For `""_(92)^(238)U` the binding energy per nucleon is 7.576 MeV. What is the atomic weight of this isotope? Use the mass of neutron and proton

Answer

To find the atomic weight of the isotope \( _{92}^{238}U \), we can follow these steps:

### Step 1: Identify the number of protons and neutrons
The atomic number (Z) of uranium (U) is 92, which means it has 92 protons. The mass number (A) is 238, which is the total number of protons and neutrons in the nucleus.

To find the number of neutrons (N), we can use the formula:
\[ N = A - Z \]
Substituting the values:
\[ N = 238 - 92 = 146 \]

### Step 2: Calculate the mass defect
The mass defect (\( \Delta m \)) can be calculated using the formula:
\[ \Delta m = (Z \cdot m_p + N \cdot m_n) - A \]
Where:
- \( m_p \) is the mass of a proton (approximately 1.007276 amu)
- \( m_n \) is the mass of a neutron (approximately 1.008665 amu)

Substituting the values:
\[ \Delta m = (92 \cdot 1.007276 + 146 \cdot 1.008665) - 238 \]
Calculating the mass of protons and neutrons:
\[ \Delta m = (92 \cdot 1.007276) + (146 \cdot 1.008665) - 238 \]
\[ \Delta m = 92.669392 + 147.26579 - 238 \]
\[ \Delta m = 239.935182 - 238 = 1.935182 \text{ amu} \]

### Step 3: Calculate the binding energy
The binding energy (BE) can be calculated using the mass defect and the binding energy per nucleon:
\[ BE = \Delta m \cdot 931.5 \text{ MeV/amu} \]
Substituting the mass defect:
\[ BE = 1.935182 \cdot 931.5 \]
Calculating the binding energy:
\[ BE \approx 1806.9 \text{ MeV} \]

### Step 4: Calculate the atomic weight
The atomic weight of the isotope can be approximated as:
\[ \text{Atomic weight} \approx A - \Delta m \]
Substituting the values:
\[ \text{Atomic weight} \approx 238 - 1.935182 \approx 236.064818 \]

Thus, the atomic weight of the isotope \( _{92}^{238}U \) is approximately **236.06 amu**.