A nucleus `._(Z)^(A)X` has mass represented by `m(A, Z)`. If `m_(p)` and `m_(n)` denote the mass of proton and neutron respectively and `BE` the blinding energy (in MeV), then

Options

A. `BE=[m(A, Z)-Zm_(p)-(A-Z)m_(n)]C^(2)`
B. `BE=[Zm_(p)+(A-Z)m_(n)-m(A, Z)]C^(2)`
C. `BE=[Zm_(p)+Am_(n)-m(A, Z)]C^(2)`
D. `BE=m(A, Z)-Zm_(p)-(A-Z)m_(n)`

Answer

To solve the problem regarding the binding energy (BE) of a nucleus represented as `._(Z)^(A)X`, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Binding Energy**:
Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is associated with the mass defect of the nucleus.

2. **Identify the Masses**:
- Let `m_p` be the mass of a proton.
- Let `m_n` be the mass of a neutron.
- The nucleus has `Z` protons and `A - Z` neutrons (where `A` is the mass number).

3. **Calculate the Total Mass of Individual Nucleons**:
The total mass of the individual nucleons (protons and neutrons) can be expressed as:
\[ \text{Total mass of nucleons} = Z \cdot m_p + (A - Z) \cdot m_n \]

4. **Mass of the Nucleus**:
The mass of the nucleus is given as `m(A, Z)`.

5. **Calculate the Mass Defect**:
The mass defect (`Δm`) is the difference between the total mass of the individual nucleons and the actual mass of the nucleus:
\[ Δm = (Z \cdot m_p + (A - Z) \cdot m_n) - m(A, Z) \]

6. **Relate Mass Defect to Binding Energy**:
The binding energy (BE) can be calculated using Einstein's mass-energy equivalence principle:
\[ BE = Δm \cdot c^2 \]
where `c` is the speed of light.

7. **Substituting the Expression for Mass Defect**:
Substitute the expression for `Δm` into the binding energy formula:
\[ BE = \left( (Z \cdot m_p + (A - Z) \cdot m_n) - m(A, Z) \right) \cdot c^2 \]

### Final Expression for Binding Energy:
Thus, the binding energy can be expressed as:
\[ BE = \left( Z \cdot m_p + (A - Z) \cdot m_n - m(A, Z) \right) \cdot c^2 \]