To determine the particle needed to complete the given nuclear reaction:
[tex]\[ { }_7^{14} N + \longrightarrow { }_6^{14} C + { }_1^1 H \][/tex]
we should ensure that both the atomic numbers (number of protons) and the mass numbers (total number of nucleons) are balanced on both sides of the equation.
Step 1: Balance the atomic numbers (protons):
1. The atomic number of the nitrogen [tex]\[ { }_7^{14} N \][/tex] is 7.
2. The atomic number of the carbon [tex]\[ { }_6^{14} C \][/tex] is 6.
3. The atomic number of the hydrogen [tex]\[ { }_1^1 H \][/tex] is 1.
Adding the atomic numbers on the product side:
[tex]\[ 6 \ (\text{from } { }_6^{14} C) + 1 \ (\text{from } { }_1^1 H) = 7 \][/tex]
Thus, the atomic numbers are already balanced.
Step 2: Balance the mass numbers (nucleons):
1. The mass number of the nitrogen [tex]\[ { }_7^{14} N \][/tex] is 14.
2. The mass number of the carbon [tex]\[ { }_6^{14} C \][/tex] is 14.
3. The mass number of the hydrogen [tex]\[ { }_1^1 H \][/tex] is 1.
Adding the mass numbers on the product side:
[tex]\[ 14 \ (\text{from } { }_6^{14} C) + 1 \ (\text{from } { }_1^1 H) = 15 \][/tex]
On the reactant side, we have a mass number of 14 (from [tex]\[ { }_7^{14} N \][/tex]). To balance the total mass number on both sides, we need an additional 1 nucleon on the reactant side.
Conclusion:
The additional particle needed must have 0 protons (so it doesn't affect the atomic number balance) and 1 nucleon (to balance the mass number). The particle that fits this requirement is a neutron, denoted as:
[tex]\[ { }_0^1 n \][/tex]
So, the particle needed to complete the equation is:
[tex]\[ { }_0^1 n \][/tex]
Thus the correct answer is:
[tex]\[ { }_0^1 n \][/tex]
