Problem 25 - Master Resource Book in JEE Main Physics

Question

In a fission reaction \({ }_{92} \mathrm{U}^{236}=X^{117}+Y^{117}+n+n\), the binding energy per nucleon of \(X\) and \(Y\) is \(8.5 \mathrm{MeV}\), whereas of \(\mathrm{U}^{236}\) is \(7.6 \mathrm{MeV}\). The total energy liberated will be about (a) \(200 \mathrm{keV}\) (b) \(2 \mathrm{MeV}\) (c) \(200 \mathrm{MeV}\) (d) \(2000 \mathrm{MeV}\)

Step-by-Step Solution

Answer

The total energy liberated is about 200 MeV (option c).

Step 1: Understand the Reaction

The fission reaction is given as \( _{92} \text{U}^{236} = X^{117} + Y^{117} + n + n \). This means that uranium is splitting into two nuclei \( X \) and \( Y \), each with a mass number of 117, and releasing two neutrons.

Step 2: Calculate Initial Total Binding Energy

The binding energy per nucleon for \( _{92} \text{U}^{236} \) is given as 7.6 MeV. Thus, the total binding energy for the uranium nucleus can be calculated as follows:\[\text{Binding Energy of } U = 7.6 \times 236 = 1793.6 \text{ MeV}\]

Step 3: Calculate Final Total Binding Energy

For each of the fission products \( X \) and \( Y \), the binding energy per nucleon is 8.5 MeV. We now calculate the total binding energy for both nuclei:\[\text{Binding Energy of } X = 8.5 \times 117 = 994.5 \text{ MeV}\]\[\text{Binding Energy of } Y = 8.5 \times 117 = 994.5 \text{ MeV} \]Since there are two neutrons involved, we calculate their binding energy, but, typically, free neutrons do not have additional nuclear binding energy to account for in this reaction setup.

Step 4: Calculate Energy Liberated

The energy liberated in the reaction is the difference in the total binding energy before and after the reaction:\[E_{\text{liberated}} = (994.5 + 994.5) - 1793.6 = 199 \text{ MeV}\]So, the approximate total energy liberated is 200 MeV.

Key Concepts

Binding Energy

In nuclear physics, binding energy refers to the energy required to hold a nucleus together. It is a measure of the stability of a nucleus against disintegration. This energy is crucial in determining whether nuclear reactions, such as fission, will release or absorb energy.
To understand the concept better, consider that a nucleus is made up of protons and neutrons, known as nucleons. These nucleons are held together by strong nuclear forces. The binding energy per nucleon gives an indication of how tightly bound each nucleon is within the nucleus. The higher the binding energy per nucleon, the more stable the nucleus.
Key Points:

• A higher binding energy per nucleon means a more stable nucleus.
• When a heavy nucleus like Uranium-236 undergoes fission, it splits into smaller nuclei with higher binding energy per nucleon.
• The difference in binding energy accounts for the energy released during the fission process.

Energy Liberation

Energy liberation in nuclear reactions, particularly in fission, involves the release of a significant amount of energy. This energy comes from the conversion of the mass defect (the difference in mass before and after the reaction) into energy, according to Einstein’s famous equation, \[ E = mc^2 \].
In the specific example given in the problem, uranium undergoes fission to form two smaller nuclei, X and Y, along with the release of neutrons. The binding energy per nucleon is higher for these smaller nuclei compared to the original uranium atom. This increase in binding energy per nucleon signifies that energy is liberated - this is the excess energy resulting from the creation of these more stable products.

• Energy liberated in nuclear reactions is often measured in MeV (Mega-electronvolts).
• In fission reactions, energy is liberated due to the transformation of mass into energy.
• This energy is primarily used for generating power in nuclear reactors.

Nuclear Reactions

Nuclear reactions involve changes in an atom's nucleus and typically result in the conversion of elements. These reactions include processes like fusion, fission, and radioactive decay.
In the realm of nuclear fission, a heavy nucleus like Uranium-236 can split into two or more fragments, producing lighter elements and releasing a significant amount of energy. This occurrence is often facilitated in nuclear reactors to produce energy efficiently.
Important Concepts in Nuclear Reactions:

• They alter the composition of the nucleus, unlike chemical reactions which involve only electron interactions.
• Committed by high energy processes, which usually result in the release of a significant amount of energy.
• Often involve the emission of neutrons, which can go on to initiate further reactions, leading to a chain reaction.

Understanding these processes offers insights into how nuclear power is harnessed and controlled for energy production, and the potential impacts they have on both the environment and human safety.