To solve this problem, let's analyze and balance the given nuclear fission equation step-by-step.
The given nuclear equation is:
[tex]\[
_{92}^{235} U + { }_0^1 n \rightarrow { }_{56}^{139} Ba + { }_8 C + 3 { }_0^1 n
\][/tex]
We need to find the missing element that should balance both the mass (nucleon) numbers and atomic (proton) numbers. Let's break it down:
1. Calculate the total mass number (A) before the reaction.
Before the reaction:
- Uranium-235 ([tex]\(_{92}^{235}U\)[/tex]): [tex]\( A = 235 \)[/tex]
- Neutron ([tex]\({ }_0^1n\)[/tex]): [tex]\( A = 1 \)[/tex]
So, the total mass number before the reaction is:
[tex]\[ 235 + 1 = 236 \][/tex]
2. Calculate the total atomic number (Z) before the reaction.
Before the reaction:
- Uranium-235 ([tex]\(_{92}^{235}U\)[/tex]): [tex]\( Z = 92 \)[/tex]
- Neutron ([tex]\({ }_0^1n\)[/tex]): [tex]\( Z = 0 \)[/tex]
So, the total atomic number before the reaction is:
[tex]\[ 92 + 0 = 92 \][/tex]
3. Calculate the mass number (A) of the known particles after the reaction.
After the reaction, we have:
- Barium-139 ([tex]\(_{56}^{139}Ba\)[/tex]): [tex]\( A = 139 \)[/tex]
- Three neutrons ([tex]\(3{ }_0^1 n\)[/tex]): [tex]\( A = 3 \times 1 = 3 \)[/tex]
So, the total mass number from the known products is:
[tex]\[ 139 + 3 = 142 \][/tex]
4. Calculate the atomic number (Z) of the known particles after the reaction.
After the reaction, we have:
- Barium-139 ([tex]\(_{56}^{139}Ba\)[/tex]): [tex]\( Z = 56 \)[/tex]
- Three neutrons ([tex]\( 3 { }_0^1 n \)[/tex]): [tex]\( Z = 3 \times 0 = 0 \)[/tex]
So, the total atomic number from the known products is:
[tex]\[ 56 + 0 = 56 \][/tex]
5. Determine the unknown element to balance the mass and atomic numbers.
To balance the mass numbers:
- Total mass number before the reaction: [tex]\( 236 \)[/tex]
- Mass number of known products ([tex]\(142\)[/tex])
Therefore, the mass number of the unknown element becomes:
[tex]\[ 236 - 142 = 94 \][/tex]
To balance the atomic numbers:
- Total atomic number before the reaction: [tex]\( 92 \)[/tex]
- Atomic number of known products ([tex]\(56\)[/tex])
Therefore, the atomic number of the unknown element becomes:
[tex]\[ 92 - 56 = 36 \][/tex]
6. Identify the unknown element using the periodic table.
The element with atomic number 36 is Krypton ([tex]\(_{36}Kr\)[/tex]).
Thus, the completed nuclear reaction equation is:
[tex]\[
{}_{92}^{235} U + {}_0^1 n \rightarrow {}_{56}^{139} Ba + {}_{36}^{94} Kr + 3 {}_0^1 n
\][/tex]
In summary:
- [tex]\( A \)[/tex]: [tex]\({}_{36}^{94} Kr\)[/tex]
- [tex]\( B \)[/tex]: [tex]\({}_{36}^{94} Kr\)[/tex]
- [tex]\( C \)[/tex]: [tex]\({}_{36}^{94} Kr\)[/tex]
