Problem 75 - Chemistry The Science in Context

Question

The fission of uranium produces dozens of isotopes. For each of the following fission reactions, determine the identity of the unknown nuclide: a. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{96} \mathrm{Zr}+?+2_{0}^{1} \mathrm{n}\) b. \(^235 \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{99} \mathrm{Nb}+?+4_{0}^{1} \mathrm{n}\) c. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{90} \mathrm{Rb}+?+3_{0}^{1} \mathrm{n}\)

Step-by-Step Solution

Answer

+3_{0}^{1} \mathrm{n}\) Answer: The unknown nuclide for the given fission reaction is \(^{142}\mathrm{Cs}\).

Step 1: Conserve Mass

First, we need to conserve the mass number (A) for the reaction. We know that the sum of the mass numbers on the left side of the reaction should equal the sum of the mass numbers on the right side. Mass conservation: \(235 + 1 = 96 + A_{2} + 2(1)\)

Step 2: Calculate Unknown Mass Number

Solve for the mass number of the unknown nuclide (A_{2}). \(A_{2} = 235 + 1 - 96 - 2(1) = 137\)

Step 3: Conserve Atomic Numbers

Now, let's conserve atomic number (Z) for the reaction. We know that the sum of the atomic numbers on the left side of the reaction should equal the sum of the atomic numbers on the right. Atomic number conservation: \(92 + 0 = 40 + Z_{2} + 2(0)\)

Step 4: Calculate Unknown Atomic Number

Solve for the atomic number of the unknown nuclide (Z2). \(Z_2 = 92 + 0 - 40 - 2(0) = 52\)

Step 5: Identify the Element

Now, we have the mass number and atomic number of the unknown nuclide. We can use the periodic table to find the element with the atomic number 52, which corresponds to tellurium (Te). Therefore, the unknown nuclide is \(^{137}\mathrm{Te}\). b. \(^235 \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{99} \mathrm{Nb}+?+4_{0}^{1} \mathrm{n}\)

Step 1: Conserve Mass

Mass conservation: \(235 + 1 = 99 + A_{2} + 4(1)\)

Step 2: Calculate Unknown Mass Number

Solve for the mass number of the unknown nuclide (A_{2}). \(A_{2} = 235 + 1 - 99 - 4(1) = 132\)

Step 3: Conserve Atomic Numbers

Atomic number conservation: \(92 + 0 = 41 + Z_{2} + 4(0)\)

Step 4: Calculate Unknown Atomic Number

Solve for the atomic number of the unknown nuclide (Z2). \(Z_2 = 92 + 0 - 41 - 4(0) = 51\)

Step 5: Identify the Element

Using the periodic table, we find that the element with the atomic number 51 is antimony (Sb). Therefore, the unknown nuclide is \(^{132}\mathrm{Sb}\). c. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{90} \mathrm{Rb}+?+3_{0}^{1} \mathrm{n}\)

Step 1: Conserve Mass

Mass conservation: \(235 + 1 = 90 + A_{2} + 3(1)\)

Step 2: Calculate Unknown Mass Number

Solve for the mass number of the unknown nuclide (A_{2}). \(A_{2} = 235 + 1 - 90 - 3(1) = 142\)

Step 3: Conserve Atomic Numbers

Atomic number conservation: \(92 + 0 = 37 + Z_{2} + 3(0)\)

Step 4: Calculate Unknown Atomic Number

Solve for the atomic number of the unknown nuclide (Z2). \(Z_2 = 92 + 0 - 37 - 3(0) = 55\)

Step 5: Identify the Element

Using the periodic table, we find that the element with the atomic number 55 is cesium (Cs). Therefore, the unknown nuclide is \(^{142}\mathrm{Cs}\).

Key Concepts

Mass Conservation in Nuclear Fission

In nuclear fission, the principle of mass conservation states that the total mass number before and after the reaction must remain constant. Fission involves splitting a heavy nucleus, such as uranium, into smaller fragments. When a nucleus, like \(^{235}\mathrm{U}\), absorbs a neutron (\(_{0}^{1} \mathrm{n}\)), it becomes unstable and splits into smaller nuclei, known as fission fragments, along with additional neutrons and energy.
To conserve mass, we ensure the sum of the mass numbers on the reactant side equals that of the products. For example, in the reaction \(^{235}\mathrm{U} + _{0}^{1}\mathrm{n} \rightarrow \) products, the left side's mass number totals 236 (235 + 1). If the known fission products have mass numbers of, say, 96 and include two neutrons (\(2 \times 1\)), the mass number of the unknown nuclide must be determined by subtraction:

• Total initial mass number: 236
• Mass number of known fission products and neutrons: 96 + 2 = 98
• Mass number of unknown fragment: 236 - 98 = 138

Thus, solving the equation for mass conservation helps identify unknown fragments in fission reactions.

Atomic Number Conservation in Fission Reactions

Atomic number conservation is another key principle in nuclear fission. This principle ensures the total atomic number (the number of protons) remains the same on both sides of the reaction. Let's consider an example of uranium fission:
The uranium-235 nucleus has an atomic number of 92, as it contains 92 protons. When a neutron is absorbed, it briefly becomes part of the uranium nucleus, affecting its stability but not changing the atomic number, leaving the total at 92. When uranium undergoes fission, it divides into smaller nuclei, each with its unique atomic number, and possibly releases additional neutrons.
The balance is maintained by ensuring the sum of the atomic numbers of all products (daughter nuclei and released neutrons) equals the atomic number of the original uranium nucleus plus any effect from added neutrons. For instance, if one of the products formed has an atomic number of 40, and we know that two neutrons are emitted (each with an atomic number of zero), we can determine the unknown nuclide's atomic number through calculation:

• Initial atomic number: 92
• Atomic number in known products: 40 + 0 (for neutrons)
• Unknown fragment atomic number calculation: 92 - 40 = 52

Identifying the unknown element by this atomic number completes the evaluation.

Isotope Analysis in Identifying Nuclear Products

Isotope analysis is crucial when identifying unknown products of nuclear reactions. An isotope is defined by its atomic number and mass number. In nuclear fission, analyzing isotopic data helps determine the identity of unknown fragments. Each element in the periodic table is characterized by an atomic number, which is representative of its proton count.
Once we've determined both the mass number and atomic number of an unknown nuclide through conservation equations, we use these values to pinpoint the specific isotope using a periodic table. For example, if we calculated a mass number of 137 and an atomic number of 52, we can consult a periodic table to find that this corresponds to tellurium, specifically \(^{137}\mathrm{Te}\). Similarly, other unknowns can be identified as isotopes like \(^{132}\mathrm{Sb}\) for antimony with a mass number 132 and atomic number 51.
This analysis provides powerful insight into the makeup of nuclear fission products and supports various applications in nuclear science, such as reactor design and safety assessments. Understanding isotope characteristics also aids in environmental and biological studies related to radiation exposure.