The mass of proton is `1.0073 u` and that of neutron is `1.0087 u` (`u=` atomic mass unit). The binding energy of `._(2)He^(4)` is (mass of helium nucleus `=4.0015 u`)

Options

A. 0.0305 J
B. 0.0305 erg
C. 28.4 MeV
D. 0.061 u

Explanation

To find the binding energy of the helium-4 nucleus, we can follow these steps:

### Step 1: Identify the number of protons and neutrons in helium-4
Helium-4 has:
- 2 protons (since it is helium)
- 2 neutrons (since the mass number is 4, which is the sum of protons and neutrons)

### Step 2: Calculate the total mass of the nucleons
The total mass of the nucleons (protons and neutrons) can be calculated using the given masses:
- Mass of one proton = 1.0073 u
- Mass of one neutron = 1.0087 u

The total mass of the nucleons is:
\[ \text{Total mass} = (2 \times \text{mass of proton}) + (2 \times \text{mass of neutron}) \]
\[ = (2 \times 1.0073 \, \text{u}) + (2 \times 1.0087 \, \text{u}) \]
\[ = 2.0146 \, \text{u} + 2.0174 \, \text{u} \]
\[ = 4.0320 \, \text{u} \]

### Step 3: Calculate the mass defect (Δm)
The mass defect is the difference between the total mass of the nucleons and the actual mass of the helium nucleus:
\[ \Delta m = \text{Total mass of nucleons} - \text{Mass of helium nucleus} \]
Given that the mass of the helium nucleus is 4.0015 u:
\[ \Delta m = 4.0320 \, \text{u} - 4.0015 \, \text{u} \]
\[ = 0.0305 \, \text{u} \]

### Step 4: Calculate the binding energy (BE)
The binding energy can be calculated using the formula:
\[ \text{Binding Energy} = \Delta m \times c^2 \]
Where \(c^2\) in terms of atomic mass units (u) to MeV is approximately 931.5 MeV/u. Thus:
\[ \text{Binding Energy} = 0.0305 \, \text{u} \times 931.5 \, \text{MeV/u} \]
\[ = 28.41 \, \text{MeV} \]

### Final Answer
The binding energy of helium-4 is approximately **28.4 MeV**.

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