Problem 88 - Master Resource Book in JEE Main Physics

Question

The nucleus \({ }_{6} \mathrm{C}^{2}\) absorbs an energetic neutron and emits a \(\beta\)-particle. The resulting nucleus is (a) \({ }_{7} \mathrm{~N}^{14}\) (b) \({ }_{5} \mathrm{~B}^{13}\) (c) \({ }_{7} \mathrm{~N}^{13}\) (d) \({ }_{6} \mathrm{C}^{13}\)

Step-by-Step Solution

Answer

The resulting nucleus is \( _{7} \mathrm{N}^{13} \). Option (c) is correct.

Step 1: Analyze the Initial Nucleus

The initial nucleus given is \( _{6} \mathrm{C}^{12} \). This means it is a carbon atom with an atomic number of 6 and a mass number of 12. However, the isotope notation in the question suggests \( _{6} \mathrm{C}^{2} \), which seems incorrect for the intended context. Adjustating to an expected neutron-induced beta decay, consider nucleus as \( _{6} \mathrm{C}^{13} \) instead, which corrects a likely typographical error.

Step 2: Understand the Reaction Process

The nucleus absorbs an energetic neutron. This would add one neutron to the original nucleus, changing it to \( _{6} \mathrm{C}^{13} \), assuming the neutron is captured by a normal carbon-12 nucleus.

Step 3: Emission of a \( \beta \)-particle

When a \( \beta \)-particle (an electron) is emitted, a neutron in the nucleus converts into a proton. Therefore, the atomic number will increase by 1, while the mass number remains unchanged. So, the conversion is from \( _{6} \mathrm{C}^{13} \) to \( _{7} \mathrm{N}^{13} \).

Step 4: Determine the Resulting Nucleus

The resulting nucleus after the absorption of a neutron followed by \( \beta \)-decay will have an atomic number of 7 and a mass number of 13, thus becoming nitrogen-13, \( _{7} \mathrm{N}^{13} \).

Step 5: Identify the Correct Option

Compare the calculated resulting nucleus \( _{7} \mathrm{N}^{13} \) with the given options. The correct option is (c) \( _{7} \mathrm{~N}^{13} \).

Key Concepts

Neutron Absorption

In nuclear physics, neutron absorption is a process where a nucleus captures additional neutrons from its surroundings. This phenomenon is particularly important in nuclear reactions, where it can significantly alter the characteristics of an atom.
When a neutron is absorbed by a nucleus, the atom's mass number increases by one because the added neutron contributes to the total mass of the nucleus. However, the atomic number, which defines the element, remains unchanged initially because neutrons do not have a charge.
The absorption of a neutron can lead to various nuclear reactions, including beta decay, due to the instability it may cause in the nucleus.

• Absorbed neutron increases nucleus mass number.
• Precursor to further nuclear transformations like beta decay.

Nuclear Reactions

Nuclear reactions involve a change in an atom's nucleus and can result in the transmutation of elements. These reactions are key when studying nuclear processes such as fission, fusion, and decay.
In the context of neutron absorption followed by beta decay, the nucleus undergoes a significant transformation. Initially, the energetic neutron is absorbed, increasing the nucleus' mass number. Then, a beta decay process converts a neutron into a proton, resulting in the emission of a beta particle (an electron or positron).
Thus, the nucleus changes one of its neutrons into a proton and increases its atomic number by one, which corresponds to the formation of a new element.

• Nuclear reactions can change the elemental identity of a nucleus.
• Neutron absorption followed by beta decay alters atomic number and converts a neutron into a proton.

Atomic Number Change

The atomic number change during nuclear reactions directly affects the identity of an element. This number denotes the number of protons in an atomic nucleus and thus defines the element itself.
In neutron-induced beta decay, an uncharged neutron in the nucleus is converted into a positively charged proton. This process increases the atomic number of the atom by one. However, the mass number of the atom remains constant, as protons and neutrons have nearly the same mass.
For example, when carbon-13 absorbs a neutron and undergoes beta decay, it becomes nitrogen-13. The neutron-to-proton conversion raises the atomic number from 6 (for carbon) to 7 (for nitrogen).

• Atomic number increase reflects a change in elemental identity.
• Occurs when a neutron turns into a proton during beta decay.