The binding energy of `alpha`-particle is
( if `m_p=1.00785u`, `m_n=1.00866u` and `m_alpha=4.00274u`)

Options

A. (a) 56.42 MeV
B. (b) 2.821 MeV
C. (c) 28.21 MeV
D. (d) 32.4 MeV

Explanation

To find the binding energy of an alpha particle, we will follow these steps:

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

### Step 2: Write down the masses of the nucleons
Given:
- Mass of proton, \( m_p = 1.00785 \, u \)
- Mass of neutron, \( m_n = 1.00866 \, u \)
- Mass of alpha particle, \( m_{\alpha} = 4.00274 \, u \)

### Step 3: Calculate the total mass of the constituent nucleons
The total mass of the nucleons (2 protons and 2 neutrons) can be calculated as follows:
\[ \text{Total mass of nucleons} = 2 \times m_p + 2 \times m_n \]
Substituting the values:
\[ \text{Total mass of nucleons} = 2 \times 1.00785 \, u + 2 \times 1.00866 \, u \]
\[ = 2.01570 \, u + 2.01732 \, u = 4.03302 \, u \]

### Step 4: Calculate the mass defect
The mass defect \( \Delta m \) is the difference between the total mass of the nucleons and the actual mass of the alpha particle:
\[ \Delta m = \text{Total mass of nucleons} - m_{\alpha} \]
Substituting the values:
\[ \Delta m = 4.03302 \, u - 4.00274 \, u = 0.03028 \, u \]

### Step 5: Convert the mass defect to binding energy
Binding energy \( E_b \) can be calculated using the formula:
\[ E_b = \Delta m \times c^2 \]
To convert mass defect from atomic mass units (u) to energy in mega electron volts (MeV), we use the conversion factor:
\[ 1 \, u \approx 931.5 \, \text{MeV} \]
Thus:
\[ E_b = 0.03028 \, u \times 931.5 \, \text{MeV/u} \]
Calculating this gives:
\[ E_b \approx 28.21 \, \text{MeV} \]

### Final Answer
The binding energy of the alpha particle is approximately **28.21 MeV**.

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