To predict the missing component in the nuclear equation
[tex]$X \rightarrow{ }_{91}^{231} \text{Pa} + { }_{-1}^0 e,$[/tex]
we need to ensure that both atomic numbers and mass numbers are conserved during the decay process.
Here’s how to approach solving this problem:
1. Identify the elements and particles involved:
- Pa-231 (Protactinium-231): This isotope has an atomic number of 91 and a mass number of 231.
- Electron (e): The electron has an atomic number of -1 and a mass number of 0.
2. Set up the conservation equations:
- Atomic Number Conservation:
The atomic number of the initial element, X, must be equal to the sum of the atomic numbers of the products Pa-231 and the electron.
[tex]\[
\text{Atomic number of } X = 91 + (-1) = 90
\][/tex]
- Mass Number Conservation:
The mass number of the initial element, X, must be equal to the sum of the mass numbers of the products Pa-231 and the electron.
[tex]\[
\text{Mass number of } X = 231 + 0 = 231
\][/tex]
3. Determine the missing component:
From the conservation equations, we have:
[tex]\[
\text{Atomic number of } X = 90 \quad \text{and} \quad \text{Mass number of } X = 231
\][/tex]
4. Match with given options:
- Option 1: [tex]\({ }_{92}^{231} \text{U}\)[/tex] (Atomic number = 92)
- Option 2: [tex]\({ }_{90}^{230} \text{Th}\)[/tex] (Atomic number = 90)
- Option 3: [tex]\({ }_{90}^{231} \text{Th}\)[/tex] (Atomic number = 90)
Comparing these with our results, we see:
- Option 1 has the correct mass number but incorrect atomic number.
- Option 2 has the correct atomic number but incorrect mass number.
- Option 3 has both the correct atomic number (90) and the correct mass number (231).
Therefore, the missing component in the nuclear equation is:
[tex]\[
\boxed{{ }_{90}^{231} \text{Th}}
\][/tex]
