Let's solve this step by step to determine the missing parent element in the alpha decay reaction that results in [tex]\(_{88}^{228} \text{Ra} + {_2^4}\text{He}\)[/tex].
### Step 1: Understand Alpha Decay
In alpha decay, a parent nucleus emits an alpha particle ([tex]\(_2^4\text{He}\)[/tex]), leading to the formation of a daughter nucleus. This particle has:
- Atomic number [tex]\(Z_{\alpha} = 2\)[/tex]
- Mass number [tex]\(A_{\alpha} = 4\)[/tex]
### Step 2: Write the General Reaction
The general reaction for alpha decay can be written as:
[tex]\[
{ }_Z^A \text{Parent} \longrightarrow { }_{Z-2}^{A-4} \text{Daughter} + { }_{2}^{4} \text{He}
\][/tex]
### Step 3: Identify Known Quantities
Given:
- Daughter element: [tex]\(_{88}^{228} \text{Ra}\)[/tex]
- Alpha particle: [tex]\(_2^4 \text{He}\)[/tex]
Here,
- Atomic number of daughter ([tex]\(Z_{\text{Ra}}\)[/tex]) = 88
- Mass number of daughter ([tex]\(A_{\text{Ra}}\)[/tex]) = 228
### Step 4: Determine the Parent Element
From the given reaction:
[tex]\[
\longrightarrow{ }_{88}^{228} \text{Ra} +{ }_2^4 \text{He}
\][/tex]
We need to find the parent element [tex]\(_Z^A \text{Parent}\)[/tex].
Using the law of conservation of mass and atomic numbers:
- Atomic number of parent ([tex]\(Z_{\text{Parent}}\)[/tex]) = Atomic number of daughter + Atomic number of alpha particle
[tex]\[
Z_{\text{Parent}} = Z_{\text{Ra}} + Z_{\alpha} = 88 + 2 = 90
\][/tex]
- Mass number of parent ([tex]\(A_{\text{Parent}}\)[/tex]) = Mass number of daughter + Mass number of alpha particle
[tex]\[
A_{\text{Parent}} = A_{\text{Ra}} + A_{\alpha} = 228 + 4 = 232
\][/tex]
Therefore, the missing parent element's atomic number is 90 and its mass number is 232.
### Step 5: Match with Given Options
Let's check the options:
A. [tex]\(\quad{ }_{96}^{224} \text{Rn}\)[/tex]
B. [tex]\(\quad{ }^{232} \text{Th}\)[/tex]
C. [tex]\(\quad{ }_{92}^{230} \text{U}\)[/tex]
D. [tex]\(\quad{ }_{92}^{235} \text{U}\)[/tex]
### Step 6: Identify the Correct Answer
The correct parent element should have:
- Atomic number [tex]\(Z = 90\)[/tex]
- Mass number [tex]\(A = 232\)[/tex]
Using the periodic table, we know that the element with atomic number 90 is Thorium ([tex]\(\text{Th}\)[/tex]).
So, the correct option is:
[tex]\[ \boxed{\text{B}} \][/tex]
