The masses of neutron and proton are `1.0087` a.m.u. and `1.0073` a.m.u. respectively. If the neutrons and protons combine to form a helium nucleus (alpha particle) of mass `4.0015`a.m.u. The binding energy of the helium nucleus will be `(1 a.m.u. = 931 MeV)`.

Options

A. 28.4 MeV
B. 20.8 MeV
C. 27.3 MeV
D. 14.2 MeV

Explanation

To find the binding energy of the helium nucleus, we will follow these steps:

### Step 1: Calculate the total mass of the constituents (neutrons and protons)
The helium nucleus consists of 2 protons and 2 neutrons. We can calculate the total mass of these particles.

\[ \text{Total mass of protons} = 2 \times \text{mass of proton} = 2 \times 1.0073 \, \text{a.m.u.} = 2.0146 \, \text{a.m.u.} \]

\[ \text{Total mass of neutrons} = 2 \times \text{mass of neutron} = 2 \times 1.0087 \, \text{a.m.u.} = 2.0174 \, \text{a.m.u.} \]

\[ \text{Total mass of constituents} = 2.0146 \, \text{a.m.u.} + 2.0174 \, \text{a.m.u.} = 4.0320 \, \text{a.m.u.} \]

### Step 2: Calculate the mass defect
The mass defect (\(\Delta m\)) is the difference between the total mass of the individual nucleons and the actual mass of the helium nucleus.

\[ \Delta m = \text{Total mass of constituents} - \text{mass of helium nucleus} \]

Given that the mass of the helium nucleus is \(4.0015 \, \text{a.m.u.}\):

\[ \Delta m = 4.0320 \, \text{a.m.u.} - 4.0015 \, \text{a.m.u.} = 0.0305 \, \text{a.m.u.} \]

### Step 3: Convert mass defect to energy
The binding energy (BE) can be calculated using the mass defect and the conversion factor \(1 \, \text{a.m.u.} = 931 \, \text{MeV}\).

\[ \text{Binding Energy} = \Delta m \times c^2 = \Delta m \times 931 \, \text{MeV} \]

Substituting the value of \(\Delta m\):

\[ \text{Binding Energy} = 0.0305 \, \text{a.m.u.} \times 931 \, \text{MeV/a.m.u.} = 28.4 \, \text{MeV} \]

### Final Answer
The binding energy of the helium nucleus is approximately \(28.4 \, \text{MeV}\).

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