Question

The mass of a proton is $1.0073\text{u}$ and that of a neutron is $1.0087\text{u}$. The binding energy of $_2^4\text{He}$ is approximately (Given: helium nucleus mass = $4.0015\text{u}$)}

• A. $24.8 \text{ MeV}$
• B. $24.4 \text{ MeV}$
• C. $2.48 \text{ MeV}$
• D. $2.84 \text{ MeV}$
• E. $28.4 \text{ MeV}$

Hint:

Mass defect is always positive for a stable nucleus. In calculations, using $1 \text{ u} = 931 \text{ MeV}$ is usually sufficient for competitive exams.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Binding energy is the energy required to split a nucleus into its constituent nucleons. It is equivalent to the mass defect ($\Delta m$) of the nucleus.
Key Formula or Approach:
1. Mass defect: \( \Delta m = [Z m_p + (A-Z) m_n] - M_{nucleus} \).
2. Binding Energy: \( BE = \Delta m \times 931.5 \text{ MeV/u} \).

Step 2: Detailed Explanation:
For helium nucleus $_2^4\text{He}$: $Z = 2$ (protons), $A-Z = 2$ (neutrons).
1. Calculate expected mass of nucleons:
Mass of protons $= 2 \times 1.0073 = 2.0146 \text{ u}$.
Mass of neutrons $= 2 \times 1.0087 = 2.0174 \text{ u}$.
Total $= 2.0146 + 2.0174 = 4.0320 \text{ u}$.
2. Calculate mass defect ($\Delta m$):
\( \Delta m = 4.0320 - 4.0015 = 0.0305 \text{ u} \).
3. Calculate binding energy:
\( BE = 0.0305 \times 931.5 \approx 28.41 \text{ MeV} \).

Step 3: Final Answer:
The binding energy is approximately $28.4 \text{ MeV}$.