Calculte the binding energy per nucleon for `._(10)^(20)Ne,._(26)^(56)Fe` and `._(98)^(238)U`. Given that mass of neutron is `1.008665 am u`, mass of proton is `1.007825 am u`, mass of `._(10)^(20)Ne` is `19.9924 am u`, mass of `._(26)^(56)Fe` is `55.93492 am u` and mass of `._(92)^(238)U` is `238.050783 am u`.

Explanation

To calculate the binding energy per nucleon for the isotopes \( _{10}^{20}\text{Ne} \), \( _{26}^{56}\text{Fe} \), and \( _{92}^{238}\text{U} \), we will follow these steps:

### Step 1: Identify Given Data
- Mass of neutron (\( m_n \)) = 1.008665 amu
- Mass of proton (\( m_p \)) = 1.007825 amu
- Mass of \( _{10}^{20}\text{Ne} \) = 19.9924 amu
- Mass of \( _{26}^{56}\text{Fe} \) = 55.93492 amu
- Mass of \( _{92}^{238}\text{U} \) = 238.050783 amu
- \( 1 \text{ amu} = 931.5 \text{ MeV/c}^2 \)

### Step 2: Calculate Binding Energy for \( _{10}^{20}\text{Ne} \)
1. **Determine A and Z**:
- \( A = 20 \) (total nucleons)
- \( Z = 10 \) (number of protons)

2. **Calculate Mass Defect**:
\[ \text{Mass Defect} = (A - Z) \cdot m_n + Z \cdot m_p - \text{mass of } _{10}^{20}\text{Ne} \]
\[ = (20 - 10) \cdot 1.008665 + 10 \cdot 1.007825 - 19.9924 \]
\[ = 10 \cdot 1.008665 + 10.07825 - 19.9924 \]
\[ = 10.08665 + 10.07825 - 19.9924 = 0.1725 \text{ amu} \]

3. **Calculate Binding Energy**:
\[ \text{Binding Energy} = \text{Mass Defect} \cdot 931.5 \text{ MeV} \]
\[ = 0.1725 \cdot 931.5 \approx 160.68 \text{ MeV} \]

4. **Calculate Binding Energy per Nucleon**:
\[ \text{Binding Energy per Nucleon} = \frac{\text{Binding Energy}}{A} \]
\[ = \frac{160.68}{20} \approx 8.03 \text{ MeV/nucleon} \]

### Step 3: Calculate Binding Energy for \( _{26}^{56}\text{Fe} \)
1. **Determine A and Z**:
- \( A = 56 \)
- \( Z = 26 \)

2. **Calculate Mass Defect**:
\[ \text{Mass Defect} = (A - Z) \cdot m_n + Z \cdot m_p - \text{mass of } _{26}^{56}\text{Fe} \]
\[ = (56 - 26) \cdot 1.008665 + 26 \cdot 1.007825 - 55.93492 \]
\[ = 30 \cdot 1.008665 + 26.20345 - 55.93492 \]
\[ = 30.25995 + 26.20345 - 55.93492 = 0.52858 \text{ amu} \]

3. **Calculate Binding Energy**:
\[ \text{Binding Energy} = 0.52858 \cdot 931.5 \approx 492.0 \text{ MeV} \]

4. **Calculate Binding Energy per Nucleon**:
\[ \text{Binding Energy per Nucleon} = \frac{492.0}{56} \approx 8.79 \text{ MeV/nucleon} \]

### Step 4: Calculate Binding Energy for \( _{92}^{238}\text{U} \)
1. **Determine A and Z**:
- \( A = 238 \)
- \( Z = 92 \)

2. **Calculate Mass Defect**:
\[ \text{Mass Defect} = (A - Z) \cdot m_n + Z \cdot m_p - \text{mass of } _{92}^{238}\text{U} \]
\[ = (238 - 92) \cdot 1.008665 + 92 \cdot 1.007825 - 238.050783 \]
\[ = 146 \cdot 1.008665 + 92.7199 - 238.050783 \]
\[ = 147.27069 + 92.7199 - 238.050783 = 1.939807 \text{ amu} \]

3. **Calculate Binding Energy**:
\[ \text{Binding Energy} = 1.939807 \cdot 931.5 \approx 1802.0 \text{ MeV} \]

4. **Calculate Binding Energy per Nucleon**:
\[ \text{Binding Energy per Nucleon} = \frac{1802.0}{238} \approx 7.57 \text{ MeV/nucleon} \]

### Final Results
- Binding Energy per Nucleon for \( _{10}^{20}\text{Ne} \): **8.03 MeV/nucleon**
- Binding Energy per Nucleon for \( _{26}^{56}\text{Fe} \): **8.79 MeV/nucleon**
- Binding Energy per Nucleon for \( _{92}^{238}\text{U} \): **7.57 MeV/nucleon**