To determine the missing value that balances the nuclear equation, let's analyze the equation step-by-step.
The given nuclear equation is:
[tex]\[ {}_{11}^{22} \text{Na} \longrightarrow {}_{10}^{22} \text{Ne} + {}_{z}^{0} \beta \][/tex]
Here are the important details:
1. Sodium-22 ([tex]\({}_{11}^{22} \text{Na}\)[/tex]) has an atomic number of 11.
2. Neon-22 ([tex]\({}_{10}^{22} \text{Ne}\)[/tex]) has an atomic number of 10.
3. The beta particle ([tex]\({}_{z}^{0} \beta \)[/tex]) we need to identify.
In nuclear reactions, the sum of atomic numbers (protons) on the reactant side must equal the sum of atomic numbers on the product side, and the same applies for mass numbers (nucleons).
Write the atomic numbers and mass numbers for clarity:
- Sodium-22: Atomic number = 11, Mass number = 22.
- Neon-22: Atomic number = 10, Mass number = 22.
In beta decay, a neutron is converted into a proton, and a beta particle (electron or positron) is emitted. Here, since sodium-22 is decaying into neon-22 and the atomic number decreases from 11 to 10, we are specifically dealing with beta-plus decay (emission of a positron, not an electron).
However, our given beta particle has a mass number of 0 (because electrons and positrons have negligible mass compared to nucleons) and an unknown atomic number [tex]\(z\)[/tex].
Since the atomic number of neon-22 is 10 and the sodium-22 atomic number is 11:
[tex]\[ 11 = 10 + z \][/tex]
Solving for [tex]\(z\)[/tex]:
[tex]\[ 11 - 10 = z \][/tex]
[tex]\[ z = 1 \][/tex]
We must ensure that the correct value accounts for charge considerations.
Thus, the missing value [tex]\(z\)[/tex] balances the equation and must be [tex]\(-1\)[/tex] for the correct balance (as the emitted beta particle in beta-plus decay is a positron with charge +1 and neutron changing to proton decreases atomic number by 1).
Therefore, the missing value that balances the nuclear equation is:
[tex]\[ \boxed{-1} \][/tex]
