Question

Complete and balance the nuclear equations for the following fission reactions: (a) \({ }_{92}^{235} \mathrm{U}+{ }_{0} \mathrm{n} \cdots \rightarrow{ }_{62}^{160} \mathrm{Sm}+{ }_{30}^{72} \mathrm{Zn}+\ldots_{0}^{1} \mathrm{n}\) (b) \({ }_{94}^{239} \mathrm{Pu}+{ }_{0}^{1} \mathrm{n} \rightarrow{ }_{58}^{144} \mathrm{Ce}+\underline{+2}_{0}^{1} \mathrm{n}\)

Short answer

The completed and balanced nuclear fission reactions are: (a) \({}_{92}^{235}\mathrm{U} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{62}^{160}\mathrm{Sm} + { }_{30}^{72}\mathrm{Zn} + 3_{0}^{1}\mathrm{n}\) (b) \({}_{94}^{239}\mathrm{Pu} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{58}^{144}\mathrm{Ce} + { }_{36}^{96}\mathrm{Kr} + 2_{0}^{1}\mathrm{n}\)

Step-by-step solution

(a) Balancing Mass and Protons

For reaction (a), we will first complete the equation: \({}_{92}^{235}\mathrm{U} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{62}^{160}\mathrm{Sm} + { }_{30}^{72}\mathrm{Zn} + x_{0}^{1}\mathrm{n}\) Now, we will balance the mass and protons for reaction (a). The mass number conservation equation is given by: \[235 + 1 = 160 + 72 + x\] And the atomic number conservation equation is: \[92 = 62 + 30\]

(a) Solve for x

Solving the atomic number conservation equation gives us: \[x = 235 - 160 - 72\] \[x = 3\] Therefore, the completed and balanced nuclear fission reaction for (a) is: \({}_{92}^{235}\mathrm{U} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{62}^{160}\mathrm{Sm} + { }_{30}^{72}\mathrm{Zn} + 3_{0}^{1}\mathrm{n}\)

(b) Balancing Mass and Protons

For reaction (b), we will first complete the equation: \({}_{94}^{239}\mathrm{Pu} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{58}^{144}\mathrm{Ce} + y_{36}^{96}\mathrm{Kr} + 2_{0}^{1}\mathrm{n}\) Now, we will balance the mass and protons for reaction (b). The mass number conservation equation is given by: \[239 + 1 = 144 + 96 + 2\] And the atomic number conservation equation is: \[94 = 58 + 36\]

(b) Checking Balancing

Since the atomic number conservation equation is satisfied and the mass numbers sum up correctly, the completed and balanced nuclear fission reaction for (b) is: \({}_{94}^{239}\mathrm{Pu} + { }_{0}^{1}\mathrm{n} \rightarrow { }_{58}^{144}\mathrm{Ce} + { }_{36}^{96}\mathrm{Kr} + 2_{0}^{1}\mathrm{n}\)

Key concepts

Nuclear Equation Balancing

Understanding how to balance a nuclear equation is crucial for anyone studying nuclear chemistry. Similar to chemical reactions, nuclear reactions must also obey certain laws of conservation. The first step in balancing nuclear equations is identifying the reactants and products involved in the reaction. In fission reactions, a heavy nucleus splits into lighter nuclei, often releasing additional neutrons and energy.

When balancing a nuclear equation, two main rules apply: the conservation of mass number and the conservation of atomic number. The mass numbers (the superscripts) must add up equally on both sides of the equation, and the same must be true for the atomic numbers (the subscripts). Additional elements or particles such as neutrons (0^{1}) may be included to balance the equation. Once the equation is set up with equal mass numbers and atomic numbers on each side, you've successfully balanced a nuclear equation.

Mass Number Conservation

The law of conservation of mass is pivotal in nuclear physics just as it is in classical chemistry. In a nuclear reaction, the sum of the mass numbers before the reaction must be equal to the sum after the reaction. The mass number, shown as the superscript to the left of the elemental symbol, represents the total number of protons and neutrons in the nucleus.

For example, when uranium-235 undergoes fission upon capturing a neutron and produces samarium-160, zinc-72, and additional neutrons, the total mass number of the products must equal the initial mass number of uranium-235 plus the neutron. If this is not the case, additional particles must be added to the product side to balance the equation.

Atomic Number Conservation

Similar to the conservation of mass number, the conservation of atomic number is another fundamental concept in nuclear reactions. The atomic number, indicated by the subscript to the left of the elemental symbol, refers to the number of protons in the nucleus and determines the chemical element's identity.

In balancing nuclear equations, the total atomic number must remain consistent before and after the reaction. This principle ensures that charge is conserved during the nuclear process. If the atomic number is not conserved initially, it indicates that the equation is imbalanced, and you'll need to adjust it to reflect the right number of protons by altering the products or reactants.

Nuclear Chemistry

Nuclear chemistry focuses on the processes and changes that occur within the nucleus of atoms. It includes various types of reactions such as nuclear fission, where a nucleus splits into smaller nuclei, and nuclear fusion, where smaller nuclei combine into a larger nucleus. These processes are governed by principles different from those of standard chemical reactions, as they involve changes in the nucleus rather than electron interactions.

The study of nuclear chemistry also involves understanding the emission of radioactivity and the application of radioactive substances in medicine, energy production, and other fields. Grasping the fundamentals of nuclear reactions, including the balancing of nuclear equations and the conservation laws, provides a foundation for comprehending the more in-depth and complex aspects of nuclear chemistry.