Problem 77 - Chemistry The Science in Context

Question

For each of the following fission reactions, determine th identity of the unknown nuclide: a. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{131} \mathrm{I}+?+2_{0}^{1} \mathrm{n}\) b. \(^{233} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{103} \mathrm{Ru}+?+3_{0}^{1} \mathrm{n}\) c. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{95} \mathrm{Zr}+?+3_{0}^{1} \mathrm{n}\)

Step-by-Step Solution

Answer

+2_{0}^{1}\mathrm{n}\) b. \(^{233}\mathrm{U}+_{0}^{1}\mathrm{n} \rightarrow^{103}\mathrm{Ru}+?+3_{0}^{1}\mathrm{n}\) c. \(^{235}\mathrm{U}+_{0}^{1}\mathrm{n} \rightarrow^{95}\mathrm{Zr}+?+3_{0}^{1}\mathrm{n}\) Answer: a. \(_{39}^{103}\mathrm{Y}\) b. \(_{48}^{128}\mathrm{Cd}\) c. \(_{52}^{138}\mathrm{Te}\)

Step 1: Conservation of mass numbers A

Add up the mass numbers on the left side (235 and 1) and the known mass numbers on the right side (131 and 2). The difference in mass numbers will give us the mass number of the unknown nuclide. Mass numbers on the left side: \(235 + 1 = 236\) Mass numbers on the right side: \(131 + 2 + A_{?} = 236\) Now, let's solve for \(A_?\): \(A_{?} = 236 - 131 - 2 = 103\)

Step 2: Conservation of atomic numbers Z

Add up the atomic numbers on the left side (92 and 0) and the known atomic numbers on the right side (53). The difference in atomic numbers will give us the atomic number of the unknown nuclide. Atomic numbers on the left side: \(92 + 0 = 92\) Atomic numbers on the right side: \(53 + Z_{?} = 92\) Now, let's solve for \(Z_?\): \(Z_{?} = 92 - 53 = 39\) The unknown nuclide has a mass number of 103 and atomic number of 39. We can identify this as Yttrium (Y): \(_{39}^{103}\mathrm{Y}\) b. \(^{233}\mathrm{U}+_{0}^{1}\mathrm{n} \rightarrow^{103}\mathrm{Ru}+?+3_{0}^{1}\mathrm{n}\)

Step 1: Conservation of mass numbers A

Add up the mass numbers on the left side (233 and 1) and the known mass numbers on the right side (103 and 3). The difference in mass numbers will give us the mass number of the unknown nuclide. Mass numbers on the left side: \(233 + 1 = 234\) Mass numbers on the right side: \(103 + 3 + A_{?} = 234\) Now, let's solve for \(A_?\): \(A_{?} = 234 - 103 - 3 = 128\)

Step 2: Conservation of atomic numbers Z

Add up the atomic numbers on the left side (92 and 0) and the known atomic numbers on the right side (44). The difference in atomic numbers will give us the atomic number of the unknown nuclide. Atomic numbers on the left side: \(92 + 0 = 92\) Atomic numbers on the right side: \(44 + Z_{?} = 92\) Now, let's solve for \(Z_?\): \(Z_{?} = 92 - 44 = 48\) The unknown nuclide has a mass number of 128 and an atomic number of 48. We can identify this as Cadmium (Cd): \(_{48}^{128}\mathrm{Cd}\) c. \(^{235}\mathrm{U}+_{0}^{1}\mathrm{n} \rightarrow^{95}\mathrm{Zr}+?+3_{0}^{1}\mathrm{n}\)

Step 1: Conservation of mass numbers A

Add up the mass numbers on the left side (235 and 1) and the known mass numbers on the right side (95 and 3). The difference in mass numbers will give us the mass number of the unknown nuclide. Mass numbers on the left side: \(235 + 1 = 236\) Mass numbers on the right side: \(95 + 3 + A_{?} = 236\) Now, let's solve for \(A_?\): \(A_{?} = 236 - 95 - 3 = 138\)

Step 2: Conservation of atomic numbers Z

Add up the atomic numbers on the left side (92 and 0) and the known atomic numbers on the right side (40). The difference in atomic numbers will give us the atomic number of the unknown nuclide. Atomic numbers on the left side: \(92 + 0 = 92\) Atomic numbers on the right side: \(40 + Z_{?} = 92\) Now, let's solve for \(Z_?\): \(Z_{?} = 92 - 40 = 52\) The unknown nuclide has a mass number of 138 and an atomic number of 52. We can identify this as Tellurium (Te): \(_{52}^{138}\mathrm{Te}\)

Key Concepts

Mass Number Conservation

In nuclear fission reactions, mass number conservation is key. This principle helps us balance nuclear equations. Each atomic nucleus has a certain mass number, which is the sum of its protons and neutrons. When a reaction occurs, the total mass number before the reaction must equal the total mass number after the reaction. For example, in a typical fission reaction involving uranium and a neutron, you may have a left side mass number of 236 (e.g., 235 from the uranium nucleus and 1 from a neutron).
To find the mass number of the unknown nuclide, we calculate based on

• The total mass on the left (reactants)
• Known masses on the right (products and neutrons)
• The remaining balance as the mass of the unknown nuclide

In practice, this means solving an equation like: 236 = 131 + 2 + A, where A represents the mass number of the unknown nuclide. Ensuring mass number conservation helps maintain accurate calculations in nuclear fission equations.

Atomic Number Conservation

The atomic number indicates the number of protons in an atom’s nucleus and determines the chemical element. In nuclear equations, atomic number conservation ensures that the total number of protons remains constant before and after a reaction. This concept is similar to the conservation of mass numbers but focuses on balancing protons.
In our example reactions, we start with uranium, which has an atomic number of 92. By adding up the atomic numbers of the known products (for instance, iodine with 53 or ruthenium with 44) and subtracting them from the original uranium plus the neutron (92), we can find the atomic number of the unknown nuclide.
This calculation allows us to identify the element of the unknown nuclide, as elements are defined by their atomic number. For example, if after calculation the remaining atomic number is 39, we find that the unknown nuclide is yttrium.

Nuclear Equations

Nuclear equations are representations of nuclear reactions, showing how the atomic nuclei of elements change during reactions like fission. Each component in the equation is typically shown with its mass number and atomic number. This detailed representation helps us verify the conservation laws.
Writing a correct nuclear equation involves the proper placement of reactants and products, ensuring that both mass numbers and atomic numbers are conserved as reactions proceed. Start by writing the known elements and add a neutron to simulate typical fission conditions. Next, identify all known products and additional neutrons released, calculating any unknowns using conservation methods.
Nuclear equations are powerful tools for displaying the changes in nuclear systems. The balance achieved reflects the complexity of nuclear interactions, giving clarity and predictability to the process.

Neutron Calculations

During nuclear fission, neutrons play a crucial role. They initiate the reaction when they strike a nucleus, leading to its splitting. Neutrons are represented by the symbol \(_0^1n\), indicating they have a mass number of 1 and no protons.
Neutron calculations often revolve around determining how many neutrons are involved and how they influence the reaction’s outcome. In a typical fission, the nucleus splits, releasing multiple neutrons. These can further drive reactions, forming a chain reaction.
In our example exercises, the presence of neutrons is listed on both sides of the equation. You need to count them accurately to maintain the integrity of the mass and atomic number conservation. For instance, if three neutrons are added to the products side and the mass numbers must equal the reactants, these calculations are essential in determining the unknown nuclide's properties based on what's left for subtraction or balance.