A nucleus `._ZX^A` emits `9 alpha-`particles and `5p` particle. The ratio of total protons and neutrons in the final nucleus is.

Options

A. `((Z-13))/((A-36))`
B. `((Z-13))/((A-Z-13))`
C. `((Z-13))/((A-Z-23))`
D. `((Z-13))/((A-Z-23))`

Explanation

To solve the problem step by step, we will analyze the emission of alpha and beta particles from the nucleus \( _Z^A \) and calculate the final number of protons and neutrons.

### Step 1: Understand the emissions
1. **Alpha particles**: Each alpha particle (\( \alpha \)) consists of 2 protons and 2 neutrons, and has a mass number of 4. If 9 alpha particles are emitted, the total change in protons and mass number is:
- Protons lost: \( 9 \times 2 = 18 \)
- Mass number lost: \( 9 \times 4 = 36 \)

### Step 2: Calculate the new values after alpha emission
2. **New atomic number (Z)** after emitting 9 alpha particles:
\[ Z' = Z - 18 \]
3. **New mass number (A)** after emitting 9 alpha particles:
\[ A' = A - 36 \]

### Step 3: Understand beta particle emissions
4. **Beta particles**: Each beta particle (\( \beta \)) is an electron with a charge of -1 and mass of 0. Emitting a beta particle increases the atomic number by 1. If 5 beta particles are emitted:
- Protons gained: \( 5 \)

### Step 4: Calculate the new values after beta emission
5. **New atomic number (Z)** after emitting 5 beta particles:
\[ Z'' = (Z - 18) + 5 = Z - 13 \]
6. **Mass number (A)** remains unchanged after beta emissions:
\[ A'' = A - 36 \]

### Step 5: Calculate the number of neutrons
7. **Number of neutrons (N)** in the final nucleus can be calculated using the formula:
\[ N = A'' - Z'' \]
Plugging in the values we found:
\[ N = (A - 36) - (Z - 13) = A - 36 - Z + 13 = A - Z - 23 \]

### Step 6: Calculate the ratio of protons to neutrons
8. **Protons (P)** in the final nucleus:
\[ P = Z - 13 \]
9. **Ratio of protons to neutrons**:
\[ \text{Ratio} = \frac{P}{N} = \frac{Z - 13}{A - Z - 23} \]

### Final Result
The ratio of total protons to neutrons in the final nucleus is:
\[ \text{Ratio} = \frac{Z - 13}{A - Z - 23} \]