To predict the missing component in the given nuclear equation:
[tex]\[
{}_{94}^{239}\text{Pu} \rightarrow X + {}_{2}^{4}\text{He}
\][/tex]
we need to determine both the atomic number and the mass number of the missing nucleus [tex]\( X \)[/tex].
### Step-by-Step Solution:
1. Identify the atomic and mass numbers of the known elements:
- Plutonium ([tex]\( \text{Pu} \)[/tex]):
- Atomic number = 94
- Mass number = 239
- Helium ([tex]\( \text{He} \)[/tex]):
- Atomic number = 2
- Mass number = 4
2. Setup the equations for conservation of atomic number and mass number:
- Conservation of atomic number:
[tex]\[
94 = Z_X + 2
\][/tex]
- Conservation of mass number:
[tex]\[
239 = A_X + 4
\][/tex]
3. Solve the atomic number equation:
[tex]\[
94 = Z_X + 2
\][/tex]
Rearrange to find [tex]\( Z_X \)[/tex]:
[tex]\[
Z_X = 94 - 2
\][/tex]
[tex]\[
Z_X = 92
\][/tex]
4. Solve the mass number equation:
[tex]\[
239 = A_X + 4
\][/tex]
Rearrange to find [tex]\( A_X \)[/tex]:
[tex]\[
A_X = 239 - 4
\][/tex]
[tex]\[
A_X = 235
\][/tex]
5. Combine the results:
- The atomic number of the missing component [tex]\( X \)[/tex] is 92.
- The mass number of the missing component [tex]\( X \)[/tex] is 235.
Therefore, the missing component [tex]\( X \)[/tex] in the nuclear equation is:
[tex]\[
{}_{92}^{235}\text{X}
\][/tex]
Given the atomic number 92, we can identify the element as Uranium (U).
Hence, the complete nuclear equation is:
[tex]\[
{}_{94}^{239}\text{Pu} \rightarrow {}_{92}^{235}\text{U} + {}_{2}^{4}\text{He}
\][/tex]
So, the missing component is [tex]\( {}_{92}^{235}\text{U} \)[/tex].
