If a nucleus `._(Z)^(A)x` emits one `alpha`-particle and one `beta` (negative `beta`) particle in succession, then the daughter nucleus will have which of the following configurations?

Options

A. A-4 nucleons
B. A-3 nucleons
C. A-Z-3 neutrons
D. A-Z protons

Answer

To solve the problem, we need to analyze the effects of the emission of an alpha particle and a beta particle on the original nucleus with mass number \( A \) and atomic number \( Z \).

### Step-by-Step Solution:

1. **Identify the Initial Nucleus:**
The initial nucleus is represented as \( _{Z}^{A}X \), where \( A \) is the mass number and \( Z \) is the atomic number.

2. **Emission of an Alpha Particle:**
An alpha particle is represented as \( _{2}^{4}\alpha \). When an alpha particle is emitted, the mass number decreases by 4 and the atomic number decreases by 2. Therefore, after the emission of one alpha particle, the new nucleus will be:
\[ A' = A - 4 \quad \text{(mass number)} \]
\[ Z' = Z - 2 \quad \text{(atomic number)} \]

3. **Emission of a Beta Particle:**
A beta particle (negative beta) is represented as \( _{−1}^{0}\beta \). When a beta particle is emitted, the mass number remains unchanged, but the atomic number increases by 1. Therefore, after the emission of one beta particle, the new nucleus will be:
\[ A'' = A' \quad \text{(mass number remains the same)} \]
\[ Z'' = Z' + 1 = (Z - 2) + 1 = Z - 1 \quad \text{(atomic number)} \]

4. **Final Configuration of the Daughter Nucleus:**
After both emissions, the daughter nucleus will have:
\[ A'' = A - 4 \]
\[ Z'' = Z - 1 \]

5. **Conclusion:**
The final configuration of the daughter nucleus after the emissions is \( _{Z-1}^{A-4}Y \), where \( Y \) represents the new element formed.

### Final Answer:
The daughter nucleus will have the configuration \( _{Z-1}^{A-4}Y \).