If M,`m_n` and `m_p` are masses (in kg) of nucleus `X_z^A`, neutron and proton respectively, then mass defect`(Deltam)` is equal to

Options

A. `Deltam=[(A-Z)m_n+Am_p-M]`
B. `Deltam=[M-(A-Z)m_n+Am_p]`
C. `Deltam=[(A-Z)m_n+Zm_p+M]`
D. `Deltam=[(A-Z)m_n+Zm_p+M]`

Answer

To find the mass defect (\(\Delta m\)) of a nucleus \(X_z^A\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- The nucleus \(X_z^A\) consists of protons and neutrons. The mass number \(A\) represents the total number of nucleons (protons + neutrons), and the atomic number \(Z\) represents the number of protons.

2. **Determine the Number of Neutrons**:
- The number of neutrons (\(N_n\)) can be calculated using the formula:
\[ N_n = A - Z \]

3. **Write the Masses**:
- Let \(m_n\) be the mass of one neutron, \(m_p\) be the mass of one proton, and \(M\) be the mass of the nucleus.

4. **Calculate the Total Mass of Individual Particles**:
- The total mass of all the neutrons is:
\[ \text{Total mass of neutrons} = N_n \cdot m_n = (A - Z) \cdot m_n \]
- The total mass of all the protons is:
\[ \text{Total mass of protons} = Z \cdot m_p \]

5. **Sum of the Masses of Individual Particles**:
- The total mass of the nucleus in terms of its constituent particles is:
\[ \text{Total mass} = (A - Z) \cdot m_n + Z \cdot m_p \]

6. **Calculate the Mass Defect**:
- The mass defect (\(\Delta m\)) is defined as the difference between the total mass of the individual nucleons and the actual mass of the nucleus:
\[ \Delta m = \text{Total mass of nucleons} - \text{Mass of nucleus} \]
- Substituting the values, we get:
\[ \Delta m = \left((A - Z) \cdot m_n + Z \cdot m_p\right) - M \]

### Final Expression:
Thus, the mass defect can be expressed as:
\[ \Delta m = (A - Z) m_n + Z m_p - M \]