Question

Complete and balance the nuclear equations for the following fission or fusion reactions: (a) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \cdots{ }_{2}^{3} \mathrm{He}+\underline{-}\) (b) \({ }_{92}^{233} \mathrm{U}+{ }_{0} \mathrm{n} \cdots \rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{98} \mathrm{Nb}+-\mathrm{d} \mathrm{n}\)

Short answer

The short answers for the balanced nuclear equations are: (a) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \rightarrow{ }_{2}^{3}\mathrm{He}+{ }_{0}^{1}\mathrm{n}\) (b) \({ }_{92}^{233} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{98} \mathrm{Nb}+3{ }_{0}^{1} \mathrm{n}\)

Step-by-step solution

Identify the initial reactants and products

We are given the following reactants and products for this fusion reaction: Reactants: \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H}\) Products: \({ }_{2}^{3}\mathrm{He}+\underline{-}\)

Balance protons and neutrons

In order to balance the equation, we need to ensure that the total number of protons and total number of neutrons are conserved on both sides. Let's count them for each side: Reactants: - Protons: 1 + 1 = 2 - Neutrons: 1 + 1 = 2 Products: - Protons: 2 - Neutrons: 1 + x Since the number of protons is already conserved, we only need to conserve the number of neutrons. We have 1 neutron in the helium-3 nucleus, so we need 1 more neutron in the unidentified product to have a total of 2 neutrons. This product is a neutron \({ }_{0}^{1}\mathrm{n}\). The balanced fusion reaction is: \[{ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \rightarrow{ }_{2}^{3}\mathrm{He}+{ }_{0}^{1}\mathrm{n}\] (b) This is a nuclear fission reaction, which involves splitting a heavy nucleus into two lighter nuclei.

Identify the initial reactants and products

We are given the following reactants and products for this fission reaction: Reactants: \({ }_{92}^{233} \mathrm{U}+{ }_{0}^{1} \mathrm{n}\) Products: \(\rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{98} \mathrm{Nb}+-\mathrm{d} \mathrm{n}\)

Balance protons and neutrons

In order to balance the equation, we need to ensure that the total number of protons and total number of neutrons are conserved on both sides. Let's count them for each side: Reactants: - Protons: 92 - Neutrons: 141 + 1 = 142 Products: - Protons: 51 + 41 = 92 - Neutrons: 82 + 57 + d = 139 + d Since the number of protons is already conserved, we only need to conserve the number of neutrons. We have 139 neutrons in the two product nuclei, and require 3 more neutrons to have a total of 142 neutrons. Therefore, d = 3. The balanced fission reaction is: \[{ }_{92}^{233} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{98} \mathrm{Nb}+3{ }_{0}^{1} \mathrm{n}\]

Key concepts

Nuclear Fission

Nuclear fission is a reaction in which a heavy atomic nucleus, such as uranium or plutonium, splits into two smaller nuclei, releasing a significant amount of energy in the process. This energy release can be harnessed for various purposes, including electricity generation in nuclear reactors and explosiveness in nuclear weapons.

The process begins when a neutron collides with a heavy nucleus, causing it to become unstable and split. The fission reaction not only produces smaller nuclei but also free neutrons and gamma radiation. These newly released neutrons can initiate further fission reactions, leading to a chain reaction if conditions allow. In the exercise provided, the fission of Uranium-233 is shown, where it splits into Antimony-133 and Niobium-98, releasing three more neutrons. Balancing the equation requires the conservation of both protons and neutrons, which aligns with the principle of conservation of nucleons.

Nuclear Fusion

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus, accompanied by the release of energy. This is the same process that powers the sun and other stars, where hydrogen atoms fuse to form helium, emitting light and heat.

Fusion requires extremely high temperatures and pressures to overcome the electrostatic forces repelling the positively charged nuclei. In the example given, two deuterium nuclei ({ }_{1}^{2}H), each consisting of one proton and one neutron, fuse to create helium-3 ({ }_{2}^{3}He) and a free neutron. This reaction needs to respect the conservation of nucleons, which means that the number of protons and neutrons must be the same before and after the reaction. The balanced fusion equation is evidence of this conservation.

Balance Nuclear Equations

To balance a nuclear equation, just like in chemical reactions, the principle of conservation of mass and energy must be followed. In nuclear reactions, this is specifically known as the conservation of nucleons, which states that the total number of protons and neutrons must remain constant.

The balancing process involves ensuring that the sum of atomic and mass numbers on both sides of the equation is equal. Atomic numbers (the number of protons) give the identity of the element and must be balanced to maintain the identity of the reactants and products. Mass numbers (the sum of protons and neutrons) govern the total mass before and after the reaction. In the examples, this method is applied by counting and comparing the number of nucleons on each side and adding missing particles to the products until equilibrium is achieved.

Conservation of Nucleons

The conservation of nucleons is a fundamental principle in nuclear physics. It states that the total number of protons and neutrons (together called nucleons) in a nuclear reaction must remain unchanged before and after the reaction occurs.

This principle is crucial when dealing with nuclear reactions because it ensures that the mass and atomic number are conserved through any transformation, whether it's fission, fusion, or radioactive decay. As a result, in the provided exercise, after balancing the equations for both fission and fusion processes, the number of nucleons on both sides of the equation remains equal. This conservation is analogous to the conservation of matter in chemical reactions and is a cornerstone in the understanding and prediction of nuclear reaction outcomes.