To solve the problem of predicting the missing component in the nuclear equation:
[tex]\[
{ }_{90}^{230} Th \rightarrow{ }_{88}^{226} Ra + X
\][/tex]
we need to ensure that both the atomic number (subscript) and the mass number (superscript) are balanced on both sides of the equation.
Step-by-Step Solution:
1. Identify the given quantities:
- Thorium (Th) has an atomic number of 90 and a mass number of 230.
- Radium (Ra) has an atomic number of 88 and a mass number of 226.
2. Write the equation with placeholders:
[tex]\[
{ }_{90}^{230} Th \rightarrow{ }_{88}^{226} Ra + { }_{Z}^{A} X
\][/tex]
Here, [tex]\( Z \)[/tex] represents the atomic number and [tex]\( A \)[/tex] represents the mass number of the unknown component [tex]\( X \)[/tex].
3. Balance the atomic numbers:
- On the left side, the atomic number is 90 (from Thorium).
- On the right side, the atomic number should also sum to 90. Radium contributes an atomic number of 88.
[tex]\[
90 = 88 + Z
\][/tex]
Solving for [tex]\( Z \)[/tex]:
[tex]\[
Z = 90 - 88 = 2
\][/tex]
4. Balance the mass numbers:
- On the left side, the mass number is 230 (from Thorium).
- On the right side, the mass number should also sum to 230. Radium contributes a mass number of 226.
[tex]\[
230 = 226 + A
\][/tex]
Solving for [tex]\( A \)[/tex]:
[tex]\[
A = 230 - 226 = 4
\][/tex]
5. Identify the unknown component [tex]\( X \)[/tex] by its atomic number and mass number:
- The atomic number [tex]\( Z \)[/tex] is 2.
- The mass number [tex]\( A \)[/tex] is 4.
- An element with atomic number 2 is Helium (He).
Conclusion:
The unknown component [tex]\( X \)[/tex] is Helium. Thus, the missing component in the nuclear equation is:
[tex]\[
{ }_2^4 He
\][/tex]
