Problem 78 - Chemistry The Science in Context

Question

For each of the following fission reactions, determine the identity of the unknown nuclide: a. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{147} \mathrm{Pm}+?+2_{0}^{1} \mathrm{n}\) b. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{94} \mathrm{Kr}+?+2_{0}^{1} \mathrm{n}\) c. \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{95} \mathrm{Sr}+?+3_{0}^{1} \mathrm{n}\)

Step-by-Step Solution

Answer

a) \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{147} \mathrm{Pm}+?+2_{0}^{1} \mathrm{n}\) Answer: \(_{31}^{87} \mathrm{Ga}\) (Gallium) b) \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{94} \mathrm{Kr}+?+2_{0}^{1} \mathrm{n}\) Answer: \(_{56}^{140} \mathrm{Ba}\) (Barium) c) \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{95} \mathrm{Sr}+?+3_{0}^{1} \mathrm{n}\) Answer: \(_{54}^{138} \mathrm{Xe}\) (Xenon)

Step 1: Write down the given reaction

The given reaction is: \(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{147} \mathrm{Pm}+?+2_{0}^{1} \mathrm{n}\)

Step 2: Analyzing the reaction

Through this analysis, conservation of mass numbers and atomic numbers is needed. Let's denote the unknown nuclide as \(_{Z}^{A} \mathrm{X}\).

Step 3: Conservation of mass numbers

To find the mass number, A, for the unknown nuclide, we use the conservation of mass numbers. This means that the total mass number on the left side of the reaction must equal the total mass number on the right side: \(235 + 1 = 147 + A + 2(1)\) Solve for A: \(A = 235 + 1 - 147 - 2 = 87\)

Step 4: Conservation of atomic numbers

To find the atomic number, Z, for the unknown nuclide, we use conservation of atomic numbers. This means that the total atomic number on the left side of the reaction must equal the total atomic number on the right side: \(92 + 0 = 61 + Z + 2(0)\) Solve for Z: \(Z = 92 - 61 = 31\)

Step 5: Identify the unknown nuclide

Now that we have found the atomic number (Z=31) and mass number (A=87) for the unknown nuclide, we can use a periodic table to identify the element with atomic number 31 as Gallium (Ga). So, the unknown nuclide is \(_{31}^{87} \mathrm{Ga}\). ###Problem b:### Following the same steps as in problem a, we get:

Step 6: Given reaction for problem b

\(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{94} \mathrm{Kr}+?+2_{0}^{1} \mathrm{n}\) Mass number: A = 235 + 1 - 94 - 2 = 140 Atomic number: Z = 92 - 36 = 56 Unknown nuclide: \(_{56}^{140} \mathrm{Ba}\) (Barium) ###Problem c:### Following the same steps as in problem a, we get:

Step 7: Given reaction for problem c

\(^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{95} \mathrm{Sr}+?+3_{0}^{1} \mathrm{n}\) Mass number: A = 235 + 1 - 95 - 3 = 138 Atomic number: Z = 92 - 38 = 54 Unknown nuclide: \(_{54}^{138} \mathrm{Xe}\) (Xenon)

Key Concepts

Mass Number Conservation

In a nuclear fission reaction, the mass number conservation is a key principle. This principle states that the total mass number on the left side (reactants) of the equation must equal the total mass number on the right side (products).
For example, consider the reaction: \[^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{147} \mathrm{Pm}+?+2_{0}^{1} \mathrm{n}\]Before fission occurs, the uranium-235 and neutron have a combined mass number of 236 (235 + 1). After fission, the products include a known nuclide promethium-147, two neutrons, and an unknown nuclide.
To ensure mass number conservation, calculate the mass number of the unknown nuclide using the following equation:\[235 + 1 = 147 + A + 2(1)\]Here, A represents the mass number of the unknown nuclide. Solving the equation gives:\[A = 235 + 1 - 147 - 2 = 87\]This principle helps in identifying the often unknown pieces of a reaction, ensuring nothing in the mass balance is overlooked.
Always remember, the mass number is the sum of protons and neutrons and must remain consistent across the fission reaction.

Atomic Number Conservation

Conservation of the atomic number is another vital concept in nuclear fission reactions. The atomic number determines the element's position on the periodic table and must remain consistent between reactants and products.
Taking the same example from before: \[^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow^{147} \mathrm{Pm}+?+2_{0}^{1} \mathrm{n}\]The atomic numbers of the reactants add up to 92 (uranium has an atomic number of 92). The fission products include promethium-147 with an atomic number of 61, two neutrons that contribute 0 to the atomic number, and an unknown nuclide.
We use the following equation to find the atomic number of the unknown nuclide:\[92 + 0 = 61 + Z + 2(0)\]Solving for Z gives:\[Z = 92 - 61 = 31\]The conservation of atomic numbers helps maintain the identity of elements and ensures the reaction's integrity.
By preserving atomic number, we can accurately probe the identities of any unknowns involved in the nuclear transformation.

Unknown Nuclide Identification

Determining the identity of an unknown nuclide in a nuclear reaction involves both mass number and atomic number conservation.
After establishing these values, the periodic table is employed to find which element corresponds to the determined atomic number.
For instance, if we find that an unknown nuclide has a mass number of 87 and an atomic number of 31, the periodic table shows the element gallium (Ga) corresponds to atomic number 31.

• Mass number (A) tells us the sum of protons and neutrons.
• Atomic number (Z) gives the count of protons, thereby defining the element.

Using these numbers, one can pinpoint the correct element even before accurately measuring its mass in experimental conditions.
This method not only helps in theoretical nuclear equations but also plays a role in practical applications such as nuclear reactor studies and radioactive decay analysis. Identifying unknown nuclides is a fundamental aspect of understanding nuclear reactions and the resulting products.