To solve this problem, follow these steps to identify the parent isotope and daughter isotopes in the given nuclear reaction:
### Step 1: Analyze the Nuclear Reaction
The nuclear reaction given is:
[tex]\[ { }_{88}^{222} Ra \rightarrow{ }_{80}^{218} Rn +{ }_2^4 He \][/tex]
### Step 2: Identify the Parent Isotope
The parent isotope in a nuclear reaction is the original isotope that undergoes decay. In this reaction:
- [tex]\({ }_{88}^{222} Ra\)[/tex] is the starting isotope.
### Step 3: Identify the Daughter Isotopes
The daughter isotopes are the products of the nuclear reaction.
- [tex]\({ }_{80}^{218} Rn\)[/tex] (Radon) is one of the daughter isotopes.
- [tex]\({ }_2^4 He\)[/tex] (Helium) is the other daughter isotope.
### Step 4: Match with the Given Options
We need to identify which of the given options correspond to the parent isotope.
Given options are:
1. [tex]\({ }_{86}^{218} Rn\)[/tex]
2. [tex]\({ }_2^4 He\)[/tex]
3. [tex]\({ }_{85}^{222} Ra\)[/tex]
By understanding the reaction:
- [tex]\({ }_{86}^{218} Rn\)[/tex] can't be the parent because its atomic number and mass number don't match the original isotope undergoing decay.
- [tex]\({ }_2^4 He\)[/tex] is a daughter isotope, not the parent.
- [tex]\({ }_{85}^{222} Ra\)[/tex] does not match the parent isotope in the reaction.
### Step 5: Conclusion
The correct parent isotope is:
[tex]\[ \({ }_{88}^{222} Ra\) \rightarrow{ }_{80}^{218} Rn +{ }_2^4 He \][/tex]
So, none of the provided options match the parent isotope exactly. Therefore, you should check and correct the given question or assumptions about the options available. The main takeaway is that the parent isotope for this reaction is [tex]\({ }_{88}^{222} Ra\)[/tex].
### Step-by-Step Numerical Confirmation:
- From the given numerical result:
- Parent Isotope: 86222
- Daughter Isotope Rn: 80218
- Daughter Isotope He: 2004
- Verified Isotope: None
None of the provided options match the parent isotope [tex]\({ }_{88}^{222} Ra\)[/tex] exactly, which translates to [tex]\(88 \times 1000 + 222 = 88222\)[/tex].
Hence, the correct answer should be [tex]\({ }_{88}^{222} Ra\)[/tex] but since this option does not exist in the provided list, reassessment of options is recommended.
Thus, the parent isotope in the given nuclear reaction is [tex]\({ }_{88}^{222} Ra\)[/tex].
