A radioactive element `X` has an atomic number of 100. It decays directly into an element `Y` which decays directly into an element `Z`. In both the processes either one `alpha` or one `beta-` particle is emitted. Which of the following statement could be true?

Options

A. `Y` has an atomic number of `102`
B. `Z` has an atmic number of `101`
C. `Z` has an atomic number of `97`
D. `Z` has an atomic number of `99`

Explanation

To solve the problem, we need to analyze the decay process of the radioactive element X with an atomic number of 100 and determine the atomic numbers of elements Y and Z based on the decay processes described.

### Step-by-Step Solution:

1. **Identify the decay of element X:**
- Element X has an atomic number of 100.
- It decays into element Y by emitting either an alpha particle or a beta particle.

2. **Consider the alpha decay of X:**
- If X emits an alpha particle (which has an atomic number of 2), the atomic number of Y will be:
\[ \text{Atomic number of Y} = 100 - 2 = 98 \]
- Therefore, Y has an atomic number of 98.

3. **Identify the decay of element Y:**
- Element Y (atomic number 98) decays into element Z by emitting either an alpha particle or a beta particle.

4. **Consider the beta decay of Y:**
- If Y emits a beta particle (which has an atomic number of -1), the atomic number of Z will be:
\[ \text{Atomic number of Z} = 98 + 1 = 99 \]
- Therefore, Z has an atomic number of 99.

5. **Consider the alpha decay of Y:**
- If Y emits an alpha particle, the atomic number of Z will be:
\[ \text{Atomic number of Z} = 98 - 2 = 96 \]
- Therefore, Z could also have an atomic number of 96.

6. **Summarize the possible atomic numbers:**
- From the decay processes, we find:
- If X decays via alpha emission:
- Y = 98 (after alpha decay)
- Z = 99 (after beta decay) or Z = 96 (after alpha decay)

7. **Evaluate the options given in the question:**
- Option A: Y has an atomic number of 102. (Incorrect, Y = 98)
- Option B: Z has an atomic number of 101. (Incorrect, Z = 99 or 96)
- Option C: Z has an atomic number of 97. (Incorrect, Z = 99 or 96)
- Option D: Z has an atomic number of 99. (Correct if Y decays via beta emission)

### Conclusion:
The only valid statement based on our calculations is that Z could have an atomic number of 99 if Y decays via beta emission.