Calculate the binding energy in MeV of Uranium 238 from the following data
Mass of `.^(1)H_(1) = 1.008142 am u`, Mass of `._(0)n^(1) = 1.008982 am u`
Mass of `._(92)U^(238) = 238.124930 am u`
Also calculate the packing fraction.

Answer

To calculate the binding energy of Uranium-238 (U-238) and its packing fraction, we will follow these steps:

### Step 1: Determine the number of neutrons in U-238
The number of neutrons (N) in U-238 can be calculated using the formula:
\[ N = A - Z \]
where:
- \( A \) is the mass number (238 for U-238)
- \( Z \) is the atomic number (92 for U-238)

Calculating:
\[ N = 238 - 92 = 146 \]

### Step 2: Calculate the total mass of protons and neutrons
The total mass of protons and neutrons can be calculated using:
\[ \text{Total mass} = Z \cdot M_p + N \cdot M_n \]
where:
- \( M_p \) is the mass of a proton (1.008142 amu)
- \( M_n \) is the mass of a neutron (1.008982 amu)

Calculating:
\[ \text{Total mass} = 92 \cdot 1.008142 + 146 \cdot 1.008982 \]
\[ = 92.746064 + 147.313012 = 240.059076 \text{ amu} \]

### Step 3: Calculate the mass defect
The mass defect (\( \Delta m \)) is the difference between the total mass of the nucleons and the actual mass of the nucleus:
\[ \Delta m = \text{Total mass} - \text{Mass of U-238} \]
Given the mass of U-238 is 238.124930 amu:
\[ \Delta m = 240.059076 - 238.124930 = 1.934146 \text{ amu} \]

### Step 4: Convert the mass defect to energy
Using Einstein's equation \( E = \Delta m \cdot c^2 \) and the conversion factor \( 1 \text{ amu} = 931.5 \text{ MeV} \):
\[ E = \Delta m \cdot 931.5 \text{ MeV} \]
Calculating:
\[ E = 1.934146 \cdot 931.5 \]
\[ = 1808.054 \text{ MeV} \]

### Step 5: Calculate the binding energy per nucleon
The binding energy per nucleon can be calculated by dividing the total binding energy by the mass number \( A \):
\[ \text{Binding energy per nucleon} = \frac{E}{A} \]
Calculating:
\[ \text{Binding energy per nucleon} = \frac{1808.054}{238} \approx 7.5714 \text{ MeV} \]

### Step 6: Calculate the packing fraction
The packing fraction (PF) is defined as:
\[ \text{Packing fraction} = \frac{\Delta m}{A} \]
Calculating:
\[ \text{Packing fraction} = \frac{1.934146}{238} \approx 0.00812 \text{ amu} \]

### Summary of Results
- **Total Binding Energy**: 1808.054 MeV
- **Binding Energy per Nucleon**: 7.5714 MeV
- **Packing Fraction**: 0.00812 amu