Problem 58 - Master Resource Book in JEE Main Physics

Question

The binding energies per nucleon of \(\mathrm{Li}^{7}\) and \(\mathrm{He}^{4}\) are \(5.6 \mathrm{MeV}\) and \(7.06 \mathrm{MeV}\) respectively, then the energy of the reaction \(\mathrm{Li}^{7}+p=2\left[{ }_{2} \mathrm{He}^{4}\right]\) will be (a) \(17.28 \mathrm{MeV}\) (b) \(39.2 \mathrm{MeV}\) (c) \(28.24 \mathrm{MeV}\) (d) \(1.46 \mathrm{MeV}\)

Step-by-Step Solution

Answer

The energy of the reaction is 17.28 MeV (Option a).

Step 1: Understand the Reaction

The nuclear reaction given is \(^7\text{Li} + p \rightarrow 2 \cdot ({}^4\text{He})\). This means a lithium-7 nucleus and a proton combine to create two helium-4 nuclei.

Step 2: Calculate Binding Energy of Reactants

The binding energy of \(^7\text{Li}\) is given as 5.6 MeV per nucleon. Since lithium-7 has 7 nucleons, the total binding energy is \(7 \times 5.6 \text{ MeV} = 39.2 \text{ MeV}\). For a proton, being a single nucleon, the binding energy contribution is 0 MeV.

Step 3: Calculate Binding Energy of Products

Helium-4 has a binding energy of 7.06 MeV per nucleon and has 4 nucleons, giving a total binding energy per helium nucleus of \(4 \times 7.06 \text{ MeV} = 28.24 \text{ MeV}\). Since there are 2 helium nuclei, the total binding energy is \(2 \times 28.24 \text{ MeV} = 56.48 \text{ MeV}\).

Step 4: Determine Energy Release

The energy released in the reaction is the difference between the total binding energy of the products and the reactants. Therefore, the energy released \(= 56.48 \text{ MeV} - 39.2 \text{ MeV} = 17.28 \text{ MeV}\).

Step 5: Confirm the Correct Answer

Comparing the calculated energy release with the options given, the correct answer is (a) 17.28 MeV.

Key Concepts

Binding Energy per Nucleon

In the world of nuclear physics, understanding the concept of binding energy per nucleon is crucial. This measurement indicates the energy required to remove a nucleon from a nucleus. It essentially explains how tightly a nucleon is bound to the rest of the nucleus. A higher binding energy per nucleon generally implies a more stable nucleus.

Binding energy per nucleon is calculated by taking the total binding energy of a nucleus and dividing it by the number of nucleons it contains. For example, if we take lithium-7 (\( \mathrm{Li}^7 \)) with a binding energy of 5.6 MeV per nucleon, and helium-4 (\( \mathrm{He}^4 \)) with 7.06 MeV, we see that helium is more tightly bound.

• This information helps predict how much energy will be released or absorbed in nuclear reactions.
• An increase in binding energy during a reaction generally means energy is released.

The knowledge of binding energy is essential in understanding why certain nuclear reactions occur spontaneously, while others require a significant input of energy.

Energy Calculation in Nuclear Reactions

Energy calculations in nuclear reactions determine the feasibility and energy yield of the reaction. The process involves determining the difference between the total binding energy of the reactants and the products involved in the reaction.

Consider a reaction involving lithium-7 and a proton producing two helium-4 nuclei. The reactant \( \mathrm{Li}^{7} + p \) has a calculated total binding energy of 39.2 MeV (from lithium). Protons, being free nucleons, contribute no binding energy. The products, two \( \mathrm{He}^{4} \) nuclei, have a combined binding energy of 56.48 MeV.
Thus, the energy released can be calculated as:
\[ \text{Energy Released} = \text{Binding Energy of Products} - \text{Binding Energy of Reactants} \]In this example:

• Binding energy of products (2 helium nuclei): 56.48 MeV
• Binding energy of reactants: 39.2 MeV
• Energy released: 17.28 MeV

Such calculations are vital for predicting the results and designing processes for both energy generation and nuclear waste management.

Nuclear Fusion

Nuclear fusion is a process where two light atomic nuclei combine to form a heavier nucleus, releasing significant energy in the process. This is the same process that powers the sun and other stars, making it a potential candidate for clean energy on Earth.

In the context of the reaction involving lithium and a proton to form helium, fusion is at play. This process combines smaller elements to form a larger element, releasing energy due to the rearrangement and binding of nucleons.

• Unlike nuclear fission, which involves splitting a heavy nucleus, fusion combines lighter ones.
• Energy from fusion arises because the binding energy per nucleon in the resulting nucleus is often higher than that of the original nuclei.
• This results in a reduction in total mass (as per Einstein's \( E=mc^2 \)) and a corresponding release of energy.

Nuclear fusion holds the promise of producing large amounts of energy with minimal radioactive waste and virtually an unlimited fuel supply. However, achieving the conditions required for commercially viable fusion power remains a significant scientific and technological challenge.