To determine the particle needed to complete the nuclear reaction:
[tex]$
{ }_{86}^{222} Rn \rightarrow{ }_{84}^{218} Po + ?
$[/tex]
we need to balance both the atomic number (Z) and the mass number (A) on both sides of the equation.
1. Balancing the Mass Number (A):
The mass number on the left side of the reaction is 222 (from [tex]\( ^{222}_{86}Rn \)[/tex]).
The mass number on the right side from [tex]\( ^{218}_{84}Po \)[/tex] is 218. Let's denote the mass number of the unknown particle as [tex]\( A_x \)[/tex].
To balance the mass number:
[tex]\[
222 = 218 + A_x
\][/tex]
Solving for [tex]\( A_x \)[/tex]:
[tex]\[
A_x = 222 - 218 = 4
\][/tex]
2. Balancing the Atomic Number (Z):
The atomic number on the left side of the reaction is 86 (from [tex]\( _{86}^{222}Rn \)[/tex]).
The atomic number on the right side from [tex]\( _{84}^{218}Po \)[/tex] is 84. Let's denote the atomic number of the unknown particle as [tex]\( Z_x \)[/tex].
To balance the atomic number:
[tex]\[
86 = 84 + Z_x
\][/tex]
Solving for [tex]\( Z_x \)[/tex]:
[tex]\[
Z_x = 86 - 84 = 2
\][/tex]
Therefore, the particle needed must have a mass number of 4 and an atomic number of 2. This corresponds to the alpha particle, which is represented as [tex]\( ^4_2He \)[/tex].
So, the correct particle to complete the nuclear reaction is:
[tex]\[
{ }_2^4 \text{He}
\][/tex]
Thus, the third option [tex]\( { }_2^4He \)[/tex] is correct.
[tex]\[
\boxed{3}
\][/tex]
