Calculate the packing fraction of `alpha`-particle from the following data :
Mass of helium nucleus =4.0028 amu
Mass of free proton =1.00758 amu
Mass of free neutron =1.00897 amu

Explanation

To calculate the packing fraction of an alpha particle, we will follow these steps:

### Step 1: Identify the Components of the Alpha Particle
An alpha particle is the nucleus of helium, which consists of:
- 2 protons
- 2 neutrons

### Step 2: Calculate the Mass of Reactants
The mass of the reactants (the individual protons and neutrons) can be calculated as follows:
- Mass of 2 protons = 2 × mass of proton = 2 × 1.00758 amu
- Mass of 2 neutrons = 2 × mass of neutron = 2 × 1.00897 amu

Calculating these values:
- Mass of 2 protons = 2 × 1.00758 = 2.01516 amu
- Mass of 2 neutrons = 2 × 1.00897 = 2.01794 amu

Total mass of reactants = Mass of 2 protons + Mass of 2 neutrons
Total mass of reactants = 2.01516 amu + 2.01794 amu = 4.03310 amu

### Step 3: Calculate the Mass Defect
The mass defect (Δm) is the difference between the total mass of the reactants and the mass of the product (the alpha particle):
- Mass of helium nucleus (alpha particle) = 4.0028 amu

Now, calculate the mass defect:
Δm = Total mass of reactants - Mass of helium nucleus
Δm = 4.03310 amu - 4.0028 amu = 0.0303 amu

### Step 4: Calculate the Packing Fraction
The packing fraction (PF) is given by the formula:
\[ \text{Packing Fraction} = \frac{\Delta m}{A} \]
where \( A \) is the mass number (number of nucleons).

For an alpha particle, \( A = 4 \):
\[ \text{Packing Fraction} = \frac{0.0303 \text{ amu}}{4} = 0.007575 \text{ amu per nucleon} \]

### Final Result
The packing fraction of the alpha particle is approximately:
\[ \text{Packing Fraction} \approx 0.00758 \text{ amu per nucleon} \]

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