An alpha particle `(.^(4)He)` has a mass of 4.00300 amu. A proton has a mass of 1.00783 amu and a neutron has a mass of 1.00867 amu respectively. The binding energy of alpha particle estimated from these data is the closest to

Options

A. 27.9 MeV
B. 22.3 MeV
C. 35.0 MeV
D. 20.4 MeV

Explanation

To find the binding energy of an alpha particle (Helium-4), we can follow these steps:

### Step 1: Identify the components of the alpha particle
An alpha particle consists of:
- 2 protons
- 2 neutrons

### Step 2: Calculate the total mass of the nucleons
Using the given masses:
- Mass of a proton (mp) = 1.00783 amu
- Mass of a neutron (mn) = 1.00867 amu

The total mass of the nucleons (mass of 2 protons and 2 neutrons) can be calculated as:
\[ \text{Total mass of nucleons} = 2 \times m_p + 2 \times m_n \]
Substituting the values:
\[ \text{Total mass of nucleons} = 2 \times 1.00783 + 2 \times 1.00867 \]
\[ = 2.01566 + 2.01734 = 4.03300 \text{ amu} \]

### Step 3: Find the mass defect
The mass defect (Δm) is the difference between the total mass of the nucleons and the actual mass of the alpha particle.
- Mass of the alpha particle (He-4) = 4.00300 amu

Calculating the mass defect:
\[ \Delta m = \text{Total mass of nucleons} - \text{Mass of alpha particle} \]
\[ \Delta m = 4.03300 - 4.00300 = 0.03000 \text{ amu} \]

### Step 4: Convert the mass defect to energy
Using Einstein's equation \(E = \Delta m c^2\), we can convert the mass defect into energy. The conversion factor from amu to MeV is approximately 931.5 MeV/amu.
\[ E = \Delta m \times 931.5 \text{ MeV/amu} \]
Substituting the mass defect:
\[ E = 0.03000 \times 931.5 = 27.795 \text{ MeV} \]

### Step 5: Round the result
The binding energy of the alpha particle is approximately:
\[ E \approx 27.8 \text{ MeV} \]

### Step 6: Conclusion
The closest value for the binding energy of the alpha particle is:
\[ \text{Binding Energy} \approx 27.9 \text{ MeV} \]