Question

The binding energy per nucleon of \(^{209} \text{Bi}\)  is _______ MeV. \[ \text{Take } m(^{209} \text{Bi}) = 208.98038 \, \text{u}, \, m_p = 1.007825 \, \text{u}, \, m_n = 1.008665 \, \text{u}, \, 1 \, \text{u} = 931 \, \text{MeV}/c^2. \]

• A. 7.48
• B. 7.84
• C. 8.79
• D. 6.94

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons.
It is calculated by finding the mass defect (the difference between the sum of the masses of individual nucleons and the actual mass of the nucleus) and converting it to energy.
Finally, dividing by the total number of nucleons gives the binding energy per nucleon.
Step 2: Key Formula or Approach:
Number of protons \(Z = 83\), Number of neutrons \(N = A - Z = 209 - 83 = 126\).
Mass defect: \(\Delta m = Z m_p + N m_n - M_{nucleus}\).
Binding Energy: \(BE = \Delta m \times 931 \text{ MeV}\).
Binding Energy per nucleon: \(BE/A = BE / 209\).
Step 3: Detailed Explanation:
Calculate the total mass of the individual constituent nucleons:
Mass of 83 protons = \(83 \times 1.007825 \text{ u} = 83.649475 \text{ u}\).
Mass of 126 neutrons = \(126 \times 1.008665 \text{ u} = 127.091790 \text{ u}\).
Total constituent mass = \(83.649475 + 127.091790 = 210.741265 \text{ u}\).
Now, subtract the actual mass of the Bismuth nucleus to find the mass defect:
\[ \Delta m = 210.741265 \text{ u} - 208.980388 \text{ u} = 1.760877 \text{ u} \] Convert the mass defect into energy in MeV:
\[ BE = 1.760877 \text{ u} \times 931 \text{ MeV/u} = 1639.376487 \text{ MeV} \] Calculate the binding energy per nucleon by dividing by \(A = 209\):
\[ \frac{BE}{A} = \frac{1639.376487}{209} \approx 7.8439 \text{ MeV/nucleon} \] This rounds to \(7.84 \text{ MeV}\).
Step 4: Final Answer:
The binding energy per nucleon is \(7.84 \text{ MeV}\).