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 (Δm) of the nucleus \( X_Z^A \), we can follow these steps:

### Step 1: Identify the components of the nucleus
The nucleus \( X_Z^A \) consists of:
- \( Z \) protons
- \( A - Z \) neutrons

Where:
- \( Z \) is the atomic number (number of protons)
- \( A \) is the mass number (total number of nucleons, protons + neutrons)

### Step 2: Write the mass of the components
The masses of the components are given as:
- Mass of one proton = \( m_p \)
- Mass of one neutron = \( m_n \)

Thus, the total mass of the protons and neutrons can be calculated as:
- Total mass of protons = \( Z \cdot m_p \)
- Total mass of neutrons = \( (A - Z) \cdot m_n \)

### Step 3: Calculate the total mass of the nucleus
The total mass of the nucleus \( M \) can be expressed as:
\[ M = Z \cdot m_p + (A - Z) \cdot m_n \]

### Step 4: Define the mass defect
The mass defect \( \Delta m \) is defined as the difference between the total mass of the individual nucleons (protons and neutrons) and the actual mass of the nucleus. Therefore, we can write:
\[ \Delta m = \text{(mass of protons + mass of neutrons)} - \text{mass of nucleus} \]

### Step 5: Substitute the expressions
Substituting the expressions from Step 2 and Step 3 into the equation for mass defect:
\[ \Delta m = \left( Z \cdot m_p + (A - Z) \cdot m_n \right) - M \]

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

### Conclusion
The final expression for the mass defect \( \Delta m \) of the nucleus \( X_Z^A \) is:
\[ \Delta m = (A - Z) m_n + Z m_p - M \]