If `m,m_n` and `m_p` are masses of `._Z X^A` nucleus, neutron and proton respectively.

Options

A. `M=(A-Z)m_(n)+Zm_(p)`
B. `M=Zm_(n)+(A-Z)m_(p)`
C. `M lt (A-Z) m_(n)+Zm_(p)`
D. `M gt (A-Z) +Zm_(p)`

Answer

To solve the problem regarding the masses of the nucleus, neutron, and proton, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Components:**
- Let \( M \) be the mass of the nucleus \( _Z X^A \).
- Let \( m_n \) be the mass of a neutron.
- Let \( m_p \) be the mass of a proton.

2. **Determine the Number of Protons and Neutrons:**
- The atomic number \( Z \) represents the number of protons in the nucleus.
- The mass number \( A \) represents the total number of nucleons (protons + neutrons).
- Therefore, the number of neutrons \( N \) can be calculated as:
\[ N = A - Z \]

3. **Calculate the Predicted Mass of the Nucleus:**
- The predicted mass of the nucleus can be expressed as the sum of the masses of the protons and neutrons:
\[ \text{Predicted mass} = Z \cdot m_p + (A - Z) \cdot m_n \]
- This can be simplified to:
\[ \text{Predicted mass} = Z \cdot m_p + (A - Z) \cdot m_n \]

4. **Understanding Mass Defect:**
- The actual mass \( M \) of the nucleus is less than the predicted mass due to the binding energy, which is the energy required to disassemble the nucleus into its individual nucleons.
- This phenomenon leads to a mass defect, which is the difference between the predicted mass and the actual mass.

5. **Establishing the Relationship:**
- Thus, we can write:
\[ M < Z \cdot m_p + (A - Z) \cdot m_n \]
- This indicates that the actual mass \( M \) of the nucleus is less than the predicted mass.

6. **Conclusion:**
- The correct statement is that the actual mass of the nucleus \( M \) is less than the predicted mass \( Z \cdot m_p + (A - Z) \cdot m_n \) due to the mass defect caused by binding energy.

### Final Answer:
The correct relationship is:
\[ M < (A - Z) m_n + Z m_p \]