Problem 83 - Chemistry: An Atoms-Focused Approach

Question

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

Step-by-Step Solution

Answer

a. \(^{235}\mathrm{U} + _{0}^{1}\mathrm{n} \rightarrow ^{131}\mathrm{I} + ? + 2_{0}^{1}\mathrm{n}\) b. \(^{235}\mathrm{U} + _{0}^{1}\mathrm{n} \rightarrow ^{103}\mathrm{Ru} + ? + 3_{0}^{1}\mathrm{n}\) c. \(^{235}\mathrm{U} + _{0}^{1}\mathrm{n} \rightarrow ^{95}\mathrm{Zr} + ? + 3_{0}^{1}\mathrm{n}\) Answer: a. Yttrium-102 (\(^{102}\mathrm{Y}\)) b. Cadmium-130 (\(^{130}\mathrm{Cd}\)) c. Tellurium-138 (\(^{138}\mathrm{Te}\))

Step 1: a. Identify the unknown nuclide in the first fission reaction.

First, let's write down the fission reaction: \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{131} \mathrm{I}+?+2_{0}^{1} \mathrm{n}\) Conservation of mass numbers: \(235 + 1 = 131 + A + 2(1)\) Where A is the mass number of the unknown nuclide. Now solve for A: \(A = 235 + 1 - 131 - 2 = 102\) Conservation of atomic numbers: \(92 + 0 = 53 + Z + 2(0)\) Where Z is the atomic number of the unknown nuclide. Now solve for Z: \(Z = 92 + 0 - 53 = 39\) So, the unknown nuclide is Yttrium-102 (\(^{102}\mathrm{Y}\)).

Step 2: b. Identify the unknown nuclide in the second fission reaction.

First, let's write down the fission reaction: \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{103} \mathrm{Ru}+?+3_{0}^{1} \mathrm{n}\) Conservation of mass numbers: \(235 + 1 = 103 + B + 3(1)\) Where B is the mass number of the unknown nuclide. Now solve for B: \(B = 235 + 1 - 103 - 3 = 130\) Conservation of atomic numbers: \(92 + 0 = 44 + X + 3(0)\) Where X is the atomic number of the unknown nuclide. Now solve for X: \(X = 92 - 44 = 48\) So, the unknown nuclide is Cadmium-130 (\(^{130}\mathrm{Cd}\)).

Step 3: c. Identify the unknown nuclide in the third fission reaction.

First, let's write down the fission reaction: \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{95} \mathrm{Zr}+?+3_{0}^{1} \mathrm{n}\) Conservation of mass numbers: \(235 + 1 = 95 + C + 3(1)\) Where C is the mass number of the unknown nuclide. Now solve for C: \(C = 235 + 1 - 95 - 3 = 138\) Conservation of atomic numbers: \(92 + 0 = 40 + Y + 3(0)\) Where Y is the atomic number of the unknown nuclide. Now solve for Y: \(Y = 92 - 40 = 52\) So, the unknown nuclide is Tellurium-138 (\(^{138}\mathrm{Te}\)).

Key Concepts

Conservation of Mass Numbers

In fission reactions, the conservation of mass numbers is crucial. This principle states that the total mass number of the reactants must be equal to the total mass number of the products. This is like keeping the mass intact, even though the nuclei split into smaller parts. For example, consider the reaction: - In the fission of Uranium-235 - A neutron is added, resulting in an intermediate that splitsIf it results in an Iodine-131, an unknown nuclide, and two neutrons, the mass numbers add up to track where the mass went. The formula to use is:\[ 235 + 1 = 131 + A + 2 \times 1 \] Solving for \(A\), the mass number of the unknown, involves moving terms:\[ A = 235 + 1 - 131 - 2 = 102 \] Thus, the unknown nuclide in this reaction has a mass number of 102.

Conservation of Atomic Numbers

Just like mass numbers, atomic numbers must also be conserved in fission reactions. Atomic numbers represent the number of protons in the nucleus and define the element's identity. For example, when Uranium-235 combines with a neutron:- The total atomic number before the reaction is the sum of Uranium's atomic number (92) plus the neutron's (0)Using conservation, this should equal the sum of the atomic numbers of all products, including an unknown nuclide:\[ 92 = 53 + Z + 2 \times 0 \] Solving for \(Z\):\[ Z = 92 - 53 = 39 \] This leads to an atomic number of 39 for the unknown. Since the atomic number of 39 corresponds to Yttrium, the unknown nuclide in this fission reaction scenario must be Yttrium-102.

Unknown Nuclide Identification

Identifying unknown nuclides is an essential skill in nuclear reactions. Once you've conserved mass and atomic numbers, you can deduce the identity of unknown nuclides effectively. Let's apply this knowledge: - Assume you know a reaction sequence with fission products - You use conservation laws to first calculate the mass number (A) and atomic number (Z) Take the example of the second fission reaction: - You calculated a mass number (B) as 130 and an atomic number (X) as 48 from the earlier steps. A periodic table then helps identify the element corresponding to an atomic number of 48, which is Cadmium. Therefore, the unknown nuclide must be Cadmium-130. By following these systematic steps, identifying unknown nuclides becomes much easier, avoiding guesswork and focusing on calculated results.