Question

Write balanced equations for each of the following nuclear reactions: (a) \({ }_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma){ }_{92}^{239} \mathrm{U},(\mathrm{b}){ }_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha){ }_{7}^{13} \mathrm{~N}\), (c) \({ }_{8}^{18} \mathrm{O}(\mathrm{n}, \beta){ }_{9}^{19} \mathrm{~F}\). Rates of Radioactive Decay (Section 21.4)

Short answer

The balanced equations for the given nuclear reactions are: 1. \( {}_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma){ }_{92}^{239} \mathrm{U}\) 2. \( { }_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha) { }_{7}^{13} \mathrm{N} \) 3. \( { }_{8}^{18} \mathrm{O}(\mathrm{n}, \beta) { }_{9}^{19} \mathrm{F} \)

Step-by-step solution

Conservation Laws

To balance nuclear reactions, we must ensure that conservation laws are obeyed, which includes: 1. Conservation of nucleons (total number of protons and neutrons) 2. Conservation of charge (total charge should remain the same) Armed with this knowledge, we can now proceed to balance the given nuclear reactions. Step 2: Balancing Reaction (a)

Balancing Reaction (a) - Uranium capturing a neutron and emitting gamma.

In this reaction, we have: \({}_{92}^{238} \mathrm{U} + \mathrm{n} \longrightarrow {}_{92}^{239} \mathrm{U} + \gamma\) The reaction is already balanced in terms of nucleons and charge: - Nucleons: \(238 + 1 = 239\) - Charge: \(92 = 92\) So, the balanced equation for reaction (a) is: \( {}_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma){ }_{92}^{239} \mathrm{U}\) Step 3: Balancing Reaction (b)

Balancing Reaction (b) - Oxygen capturing a proton and releasing an alpha particle

In this reaction, we have: \({ }_{8}^{16} \mathrm{O} + \mathrm{p} \longrightarrow { }_{7}^{13} \mathrm{N} + \alpha\) Now we must balance the equation in terms of nucleons and charge: - Nucleons: \(16 + 1 = 13 + X\) - Charge: \(8 + 1 = 7 + Y\) Solving for X and Y, we find that: - \(X = 4\) - \(Y = 2\) Since an alpha particle consists of 2 protons and 2 neutrons, this matches with the given reaction output, which means it is balanced. So, the balanced equation for reaction (b) is: \( { }_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha) { }_{7}^{13} \mathrm{N} \) Step 4: Balancing Reaction (c)

Balancing Reaction (c) - Oxygen capturing a neutron and releasing a beta particle

In this reaction, we have: \({ }_{8}^{18} \mathrm{O} + \mathrm{n} \longrightarrow { }_{9}^{19} \mathrm{F} + \beta\) Now we must balance the equation in terms of nucleons and charge: - Nucleons: \(18 + 1 = 19 + X\) - Charge: \(8 = 9 + Y\) Solving for X and Y, we find that: - \(X = 0\) (no additional nucleon) - \(Y = -1\) (which is the charge of a beta particle, also known as an electron) So, the balanced equation for reaction (c) is: \( { }_{8}^{18} \mathrm{O}(\mathrm{n}, \beta) { }_{9}^{19} \mathrm{F} \) In conclusion, the balanced equations for the given nuclear reactions are: 1. \( {}_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma){ }_{92}^{239} \mathrm{U}\) 2. \( { }_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha) { }_{7}^{13} \mathrm{N} \) 3. \( { }_{8}^{18} \mathrm{O}(\mathrm{n}, \beta) { }_{9}^{19} \mathrm{F} \)

Key concepts

Conservation Laws in Nuclear Reactions

Conservation laws are fundamental principles in physics, especially when studying nuclear reactions. These laws ensure that certain properties remain constant throughout the reaction. In nuclear reactions, we primarily look at two major conservation laws:

• Conservation of Nucleons: This law states that the total number of nucleons (protons + neutrons) remains the same before and after the reaction. In simpler terms, the sum of the nucleons in the reactants must equal the sum of the nucleons in the products.
• Conservation of Charge: This is another crucial law, which ensures that the total charge is conserved in a nuclear reaction. The sum of the atomic numbers (charge) in the reactants should equal the sum of the atomic numbers in the products.

Applying these principles helps us balance nuclear equations by making sure that no nucleons or charges mysteriously appear or disappear during the reaction. Understanding these laws allows us to predict and balance nuclear reactions efficiently.

Understanding Nucleons

Nucleons are the core components of an atomic nucleus, comprising protons and neutrons. These particles are fundamental to the structure of matter and play a vital role in nuclear reactions.

• The number of nucleons in a nucleus is also known as the mass number. It is crucial in identifying the stability and properties of an element.
• Protons: These positively charged particles determine the identity of an element. For instance, all oxygen atoms have 8 protons.
• Neutrons: Neutrons, with no charge, contribute to the weight of the nucleus and can affect the stability of an atom. Different isotopes of an element have variations in neutron number.

During nuclear reactions, nucleons can be rearranged but not created or destroyed, respecting the conservation of nucleons. This fundamental balance is what underlies the stability of elements and isotopes.

Alpha Particles in Nuclear Reactions

Alpha particles play a significant role in nuclear reactions. They are composed of two protons and two neutrons, symbolizing a helium nucleus. This makes them relatively heavy and positively charged.

• Emission: In many radioactive decay processes, an unstable nucleus may eject an alpha particle to become more stable. This is known as alpha decay.
• Balancing Equations: When an alpha particle is involved in a reaction, it adds two protons and two neutrons to the product side. It’s crucial to account for this when balancing nuclear reactions.
• Impact: Alpha particles have lower penetration depth but can be highly ionizing. They can cause significant damage to biological tissues at close range.

Understanding the role of alpha particles helps in predicting the outcomes of nuclear reactions and ensuring equilibrium in nuclear equations.

Beta Particles and Their Characteristics

Beta particles are another key component in some nuclear reactions. They can be either electrons or positrons, resulting from beta decay in an unstable nucleus.

• Beta Minus Decay: This occurs when a neutron in an unstable nucleus transforms into a proton and an electron (beta particle) is emitted. Beta minus particles are negatively charged.
• Beta Plus Decay: Also known as positron emission, this happens when a proton converts into a neutron with the emission of a positron (the beta plus particle). Positrons are positively charged.
• Balancing Equations: Including a beta particle in a nuclear reaction affects the charge balance. For beta minus decay, the atomic number of the product element increases by one, while for beta plus decay, it decreases by one.
• Peneranting Ability: Beta particles carry moderate energy and can penetrate materials more deeply than alpha particles. However, they are less ionizing.

Comprehending beta particles aids in visualizing complex nuclear reactions and understanding changes in atomic structures.