Calculate the binding energy of an `alpha`-particle in MeV Given : `m_(p)` (mass of proton) = 1.007825 amu, `m_(n)` (mass of neutron) = 1.008665 amu Mass of the nucleus `=4.002800 amu, 1 amu = 931 MeV.

Options

A. 28.097 eV
B. 28.097 MeV
C. 38.097 eV
D. 48.097 Mev

Answer

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

### Step 1: Understand the concept of mass defect
The binding energy of a nucleus can be calculated using the mass defect, which is the difference between the mass of the individual nucleons (protons and neutrons) and the mass of the nucleus itself. The formula for binding energy (BE) is given by:

\[ BE = \Delta m \times 931 \text{ MeV} \]

where \(\Delta m\) is the mass defect in atomic mass units (amu).

### Step 2: Identify the components for the alpha particle
An alpha particle is a helium nucleus, which consists of:
- Atomic number (Z) = 2 (2 protons)
- Mass number (A) = 4 (2 protons + 2 neutrons)

From the given data:
- Mass of proton (\(m_p\)) = 1.007825 amu
- Mass of neutron (\(m_n\)) = 1.008665 amu
- Mass of nucleus (alpha particle) = 4.002800 amu

### Step 3: Calculate the mass defect (\(\Delta m\))
The mass defect can be calculated using the formula:

\[ \Delta m = (Z \cdot m_p + (A - Z) \cdot m_n) - m_{\text{nucleus}} \]

Substituting the values:

\[ \Delta m = (2 \cdot 1.007825 + 2 \cdot 1.008665) - 4.002800 \]

Calculating the individual terms:

\[ \Delta m = (2.01565 + 2.01733) - 4.002800 \]

Adding the masses of the protons and neutrons:

\[ \Delta m = 4.03298 - 4.002800 \]

Calculating the mass defect:

\[ \Delta m = 0.03018 \text{ amu} \]

### Step 4: Calculate the binding energy (BE)
Now, substituting the mass defect into the binding energy formula:

\[ BE = 0.03018 \times 931 \text{ MeV} \]

Calculating the binding energy:

\[ BE \approx 28.097 \text{ MeV} \]

### Final Answer
The binding energy of the alpha particle is approximately **28.097 MeV**.

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