A radiaoactive nucleus (initial mass number `A` and atomic number `Z` emits `3 alpha`- particles and 2 positrons The ratio of number of neutrons to that of proton in the final nucleus will be

Options

A. `(A-Z-8)/(Z-4)`
B. `(A-Z-4)/(Z-8)`
C. `(A-Z-12)/(Z-4)`
D. `(A-Z-4)/(Z-2)`

Answer

To solve the problem step by step, let's analyze the emission of particles from the radioactive nucleus.

### Step 1: Understand the Initial Conditions
- The initial nucleus has a mass number \( A \) and an atomic number \( Z \).
- The mass number \( A \) represents the total number of nucleons (protons + neutrons).
- The atomic number \( Z \) represents the number of protons.

### Step 2: Calculate Changes Due to Emission of Alpha Particles
- An alpha particle has a mass number of 4 and an atomic number of 2.
- When 3 alpha particles are emitted, the total change in mass number is:
\[ \Delta A = 3 \times 4 = 12 \]
Therefore, the new mass number after the emission of 3 alpha particles will be:
\[ A' = A - 12 \]
- The total change in atomic number is:
\[ \Delta Z = 3 \times 2 = 6 \]
Therefore, the new atomic number after the emission of 3 alpha particles will be:
\[ Z' = Z - 6 \]

### Step 3: Calculate Changes Due to Emission of Positrons
- A positron has an atomic number of +1 but no mass (mass number = 0).
- When 2 positrons are emitted, the total change in atomic number is:
\[ \Delta Z = 2 \times 1 = 2 \]
Therefore, the new atomic number after the emission of 2 positrons will be:
\[ Z'' = Z' - 2 = (Z - 6) - 2 = Z - 8 \]
- The mass number remains unchanged due to the emission of positrons:
\[ A'' = A' = A - 12 \]

### Step 4: Determine the Final Number of Protons and Neutrons
- The final number of protons (after all emissions) is:
\[ \text{Number of Protons} = Z'' = Z - 8 \]
- The final number of neutrons can be calculated as:
\[ \text{Number of Neutrons} = A'' - Z'' = (A - 12) - (Z - 8) = A - 12 - Z + 8 = A - Z - 4 \]

### Step 5: Calculate the Ratio of Neutrons to Protons
- The ratio of the number of neutrons to the number of protons is given by:
\[ \text{Ratio} = \frac{\text{Number of Neutrons}}{\text{Number of Protons}} = \frac{A - Z - 4}{Z - 8} \]

### Final Answer
The ratio of the number of neutrons to the number of protons in the final nucleus is:
\[ \frac{A - Z - 4}{Z - 8} \]