Which of the following pair of species are isodiaphers?

Options

A. `C_6^14 and N_7^14`
B. `C_6^13 andC_6^14`
C. `C_6^12 and N_7^14`
D. All the these

Answer

To determine which pair of species are isodiaphers, we need to understand the concept of isodiaphers. Isodiaphers are nuclides (nuclear species) that have the same difference between the number of neutrons and the number of protons, even though they may have different atomic numbers or mass numbers.

### Step-by-Step Solution:

1. **Identify the pairs of species**: We will analyze the given pairs of species to determine their atomic numbers (Z) and mass numbers (A).

2. **Calculate the number of protons and neutrons**:
- For each species, the number of protons (Z) is equal to the atomic number.
- The number of neutrons (N) can be calculated using the formula:
\[ N = A - Z \]
where A is the mass number.

3. **Calculate the difference (N - Z)**:
- For each species, calculate the difference between the number of neutrons and the number of protons (N - Z).

4. **Compare the differences**:
- If the difference (N - Z) is the same for both species in a pair, then they are isodiaphers.

### Example Analysis:

Let's analyze the pairs one by one:

**Pair 1**:
- Species 1: Carbon (C) with atomic number 6 and mass number 14.
- Protons (Z) = 6
- Neutrons (N) = 14 - 6 = 8
- Difference (N - Z) = 8 - 6 = 2

- Species 2: Nitrogen (N) with atomic number 7 and mass number 14.
- Protons (Z) = 7
- Neutrons (N) = 14 - 7 = 7
- Difference (N - Z) = 7 - 7 = 0

Since the differences are not equal, this pair is not isodiaphers.

**Pair 2**:
- Species 1: Carbon (C) with atomic number 6 and mass number 13.
- Protons (Z) = 6
- Neutrons (N) = 13 - 6 = 7
- Difference (N - Z) = 7 - 6 = 1

- Species 2: Carbon (C) with atomic number 6 and mass number 14.
- Protons (Z) = 6
- Neutrons (N) = 14 - 6 = 8
- Difference (N - Z) = 8 - 6 = 2

Since the differences are not equal, this pair is not isodiaphers.

**Pair 3**:
- Species 1: Carbon (C) with atomic number 6 and mass number 12.
- Protons (Z) = 6
- Neutrons (N) = 12 - 6 = 6
- Difference (N - Z) = 6 - 6 = 0

- Species 2: Nitrogen (N) with atomic number 7 and mass number 14.
- Protons (Z) = 7
- Neutrons (N) = 14 - 7 = 7
- Difference (N - Z) = 7 - 7 = 0

Since the differences are equal (both are 0), this pair is isodiaphers.

### Conclusion:
The correct answer is that the third pair of species (Carbon-12 and Nitrogen-14) are isodiaphers.