Question

The atomic mass of an element $_{10}\text{X}^{20}$ is $198170\text{ amu}$. The binding energy per nucleon of that element is: (Given mass of neutron $= 00867\text{ amu}$, mass of proton $= 00783\text{ amu}$, and $1\text{ amu} = 931\text{ MeV}$)}

• A. $8.533\text{ MeV/nucleon}$
• B. $85.33\text{ MeV/nucleon}$
• C. $170.66\text{ MeV/nucleon}$
• D. $17.66\text{ MeV/nucleon}$

Hint:

Be careful not to select Option (C) by mistake! $170.66\text{ MeV}$ represents the total binding energy of the entire system. The question explicitly asks for the value per nucleon, requiring a division by the total atomic mass number of $20$.

Answer 1

The Correct Option is A

Understanding the Concept: The stable configuration of any atomic nucleus results in a total mass that is slightly less than the sum of its individual isolated constituents. This missing quantity is known as the mass defect ($\Delta m$). The binding energy ($E_b$) corresponding to this mass defect is calculated using Einstein's equivalence relation, and the binding energy per nucleon is found by dividing by the total mass number ($A$): \[ \Delta m = [Z \cdot m_p + (A - Z) \cdot m_n] - m_{\text{nucleus}} \] \[ \text{Binding Energy per Nucleon} = \frac{E_b}{A} = \frac{\Delta m \times 931\text{ MeV}}{A} \]
Step 1: Calculate the mass defect ($\Delta m$).
For the element $_{10}\text{X}^{20}$:
Number of protons ($Z$) $= 10$
Number of neutrons ($A - Z$) $= 20 - 10 = 10$
Let's find the combined constituent mass: \[ \text{Mass of protons} = 10 \times 1.00783 = 10.0783\text{ amu} \] \[ \text{Mass of neutrons} = 10 \times 1.00867 = 10.0867\text{ amu} \] \[ \text{Total individual mass} = 10.0783 + 10.0867 = 20.1650\text{ amu} \] Subtracting the actual nuclear mass value to find $\Delta m$: \[ \Delta m = 20.1650 - 19.98170 = 0.1833\text{ amu} \]
Step 2: Determine the total binding energy and divide by the nucleon count.
Convert mass defect into total energy: \[ E_b = 0.1833 \times 931\text{ MeV} \approx 170.6523\text{ MeV} \] Dividing by the total mass number $A = 20$: \[ \text{Binding Energy per Nucleon} = \frac{170.6523}{20} \approx 8.5326\text{ MeV/nucleon} \approx 8.533\text{ MeV/nucleon} \]