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

Answer

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

### Step 1: Identify the Composition of Helium-4
Helium-4 (²He⁴) consists of:
- 2 protons
- 2 neutrons

### Step 2: Calculate the Mass of Protons
The mass of one proton is given as \(1.0073 \, u\). Therefore, the total mass of 2 protons is:
\[ \text{Mass of protons} = 2 \times 1.0073 \, u = 2.0146 \, u \]

### Step 3: Calculate the Mass of Neutrons
The mass of one neutron is given as \(1.0087 \, u\). Therefore, the total mass of 2 neutrons is:
\[ \text{Mass of neutrons} = 2 \times 1.0087 \, u = 2.0174 \, u \]

### Step 4: Calculate the Total Mass of Protons and Neutrons
Now, we can find the total mass of the helium nucleus (protons + neutrons):
\[ \text{Total mass} = \text{Mass of protons} + \text{Mass of neutrons} = 2.0146 \, u + 2.0174 \, u = 4.032 \, u \]

### Step 5: 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:
\[ \Delta m = \text{Total mass} - \text{Mass of helium nucleus} = 4.032 \, u - 4.0015 \, u = 0.0305 \, u \]

### Step 6: Convert Mass Defect to Energy
To find the binding energy, we use the formula:
\[ \text{Binding Energy} = \Delta m \times c^2 \]
Where \(c^2\) in terms of atomic mass units can be converted to MeV using the conversion factor \(931 \, \text{MeV/u}\):
\[ \text{Binding Energy} = 0.0305 \, u \times 931 \, \text{MeV/u} = 28.4 \, \text{MeV} \]

### Final Answer
The binding energy of the helium-4 nucleus is approximately \(28.4 \, \text{MeV}\).

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