`m_(P)` and `m_(n)` are masses of proton and neutron respectively. An element of mass `m` has `Z` protons and `N` neutrons, then

Options

A. `m gt Zm_(p) +Nm_(n)`
B. `m=Zm_(p)+Nm_(n)`
C. `m lt Zm_(p)+Nm_(n)`
D. `m` may be greater than, less than or equal to `Zm_(p)+Nm_(n)`, depending on nature of element

Answer

To solve the problem, we need to establish the relationship between the mass of an element (M), the number of protons (Z), the number of neutrons (N), and their respective masses (m_P for protons and m_N for neutrons).

### Step-by-Step Solution:

1. **Identify the Masses**:
- Let \( m_P \) be the mass of one proton.
- Let \( m_N \) be the mass of one neutron.
- The total mass contributed by protons is \( Z \cdot m_P \).
- The total mass contributed by neutrons is \( N \cdot m_N \).

2. **Calculate the Total Mass**:
- The total mass of the nucleons (protons and neutrons) can be expressed as:
\[ M_{total} = Z \cdot m_P + N \cdot m_N \]

3. **Consider Binding Energy**:
- When protons and neutrons come together to form a nucleus, they release energy due to the strong nuclear force. This energy release is known as binding energy.
- According to the mass-energy equivalence principle (Einstein's equation \( E = mc^2 \)), the energy released results in a decrease in the mass of the nucleus.

4. **Establish the Relationship**:
- The actual mass of the nucleus (M) is less than the total mass of the individual nucleons due to the binding energy. Therefore, we can write:
\[ M < M_{total} \]
- This implies:
\[ M < Z \cdot m_P + N \cdot m_N \]

5. **Conclusion**:
- The mass of the nucleus (M) is always less than the sum of the masses of the individual protons and neutrons due to the binding energy that holds the nucleus together.

### Final Answer:
\[ M < Z \cdot m_P + N \cdot m_N \]