To balance a nuclear reaction, we need to ensure that both the atomic numbers and the mass numbers are balanced on both sides of the equation.
Given the nuclear reaction:
[tex]\[
_{98}^{249} \text{Cf} + _{2}^{?} \: ?
\][/tex]
Step-by-Step Solution:
1. Identify the known elements:
- Californium (Cf): has an atomic number of 98 and a mass number of 249.
- We need to find the unknown nucleus such that the atomic numbers and mass numbers balance out.
2. Determine the sums needed to balance the reaction:
- Suppose the product element formed on the right side of the equation needs to balance out the californium (Cf) on the left.
3. Balance the atomic numbers:
- The total atomic number on the left side should equal the total atomic number on the right side.
- Catalyst is californium, which has an atomic number of 98.
- Let's find the atomic number that must sum up to 100.
4. Atomic Number Balance:
- Given the californium (_98) element:
- We need to find the element with atomic number [tex]\( Z \)[/tex] such that [tex]\( 98 + Z = 100 \)[/tex].
5. Solve for the missing atomic number:
- [tex]\( 98 + Z = 100 \)[/tex]
- [tex]\( Z = 100 - 98 \)[/tex]
- [tex]\( Z = 2 \)[/tex]
6. Balance the mass numbers:
- The mass number of Californium is 249.
- The total mass number on the left side should equal the total mass number on the right which is 252.
7. Mass Number Balance:
- We need to find the mass number [tex]\( A \)[/tex] such that [tex]\( 249 + A = 252 \)[/tex].
8. Solve for the missing mass number:
- [tex]\( 249 + A = 252 \)[/tex]
- [tex]\( A = 252 - 249 \)[/tex]
- [tex]\( A = 3 \)[/tex]
9. Determine the missing nucleus:
- The missing nucleus should have an atomic number (Z) = 2 and a mass number (A) = 3.
- This corresponds to the isotope of Helium (_2) with 3 as the mass number.
Thus, the balanced nuclear reaction is:
[tex]\[
_{98}^{249} \text{Cf} + _{2}^{3} \text{He}
\][/tex]
Here, we added a nucleus of Helium (_2^3 He) to balance both the atomic numbers and mass numbers.
