General Principle

A nuclear reaction (fusion or fission) releases energy if the products are more stable than the reactants. This corresponds to an increase in the total binding energy. On the binding energy per nucleon (B/A) versus mass number (A) curve, energy is released when the reaction products have a higher average B/A than the reactants. The peak of the curve is at A ≈ 56 (iron), which represents the most stable nuclei.

• Fusion: Light nuclei combine to form a heavier nucleus. If the product nucleus is closer to the peak (A≈56) than the reactant nuclei, energy is released. This is generally true for the fusion of very light nuclei (on the rising part of the curve).
• Fission: A heavy nucleus splits into lighter nuclei. If the product nuclei (fragments) are closer to the peak (A≈56) than the parent nucleus, energy is released. This is generally true for the fission of very heavy nuclei (on the falling part of the curve).

Let's analyze each option based on this principle and the given graph.

Step-by-step Analysis of Options

A: Fusion of two nuclei with mass number lying in the range of 1 < A < 50 will release energy.

1. This range is on the rising part of the B/A curve. Fusing two light nuclei generally produces a heavier nucleus with a higher B/A, thus releasing energy. For example, fusing two deuterons (A=2) to form a helium nucleus (A=4) releases a significant amount of energy as we move steeply up the curve.
2. However, the statement says this will happen, implying it's true for any two nuclei in this range. Let's consider a case at the upper end of the range. Let's fuse two nuclei with A = 49. The product nucleus will have A = 98.
3. The reactant nuclei (A=49) are very close to the peak of stability (A≈56). The product nucleus (A=98) is significantly past the peak, on the downward sloping part of the curve.
4. It is possible, and indeed the case, that the B/A for A=49 is higher than the B/A for A=98. Let's denote the binding energy per nucleon for a nucleus of mass number A as $(B/A)_A$. The energy released is $Q = 98*(B/A)_98 - 2*49*(B/A)_49 = 98*((B/A)_98 - (B/A)_49)$. Since $(B/A)_49 > (B/A)_98$, Q would be negative, meaning energy is absorbed.
5. Therefore, the statement is not universally true for the entire range. Option A is incorrect.

B: Fusion of two nuclei with mass numbers lying in the range of 51 < A < 100 will release energy.

1. The reactants have mass numbers in the range 51 < A < 100. This region contains the peak of the curve and the initial part of the downward slope.
2. The product nucleus will have a mass number $A_product = A_1 + A_2$, which will lie in the range 102 < A < 200. This is further down the downward sloping part of the curve.
3. The reactant nuclei have a higher B/A than the product nucleus. For example, fusing two nuclei of A=60 gives a product with A=120. From the graph, $(B/A)_60 > (B/A)_120$.
4. This means the products are less stable than the reactants. The reaction will absorb energy, not release it.
5. Therefore, Option B is incorrect.

C: Fission of a nucleus lying in the mass range of 100 < A < 200 will release energy when broken into two equal fragments.

1. The parent nucleus has a mass number A in the range (100, 200). The fragments each have a mass number A/2, which is in the range (50, 100).
2. The region (50, 100) for the fragments contains the peak of the B/A curve (A≈56). The region (100, 200) for the parent nucleus is entirely on the downward slope, to the right of the peak.
3. This means the fragments (A/2) are always closer to the peak of stability than the parent nucleus (A). Consequently, the fragments will always have a higher B/A than the parent nucleus: $(B/A)_(A/2) > (B/A)_A$.
4. The energy released is $Q = A * ((B/A)_(A/2) - (B/A)_A)$. Since $(B/A)_(A/2) > (B/A)_A$, Q will be positive. Energy will be released.
5. Therefore, Option C is correct.

D: Fission of a nucleus lying in the mass range of 200 < A < 260 will release energy when broken into two equal fragments.

1. The parent nucleus has a mass number A in the range (200, 260). The fragments each have a mass number A/2, which is in the range (100, 130).
2. In the entire region for A > 60, the B/A curve is monotonically decreasing.
3. The parent nucleus (A) and the fragment nuclei (A/2) are both on this downward sloping part of the curve.
4. Since the fragments have a smaller mass number (A/2 < A), they will be higher up on the curve, meaning they have a higher B/A value: $(B/A)_(A/2) > (B/A)_A$.
5. The energy released $Q = A * ((B/A)_(A/2) - (B/A)_A)$ will be positive.
6. Therefore, Option D is correct.

Conclusion

Based on the analysis of the B/A curve, options C and D are the correct choices. The stored answer indicates B and D are correct. My analysis shows that option B is physically incorrect, as fusion of nuclei past the stability peak absorbs energy. Option C is correct because the fission fragments are closer to the peak of stability than the parent nucleus. There seems to be an error in the stored answer. My derived correct options are C and D.