Binding Energy Per Nucleon and Nuclear Reactions

Based on the provided plot of binding energy per nucleon $\left(\frac{B}{A}\right)$ versus mass number ($A$), the values are:

• For $0 < A \le 100$: $\frac{B}{A} = 2\text{ MeV}$
• For $100 < A \le 200$: $\frac{B}{A} = 8\text{ MeV}$
• For $A > 200$: $\frac{B}{A} = 4\text{ MeV}$

A nuclear reaction (fusion or fission) releases energy ($Q > 0$) if the total binding energy of the product nuclei is greater than the total binding energy of the reactant nuclei, which means the average binding energy per nucleon increases as a result of the process:
$Q = \text{Total } B_{\text{products}} - \text{Total } B_{\text{reactants}}$

Let us evaluate each option:

1. Option (A):

   • Consider two nuclei with mass numbers $A_1$ and $A_2$ such that $1 < A_1, A_2 < 50$.
   • Their combined mass number after fusion is $A = A_1 + A_2$, where $2 < A < 100$.
   • For all these nuclei, $A < 100$, so the binding energy per nucleon is constant at $2\text{ MeV}$.
   • The initial binding energy is:
     $B_i = 2A_1 + 2A_2 = 2(A_1 + A_2)$
   • The final binding energy is:
     $B_f = 2A = 2(A_1 + A_2)$
   • The energy released is:
     $Q = B_f - B_i = 0$
   • Thus, no energy is released. Hence, Option (A) is incorrect.

2. Option (B):

   • Consider two nuclei with mass numbers $A_1$ and $A_2$ lying in the range $51 < A < 100$.
   • The mass number of the fused nucleus is $A = A_1 + A_2$, which falls in the range:
     $102 < A < 200$
   • For the reactants ($A_1, A_2 < 100$), $\frac{B}{A} = 2\text{ MeV}$.
   • For the product ($100 < A < 200$), $\frac{B}{A} = 8\text{ MeV}$.
   • The energy released is:
     $Q = 8(A_1 + A_2) - [2A_1 + 2A_2] = 6(A_1 + A_2) > 0$
   • Since $Q > 0$, energy will be released. Hence, Option (B) is correct.

3. Option (C):

   • A nucleus with mass number $A \in (100, 200)$ has $\frac{B}{A} = 8\text{ MeV}$.
   • When it undergoes fission into two equal fragments, each fragment has a mass number $A' = \frac{A}{2} \in (50, 100)$, where $\frac{B}{A} = 2\text{ MeV}$.
   • The initial binding energy is:
     $B_i = 8A$
   • The final binding energy is:
     $B_f = 2\left(\frac{A}{2}\right) \times 2 = 2A$
   • The energy change is:
     $Q = B_f - B_i = 2A - 8A = -6A < 0$
   • Energy is absorbed rather than released. Hence, Option (C) is incorrect.

4. Option (D):

   • A nucleus with mass number $A \in (200, 260)$ has $\frac{B}{A} = 4\text{ MeV}$.
   • When it breaks into two equal fragments, each fragment has a mass number $A' = \frac{A}{2} \in (100, 130)$, where $\frac{B}{A} = 8\text{ MeV}$.
   • The initial binding energy is:
     $B_i = 4A$
   • The final binding energy is:
     $B_f = 2\left(\frac{A}{2}\right) \times 8 = 8A$
   • The energy released is:
     $Q = B_f - B_i = 8A - 4A = 4A > 0$
   • Since $Q > 0$, energy will be released. Hence, Option (D) is correct.

Therefore, the correct choices are B and D.