Atoms And Nuclei

We are given an assertion (A) and a reason (R) related to Bohr's model of the hydrogen atom. Let's analyze both statements:

1. Assertion (A): "In Bohr's model of the hydrogen atom, the angular momentum of an electron in the -th orbit is proportional to the square root of its orbit radius ."

This statement is incorrect. According to Bohr's model, the angular momentum ( ) of an electron in the -th orbit is quantized and is given by the equation:

• is the angular momentum of the electron,
• is the principal quantum number (the orbit number), and
• is the reduced Planck’s constant.

The radius of the -th orbit is proportional to , i.e.,

Therefore, the angular momentum is proportional to the quantum number , not to the square root of the radius. The assertion is therefore incorrect.

2. Reason (R): "According to Bohr's model, the electron can jump to its nearest orbits only."

This statement is correct. In Bohr's model, when an electron absorbs or emits energy, it jumps from one orbit to another. However, the electron can only jump between specific orbits that correspond to the allowed energy levels. These energy levels are quantized, and the electron can only transition between these discrete orbits (energy levels) in response to energy absorption or emission. This transition typically occurs between orbits that are closest to each other in energy. Therefore, the reason is true.

3. Evaluation of the Assertion and Reason:

While the reason (R) is correct, the assertion (A) is incorrect because, according to Bohr’s model, the angular momentum is proportional to , not the square root of the radius. Therefore, the assertion is false, even though the reason is true.

4. Conclusion:

The correct answer is: Assertion is false, and Reason is true.

When two nuclei with lower binding energy per nucleon combine to form a nucleus with a higher binding energy per nucleon, the process is called nuclear fusion.

1. Explanation of Nuclear Fusion:

In nuclear fusion, two light atomic nuclei (such as hydrogen isotopes, like deuterium and tritium) combine to form a heavier nucleus (such as helium). The key feature of this process is that the resulting nucleus has a higher binding energy per nucleon compared to the original nuclei, meaning the nucleus is more stable after fusion.

• The binding energy per nucleon is a measure of how tightly the nucleons (protons and neutrons) are bound within the nucleus. The greater the binding energy per nucleon, the more stable the nucleus.
• In the case of fusion, the binding energy per nucleon increases because the resulting nucleus is closer to the "optimal" binding energy, which is typically around iron-56 (which has the highest binding energy per nucleon).
• This release of energy during fusion is responsible for the tremendous energy output observed in stars, including the Sun, and is the principle behind hydrogen bombs and fusion reactors.

2. Why Fusion Happens with Lower Binding Energy Nuclei:

For elements lighter than iron (which have relatively lower binding energies per nucleon), fusion occurs because the resulting nucleus, which is heavier than the original nuclei, will have a higher binding energy per nucleon. This means that energy is released during the fusion process, making it energetically favorable.

3. Conclusion:

The type of nuclear reaction where two nuclei with lower binding energy per nucleon form a nucleus with more binding energy per nucleon is nuclear fusion.

Variation of Binding Energy per Nucleon with Mass Number: The binding energy per nucleon is the energy required to remove a nucleon from a nucleus. It generally increases with mass number up to iron (Fe), after which it begins to decrease. This is because larger nuclei become less tightly bound as their size increases, while smaller nuclei (like hydrogen and helium) are more tightly bound. The binding energy curve has a peak around (the mass number of iron), after which it decreases. The curve is roughly shaped like a bell, with the highest point at iron, indicating that nuclei around this mass number are the most stable.

Significance of the Binding Energy Curve:
The curve shows that nuclei with mass numbers near 56 (such as iron) are the most stable, meaning they require the most energy to break apart.
Nuclei with mass numbers greater than 56 can release energy by fission (splitting), as splitting them into smaller nuclei releases energy.
Nuclei with mass numbers less than 56 can release energy by fusion, as fusing them to form heavier nuclei also releases energy.