Problem 61 - Master Resource Book in JEE Main Physics

Question

The binding energy of two nuclei \(p^{n}\) and \(Q^{2 n}\) are \(x\) joule and \(y\) joule respectively. If \(2 x>y\), then the energy released in the reaction \(p^{n}+p^{n}=Q^{2 n}\) will be (a) \(2 x+y\) (b) \(2 x-y\) (c) \(x y\) (d) \(x+y\)

Step-by-Step Solution

Answer

The energy released is given by \(2x - y\).

Step 1: Understanding the Reaction

The reaction given is \(p^n + p^n \rightarrow Q^{2n}\). This means two nuclei \(p^n\) combine to form one nucleus \(Q^{2n}\). We need to calculate the energy released during this process.

Step 2: Determine Initial Energy

The binding energy of each \(p^n\) nucleus is \(x\) joules. Since there are two such nuclei, the total initial energy is \(2x\) joules.

Step 3: Determine Final Energy

The binding energy of \(Q^{2n}\) is given as \(y\) joules. Thus, the energy of the system after the reaction is \(y\) joules.

Step 4: Calculate Energy Released

The energy released in the reaction can be calculated as the difference in energy before and after the reaction. This is given by \(\text{Initial Energy} - \text{Final Energy} = 2x - y\).

Step 5: Verify Given Condition

We know from the condition \(2x > y\), so the energy released \(2x - y\) must be a positive value, meaning energy is indeed released and not absorbed.

Key Concepts

Chemical Reactions

Chemical reactions entail the rearrangement of atoms or nuclei to form new products. In nuclear reactions, like the one given in the exercise, nuclei of atoms are involved rather than the atoms themselves. Here, two nuclei labeled as \(p^n\) merge to form a larger nucleus \(Q^{2n}\).
This transformation requires us to understand the concept of binding energy, which is the energy that holds the nucleus together. When a reaction occurs, the difference in the binding energies before and after the reaction will determine if energy is released or absorbed.

• If the final nucleus has a higher total binding energy compared to the initial nuclei, energy is released.
• If the final binding energy is less, energy must be added for the reaction to proceed.

The given condition \(2x > y\) signifies that the total binding energy of the initial reaction components exceeds that of the final product, suggesting an energy release as per the final solution of \(2x - y\).

Energy Conservation

Energy conservation is a fundamental principle of physics, asserting that energy cannot be created or destroyed, only transformed from one form to another. In the context of nuclear reactions, this principle helps us understand how energy is transferred or released.
When two nuclei combine, the total energy before the reaction must equal the total energy afterward when accounting for any energy released. In the exercise, this is expressed by calculating \(2x - y\), where:

• \(2x\): Total initial energy from two nuclei \(p^n\).
• \(y\): Energy contained in the final nucleus \(Q^{2n}\).

By ensuring \(2x > y\), we validate that the energy appears to "disappear" from the system but is actually released as kinetic energy, usually as radiation or motion of the products. This demonstrates energy conservation through the visible release of energy.

Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. In the context of the exercise, thermodynamics helps explain the energy changes involved in nuclear binding and release.
During the reaction \(p^n + p^n \rightarrow Q^{2n}\), the concept of thermodynamics specifically applies to how energy transfers from being stored as "binding energy" to being released. This transformation is interpreted by following key thermodynamic principles:

• The First Law of Thermodynamics aligns with energy conservation, asserting that the total energy within the closed system must remain constant.
• The sign of the energy change \(2x - y\) indicates whether the reaction is exothermic (releases energy) or endothermic (absorbs energy).

In the exercise, since \(2x - y\) yields a positive value under the given conditions, the reaction is exothermic, releasing energy, which corresponds to observable phenomena such as heat or emitted light.