Nucleus `._(3)A^(7)` has binding energy per nucleon of `10 MeV`. It absorbs a proton and its mass increases by `(90)/(100)` times the mass of proton. Find the new binding energy of the nucleus so formed. [Take energy equivalent of proton `= 930 MeV`]

Explanation

To find the new binding energy of the nucleus after it absorbs a proton, we can follow these steps:

### Step 1: Understand the Initial Binding Energy
The initial binding energy per nucleon of the nucleus \( _{3}A^{7} \) is given as \( 10 \, \text{MeV} \). Since the nucleus has 7 nucleons (3 protons and 4 neutrons), the total initial binding energy can be calculated as:
\[ \text{Total Initial Binding Energy} = \text{Binding Energy per Nucleon} \times \text{Number of Nucleons} = 10 \, \text{MeV} \times 7 = 70 \, \text{MeV} \]

**Hint:** Remember that binding energy per nucleon is the total binding energy divided by the number of nucleons.

### Step 2: Determine the Mass Increase
The nucleus absorbs a proton, and its mass increases by \( \frac{90}{100} \) times the mass of a proton. The mass of a proton is approximately \( 1 \, \text{amu} \). Therefore, the increase in mass is:
\[ \Delta m = \frac{90}{100} \times 1 \, \text{amu} = 0.9 \, \text{amu} \]

**Hint:** When calculating mass increase, ensure you understand the relationship between mass and energy.

### Step 3: Calculate the Energy Equivalent of the Mass Increase
Using Einstein's equation \( E = \Delta m \cdot c^2 \), we can convert the mass increase to energy. The energy equivalent of \( 1 \, \text{amu} \) is given as \( 930 \, \text{MeV} \). Thus, the energy equivalent of the mass increase is:
\[ E = 0.9 \, \text{amu} \times 930 \, \text{MeV/amu} = 837 \, \text{MeV} \]

**Hint:** Always use the correct conversion factor for mass to energy when applying Einstein's equation.

### Step 4: Calculate the New Total Binding Energy
After the nucleus absorbs the proton, the total binding energy increases by the energy equivalent of the mass defect (which is the mass increase). Therefore, the new total binding energy is:
\[ \text{New Total Binding Energy} = \text{Total Initial Binding Energy} + E = 70 \, \text{MeV} + 837 \, \text{MeV} = 907 \, \text{MeV} \]

**Hint:** Remember to add the energy gained from the mass defect to the initial binding energy.

### Step 5: Calculate the New Binding Energy per Nucleon
After absorbing the proton, the total number of nucleons becomes \( 8 \) (7 original nucleons + 1 proton). The new binding energy per nucleon is:
\[ \text{New Binding Energy per Nucleon} = \frac{\text{New Total Binding Energy}}{\text{Number of Nucleons}} = \frac{907 \, \text{MeV}}{8} \approx 113.375 \, \text{MeV} \]

**Hint:** When calculating binding energy per nucleon, divide the total binding energy by the new total number of nucleons.

### Final Answer
The new binding energy of the nucleus after absorbing the proton is approximately \( 113.375 \, \text{MeV} \) per nucleon.