Which elements are isotopes of each other in the above sequence of reaction? `X overset(-alpha)(to)Y overset(-beta)(to)Z overset(-beta)(rarr)W`

Options

A. X and W
B. Y and Z
C. X and Z
D. None of these

Answer

To determine which elements are isotopes of each other in the given sequence of reactions, we will analyze the changes in mass number and atomic number during alpha and beta decay.

### Step-by-Step Solution:

1. **Identify the Sequence of Reactions**:
- The sequence given is:
- \( X \overset{\alpha}{\rightarrow} Y \)
- \( Y \overset{\beta}{\rightarrow} Z \)
- \( Z \overset{\beta}{\rightarrow} W \)

2. **Understand Alpha Decay**:
- In alpha decay, the mass number (A) decreases by 4 and the atomic number (Z) decreases by 2.
- If the mass number of \( X \) is \( A \) and the atomic number is \( Z \), then for \( Y \):
- Mass number of \( Y = A - 4 \)
- Atomic number of \( Y = Z - 2 \)

3. **Understand Beta Decay**:
- In beta decay, the mass number remains the same, but the atomic number increases by 1.
- For \( Y \) decaying to \( Z \):
- Mass number of \( Z = A - 4 \) (remains the same)
- Atomic number of \( Z = (Z - 2) + 1 = Z - 1 \)

4. **Continue with Beta Decay**:
- For \( Z \) decaying to \( W \):
- Mass number of \( W = A - 4 \) (remains the same)
- Atomic number of \( W = (Z - 1) + 1 = Z \)

5. **Identify Isotopes**:
- Isotopes are defined as elements with the same atomic number but different mass numbers.
- From the analysis:
- Atomic number of \( W = Z \) and atomic number of \( X = Z \).
- However, the mass number of \( X = A \) and mass number of \( W = A - 4 \).
- Since \( X \) and \( W \) have the same atomic number \( Z \) but different mass numbers \( A \) and \( A - 4 \), they are isotopes of each other.

### Conclusion:
The elements that are isotopes of each other in the given sequence are **X and W**.