To balance the given nuclear equation, we need to ensure the conservation of both the atomic number (Z) and the mass number (A) on both sides of the reaction.
The provided nuclear equation is:
[tex]\[
{}_{11}^{22} Na \longrightarrow {}_{10}^{22} Ne + {}_?^{0} \beta
\][/tex]
Let's analyze each component:
1. Sodium (Na) has an atomic number [tex]\( Z = 11 \)[/tex] and a mass number [tex]\( A = 22 \)[/tex].
2. Neon (Ne) has an atomic number [tex]\( Z = 10 \)[/tex] and a mass number [tex]\( A = 22 \)[/tex].
3. The particle we're trying to identify is represented by [tex]\( {}_?^{0} \beta \)[/tex].
To balance the nuclear equation, we need to balance both the atomic number and the mass number:
### 1. Conservation of Mass Number (A)
Since the mass number of sodium and neon are both 22, the mass number of the missing particle (β) must be:
[tex]\[
A = 22 - 22 = 0
\][/tex]
### 2. Conservation of Atomic Number (Z)
The atomic number of sodium (Na) is 11, while the atomic number of neon (Ne) is 10. Let [tex]\( Z \)[/tex] be the atomic number of the missing particle (β):
[tex]\[
11 = 10 + Z
\][/tex]
Solving for [tex]\( Z \)[/tex]:
[tex]\[
Z = 11 - 10 = 1
\][/tex]
Given that beta particles are electrons, they are represented by [tex]\( {}_{-1}^{0}\beta \)[/tex], which means the correct representation should include the negative sign.
Hence, the missing value that will balance the equation is:
[tex]\[
-1
\][/tex]
