Question

A nucleus \({ }_{n} X^{\mathrm{m}}\) emist one \(\alpha\) - Particle and two \(\beta\) -Particle. The resulting nucleus is (A) \(_{n-2} Y^{m-4}\) (B) \(_{\mathrm{n}} \mathrm{Y}^{\mathrm{m}-6}\) (C) \(\mathrm{n} \mathrm{Y}^{\mathrm{m}-4}\) (D) \(_{n-4} Y^{m-6}\)

Short answer

The resulting nucleus after emitting one alpha particle and two beta particles is \(_{\mathrm{n}} \mathrm{Y}^{\mathrm{m}-4}\), which corresponds to option (C).

Step-by-step solution

Emission of an alpha particle

When the nucleus emits an alpha particle, it loses 2 protons and 2 neutrons. So, the mass number (m) decreases by 4 and the atomic number (n) decreases by 2. After the emission of an alpha particle, the resulting nucleus is: \(_{n-2}X^{m-4}\)

First beta particle emission

In beta particle emission, if a neutron decays into a proton, the mass number remains the same while the atomic number increases by 1. After the first beta particle emission, the resulting nucleus becomes: \(_{(n-2)+1}X^{m-4} = _{n-1}X^{m-4}\)

Second beta particle emission

Similarly, for a second beta particle emission, we get: After the second beta particle emission, the resulting nucleus becomes: \(_{(n-1)+1}X^{m-4} = _{n}X^{m-4}\) However, since the nucleus has changed due to these emissions, we can't use the symbol X. Instead, we use the symbol Y to represent the new nucleus. Therefore, the resulting nucleus is: \(_{\mathrm{n}} \mathrm{Y}^{\mathrm{m}-4}\) Comparing our result with the given options, we find that our answer corresponds to option (C).

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus emits an alpha particle. Alpha particles consist of two protons and two neutrons, making them equivalent to a helium nucleus. When a nucleus undergoes alpha decay, it loses these 4 particles, which impacts both its mass number and atomic number.

• The mass number decreases by 4 because it loses two protons and two neutrons.
• The atomic number decreases by 2 due to the loss of two protons.

This process results in a significant transformation of the decaying atom. For example, if a nucleus with the symbol \(_{n}^{m}X\) undergoes alpha decay, it will become \(_{n-2}^{m-4}Y\), where Y represents the new element formed post-decay. Alpha particles are quite massive compared to other decay particles, giving them lower penetration power. This means they can be stopped by simple barriers like paper or human skin, making alpha radiation less dangerous unless ingested or inhaled.

Beta Decay

Beta decay is another crucial process of radioactive decay, where a beta particle is emitted from the nucleus. Beta particles are high-speed electrons or positrons. There are two types of beta decay: beta-minus (\(\beta^-\)) and beta-plus (\(\beta^+\)). Here's how they differ:

• In beta-minus decay, a neutron in the nucleus is transformed into a proton while emitting an electron (beta particle) and an antineutrino. The mass number remains unchanged, but the atomic number increases by 1 because a neutron, which has no charge, becomes a positively charged proton.
• Beta-plus decay occurs when a proton is converted into a neutron while emitting a positron (positive beta particle) and a neutrino. Here, the atomic number decreases by 1, while the mass number remains the same.

In the context of this exercise, after the nucleus \(_{n-2}^{m-4}Y\) undergoes two beta-minus decays, each emission increases the atomic number by 1. This results in the new nucleus \(_{n}^{m-4}Y\), as seen in step-by-step changes happening to the original nucleus. Beta particles have higher penetration abilities than alpha particles but can be stopped by materials like plastic or thin metal sheets.

Nuclear Reactions

A nuclear reaction involves the change in an atom's nucleus, leading to elements transforming or emitting particles. These reactions are foundational to the processes discussed, such as radioactive decay. Nuclear reactions can be spontaneous, like radioactive decay, or induced through interactions with particles.

• In spontaneous nuclear reactions, unstable isotopes decay naturally over time into more stable forms, emitting radiation in the process.
• Induced nuclear reactions occur when a nucleus interacts with external particles, like neutrons, causing changes in the arrangement of protons and neutrons.

These reactions play an essential role in nuclear physics, encompassing a wide range of processes beyond just decay, such as fission and fusion. Fission involves splitting heavy nuclei into lighter nuclei, releasing energy. Fusion, conversely, combines lighter nuclei into heavier ones, also releasing energy. Understanding these reactions helps in fields like energy production, medical applications, and even astrophysical processes.