are similar to the helium nucleus.

• They have a +2 charge due to the presence of two protons.
• Alpha particles have a relatively high mass, compared to other types of radioactive emissions like beta particles or gamma rays.

These particles are common in nuclear decay reactions and nuclear transformations. In the problems above, alpha particles are part of the processes that alter the structure of the nucleus, highlighting their role in nuclear equations.

Deuteron

A deuteron is a nucleus of deuterium, an isotope of hydrogen, represented as \( {}^{2}_{1} \mathrm{H} \). It consists of one proton and one neutron, making it heavier than ordinary hydrogen, which contains only one proton.

• Deuterons are stable and often used in nuclear reactions as projectiles.
• The presence of a neutron in addition to a proton gives deuterons different nuclear properties than regular hydrogen nuclei.

Deuterons play a significant role in nuclear reactions and fusion processes, serving as a building block for creating heavier elements. These reactions are crucial for balancing nuclear equations, as seen in the given exercise when aluminum reacts with a deuteron to form magnesium and an alpha particle.