Understanding Mirror Nuclei Pairs

The question asks us to identify a pair of mirror nuclei from the given options. Mirror nuclei are pairs of nuclides that have the same mass number (A) but have their number of protons (Z) and neutrons (N) swapped.

Mathematically, if one nucleus is represented as $^A_Z X$, its mirror nucleus $^A_{Z'} Y$ must satisfy these conditions:

• They must have the same mass number: $A = A$.
• The number of protons in one must equal the number of neutrons in the other, and vice versa. If nucleus 1 has Z protons and N neutrons ($N = A - Z$), then nucleus 2 must have Z' protons and N' neutrons ($N' = A - Z'$) such that $Z' = N$ and $N' = Z$.
• This implies that the sum of their atomic numbers equals the mass number: $Z + Z' = A$.

Let's analyze each option based on these criteria:

Option 1: $^{14}C$ and $^{13}C$

• $^{14}C$: Mass number $A = 14$, Atomic number $Z = 6$ (Carbon). Neutron number $N = A - Z = 14 - 6 = 8$.
• $^{13}C$: Mass number $A = 13$, Atomic number $Z = 6$ (Carbon). Neutron number $N = A - Z = 13 - 6 = 7$.
• The mass numbers (14 and 13) are different. Therefore, they are not mirror nuclei.

Option 2: $^{13}C$ and $^{14}N$

• $^{13}C$: Mass number $A = 13$, Atomic number $Z = 6$ (Carbon). Neutron number $N = 13 - 6 = 7$.
• $^{14}N$: Mass number $A = 14$, Atomic number $Z = 7$ (Nitrogen). Neutron number $N = 14 - 7 = 7$.
• The mass numbers (13 and 14) are different. Therefore, they are not mirror nuclei.

Option 3: $^{13}C$ and $^{13}N$

• $^{13}C$: Mass number $A = 13$, Atomic number $Z = 6$ (Carbon). Neutron number $N = 13 - 6 = 7$.
• $^{13}N$: Mass number $A = 13$, Atomic number $Z = 7$ (Nitrogen). Neutron number $N = 13 - 7 = 6$.
• The mass numbers are the same ($A = 13$).
• Let's check the proton and neutron counts:
  • For $^{13}C$: $Z = 6$, $N = 7$.
  • For $^{13}N$: $Z' = 7$, $N' = 6$.
• We see that $Z' = 7$ which is equal to $N = 7$ for $^{13}C$.
• We also see that $N' = 6$ which is equal to $Z = 6$ for $^{13}C$.
• Alternatively, check $Z + Z' = A$: $6 + 7 = 13$. This matches the mass number $A=13$.
• Since both conditions (same mass number and swapped proton/neutron numbers) are met, this pair are mirror nuclei.

Here is a summary table for this pair:

Option 4: $^{13}C$ and $^{12}B$

• $^{13}C$: Mass number $A = 13$, Atomic number $Z = 6$ (Carbon). Neutron number $N = 13 - 6 = 7$.
• $^{12}B$: Mass number $A = 12$, Atomic number $Z = 5$ (Boron). Neutron number $N = 12 - 5 = 7$.
• The mass numbers (13 and 12) are different. Therefore, they are not mirror nuclei.

Conclusion

Based on the analysis, the pair $^{13}C$ and $^{13}N$ is the only one that satisfies the definition of mirror nuclei.