To determine the missing value that will balance the given nuclear equation, we must analyze the conservation of atomic numbers and mass numbers in the reaction.
Here is the given nuclear equation:
[tex]\[ {}_{11}^{22} \text{Na} \longrightarrow {}_{10}^{22} \text{Ne} + {}_?^0 \beta \][/tex]
1. Conservation of Mass Numbers:
- The mass number of the sodium (Na) isotope is 22.
- The mass number of the neon (Ne) isotope is also 22.
- The beta particle ([tex]\(\beta\)[/tex]) has a mass number of 0.
Since the mass numbers must be conserved in a nuclear reaction:
[tex]\[ 22 (\text{Na}) = 22 (\text{Ne}) + 0 (\beta) \][/tex]
The mass numbers are already balanced.
2. Conservation of Atomic Numbers:
- The atomic number of sodium (Na) is 11.
- The atomic number of neon (Ne) is 10.
- Let [tex]\( x \)[/tex] be the atomic number of the beta particle ([tex]\(\beta\)[/tex]).
The atomic numbers must also be conserved:
[tex]\[ 11 (\text{Na}) = 10 (\text{Ne}) + x (\beta) \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 11 - 10 \][/tex]
[tex]\[ x = 1 \][/tex]
However, knowing that a beta particle ([tex]\(\beta\)[/tex]) is an electron, which has an atomic number of -1, we expect:
[tex]\[ 11 (\text{Na}) = 10 (\text{Ne}) + (-1) (\beta) \][/tex]
[tex]\[ 11 = 10 - 1 \][/tex]
[tex]\[ 11 = 11 \][/tex]
Therefore, the missing value that balances the atomic number in the equation is [tex]\(-1\)[/tex].
Hence, the correct answer is:
[tex]\[ -1 \][/tex]
