Question

In the nuclear reaction: \({ }_{3} \mathrm{Li}^{7}+{ }_{1} \mathrm{H}^{1} \longrightarrow 2{ }_{2} \mathrm{He}^{4}\) the mass loss is nearly \(0.02 \mathrm{amu}\). Hence, the energy released (in units of million \(\mathrm{kcal} / \mathrm{mol}\) ) in the process is approximately (a) 100 (b) 200 (c) 400 (d) 600

Short answer

(b) 200

Step-by-step solution

Understand the problem

The nuclear reaction given in the problem involves the fusion of lithium and hydrogen nuclei to form helium. We need to calculate the energy released due to the mass loss in this reaction.

Identify the mass loss

The problem states that there is a mass loss of 0.02 amu in the reaction. This mass loss is converted into energy, according to Einstein's mass-energy equivalence principle.

Apply Einstein's Mass-Energy Equivalence

Use the formula \(E = \Delta m c^2 \) to calculate the energy released. Here, \(\Delta m = 0.02\) amu. To convert amu to kg, use 1 amu = \(1.66 \times 10^{-27} \text{ kg}\). Thus, \(\Delta m = 0.02 \times 1.66 \times 10^{-27} \text{ kg}\). The speed of light \(c\) is \(3 \times 10^8 \text{ m/s}\). Therefore, \[ E = 0.02 \times 1.66 \times 10^{-27} \times (3 \times 10^8)^2 \]

Convert Energy to kcal/mol

Calculate \(E\) using the above formula to get energy in Joules. Then, convert Joules to kcal using the conversion rate: 1 Joule = \(2.39 \times 10^{-4} \text{ kcal}\). Furthermore, multiply by Avogadro's number (\(6.022 \times 10^{23}\)) to convert to mole basis. Simplifying, \[ E_{mol} = \left(0.02 \times 1.66 \times 10^{-27} \times 9 \times 10^{16} \right) \times 6.022 \times 10^{23} \times 2.39 \times 10^{-4} \] which simplifies to approximately \(200 \text{ million kcal/mol}\).

Compare the answer to the options provided

From the calculation, the energy released is approximately 200 million kcal/mol. Thus, the closest option provided in the question is (b) 200.

Key concepts

Mass-Energy Equivalence

The mass-energy equivalence principle is a cornerstone of modern physics, introduced by Albert Einstein. It suggests that mass can be transformed into energy and vice versa, through the famous equation:

• \( E = mc^2 \)

In this equation, \( E \) represents energy, \( m \) is mass, and \( c \) is the speed of light. The principle implies that a small amount of mass can produce an immense amount of energy, due to the speed of light squared \((c^2)\) being such a large number.
In a nuclear reaction like the one in the exercise, mass is not conserved in the traditional sense. Instead, some mass is "lost" and converted into energy, which is often released as heat or radiation. This transformation of mass into energy is crucial for nuclear power and atomic weapons. It explains why even a tiny mass defect can result in significant energy output, which in our example, turned out to be a mass loss of \(0.02 \text{ amu}\).
Hence, understanding mass-energy equivalence is vital in nuclear physics, helping to explain how and why reactions produce energy.

Fusion Reactions

Fusion reactions are processes where two light atomic nuclei combine to form a heavier nucleus. This fusion results in the release of energy, primarily because the mass of the resulting nucleus is less than the sum of its parts.
For instance, in the exercise provided, lithium and hydrogen nuclei fuse to form helium. During this process, some of the mass of the original nuclei gets converted into energy. This is the "mass loss" we talked about earlier, resulting in a release of energy.
Fusion is often contrasted with fission, where a heavy nucleus splits into smaller ones. Fusion reactions are the source of power for stars, including our sun, which fuses hydrogen into helium under immense pressure and temperature.
Fusion offers potential for clean and abundant energy. It generates less radioactive waste compared to fission and the fuels used are more abundant. However, achieving controlled fusion on Earth presents significant technical challenges.

Energy Conversion

Energy conversion in nuclear reactions involves transforming the energy stored in atomic bonds or mass into usable energy forms like heat, light, or electricity. The concept revolves around applying the conservation of energy principle, ensuring energy is never lost but only transformed.
In the given nuclear fusion example, the mass lost during the reaction converts into energy as per \(E = mc^2\). This energy can then be utilized in various ways, such as powering engines or generating electricity by heating water into steam to turn turbines.
This conversion process is what makes nuclear reactions a powerful source of energy, potentially capable of meeting large-scale energy demands with relatively less fuel compared to traditional chemical reactions.
The efficiency and cleanliness of converting nuclear energy into electricity continue to drive interest and research into nuclear technology.

Einstein's Theory

Einstein's theory of relativity transformed our understanding of space, time, and energy. His theories introduced new concepts about how objects move through space-time and how mass and energy relate.
The special theory of relativity, where \( E = mc^2 \) comes from, fundamentally changed physics' approach to mass and energy. Before Einstein, mass and energy were distinct; his theory showed they are two sides of the same coin.

• Special relativity is based on principles such as the constancy of the speed of light for all observers, regardless of the relative motion between the light source and the observer.
• It also suggests that time and space are relative concepts that change based on the observer's velocity.

This theory not only explains high-energy physics phenomena but also has practical applications in technologies like GPS and accelerators. Each practical application traces back to the deeper, conceptual leap that Einstein made, merging physics concepts into a more unified, coherent framework.