The particle A is converted to C via following reactions then
`A to B + "" _(2) H e ^(4) , B to C + 2 "" _(-1) e ^(0)`

Options

A. A and C are isobars
B. A and C are isotopes
C. A and B are isobars
D. A and B are isotopes

Answer

To solve the problem of particle A converting to C through the given reactions, we will analyze the reactions step by step, focusing on the conservation of charge and atomic mass.

### Step 1: Analyze the first reaction
The first reaction is:
\[ A \rightarrow B + 2 \, ^4_2He \]

Here, particle A is converting into particle B and emitting two alpha particles (\( ^4_2He \)).

- **Mass Number Conservation**:
Let the mass number of particle A be \( A_A \) and that of B be \( A_B \). Each alpha particle has a mass number of 4. Therefore, the mass number conservation gives us:
\[ A_A = A_B + 2 \times 4 \]
\[ A_A = A_B + 8 \]

- **Charge Conservation**:
Let the atomic number of particle A be \( Z_A \) and that of B be \( Z_B \). Each alpha particle has a charge of +2. Therefore, the charge conservation gives us:
\[ Z_A = Z_B + 2 \times 2 \]
\[ Z_A = Z_B + 4 \]

### Step 2: Analyze the second reaction
The second reaction is:
\[ B \rightarrow C + 2 \, e^- \]

In this reaction, particle B is converting into particle C and emitting two electrons.

- **Mass Number Conservation**:
The mass number of C, denoted as \( A_C \), must be equal to that of B since electrons have negligible mass:
\[ A_B = A_C \]

- **Charge Conservation**:
The charge of particle C must account for the loss of two electrons (each with a charge of -1):
\[ Z_B = Z_C + 2 \]

### Step 3: Combine the equations
From the first reaction, we have:
1. \( A_A = A_B + 8 \)
2. \( Z_A = Z_B + 4 \)

From the second reaction, we have:
3. \( A_B = A_C \)
4. \( Z_B = Z_C + 2 \)

Substituting \( A_B \) from equation 3 into equation 1 gives:
\[ A_A = A_C + 8 \]

Substituting \( Z_B \) from equation 4 into equation 2 gives:
\[ Z_A = (Z_C + 2) + 4 \]
\[ Z_A = Z_C + 6 \]

### Step 4: Conclusion
From the above equations, we can summarize:
- The mass number of C is \( A_C = A_A - 8 \).
- The atomic number of C is \( Z_C = Z_A - 6 \).

Since both A and C have the same atomic number \( Z \) (as they are isotopes), we conclude that A and C are isotopes of each other.