Question

Nuclei \(A\) and \(B\) form a nucleus \(C\). The binding energy per nucleon (BE/N) for \(A\), \(B\), and \(C\) are \(3\,\text{MeV}\), \(7\,\text{MeV}\), and \(6\,\text{MeV}\) respectively. Find the energy produced in the reaction: \[ 2A^3 + B^4 \rightarrow C^{10} \]

• A. \(8\,\text{MeV}\)
• B. \(12\,\text{MeV}\)
• C. \(14\,\text{MeV}\)
• D. \(10\,\text{MeV}\)

Hint:

Energy released in nuclear reactions is determined from the \textbf{change in total binding energy}. If the final nucleus has higher binding energy, the excess energy is released.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the energy released (Q-value) in a nuclear reaction.
Energy produced = (Total Binding Energy of products) - (Total Binding Energy of reactants).
Step 2: Key Formula or Approach:
Binding Energy (\(BE\)) = (Binding Energy per Nucleon, \(BE/N\)) \(\times\) (Total Nucleons, \(N\)).
Step 3: Detailed Explanation:
Reactants: Two nuclei of \(A\) (mass number 3) and one nucleus of \(B\) (mass number 4).
- \(BE\) of \(2\) nuclei of \(A = 2 \times (3 \times 3\text{ MeV}) = 18\text{ MeV}\).
- \(BE\) of \(1\) nucleus of \(B = 1 \times (4 \times 7\text{ MeV}) = 28\text{ MeV}\).
Total initial Binding Energy = \(18 + 28 = 46\text{ MeV}\).
Product: One nucleus of \(C\) (mass number 10).
- \(BE\) of nucleus \(C = 10 \times 6\text{ MeV} = 60\text{ MeV}\).
Energy produced (\(Q\)):
\[ Q = BE(\text{Product}) - BE(\text{Reactants}) \]
\[ Q = 60\text{ MeV} - 46\text{ MeV} = 14\text{ MeV} \]
Step 4: Final Answer:
The energy produced in the reaction is \(14\text{ MeV}\).