Question

In the decay process $$ \mathrm{P} \stackrel{-\alpha}{\longrightarrow} \mathrm{Q} \stackrel{-\beta}{\longrightarrow} \mathrm{R} \stackrel{-\beta}{\longrightarrow} \mathrm{S} $$ (a) \(\mathrm{P}\) and \(\mathrm{R}\) are isotones (b) \(\mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) are isobars (c) \(\mathrm{P}\) and \(\mathrm{S}\) are isotopes (d) \(\mathrm{P}\) and \(\mathrm{Q}\) are isobars

Short answer

The correct options are (a) and (b).

Step-by-step solution

Definitions of Key Terms

To solve the problem, we need to understand the definitions: **Isotones** are nuclides with the same number of neutrons. **Isobars** are nuclides with the same mass number. **Isotopes** are nuclides with the same atomic number but different mass numbers.

Analyze Decay from P to Q

The decay \( \mathrm{P} \overset{-\alpha}{\rightarrow} \mathrm{Q} \) implies that during this process, an \( \alpha \) particle (which is \( ^4_2\mathrm{He} \)) is emitted. This results in the mass number of \( \mathrm{Q} \) being \( A - 4 \) and the atomic number being \( Z - 2 \).

Analyze Decay from Q to R

The decay \( \mathrm{Q} \overset{-\beta}{\rightarrow} \mathrm{R} \) indicates that a \( \beta^- \) particle is emitted, which increases the atomic number by 1 while the mass number remains unchanged. Thus, \( \mathrm{R} \) has mass number \( A - 4 \) and atomic number \( Z - 1 \).

Analyze Decay from R to S

The decay \( \mathrm{R} \overset{-\beta}{\rightarrow} \mathrm{S} \) indicates another \( \beta^- \) decay, increasing the atomic number by 1. Thus, \( \mathrm{S} \) has the same mass number \( A - 4 \) and atomic number \( Z \).

Determine Isotone Relationship between P and R

For \( \mathrm{P} \) and \( \mathrm{R} \) to be isotones, they must have the same number of neutrons. After analysis, both have \( A - Z \) neutrons, so they are isotones. Thus claiming (a) is correct.

Verify Isobar Relationship for Q, R, and S

Isobars have the same mass number. From step analysis, \( \mathrm{Q} \), \( \mathrm{R} \), and \( \mathrm{S} \) all have mass number \( A - 4 \). So (b) is correct.

Validate Isotope Relationship for P and S

Isotopes need the same atomic number. While \( \mathrm{S} \) and \( \mathrm{P} \) have different atomic numbers according to previous steps, (c) is incorrect.

Check Isobar Relationship for P and Q

For \( \mathrm{P} \) and \( \mathrm{Q} \) to be isobars, their mass numbers should be equal. Since \( \mathrm{P} \) has \( A \) and \( \mathrm{Q} \) has \( A - 4 \), they are not isobars, making (d) incorrect.

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable atomic nucleus emits an alpha particle. An alpha particle is essentially a helium nucleus consisting of two protons and two neutrons. It is denoted as - \[^4_2\text{He}\]- When a nucleus undergoes alpha decay, it loses two protons and two neutrons, resulting in a decrease in both its atomic number and mass number by 2 and 4, respectively.
For example, when a nucleus \(\mathrm{P}\) decays to \(\mathrm{Q}\):

• The mass number decreases from \(A\) to \(A - 4\).
• The atomic number decreases from \(Z\) to \(Z - 2\).
• This generally produces a new element that is two places back on the periodic table.

Alpha decay is usually observed in heavier elements and is the process through which exceptionally heavy elements shed mass.

Beta Decay

Beta decay is a radioactive process by which an unstable nucleus transforms itself into a more stable one by altering its neutron-to-proton ratio. This transformation comes in two types, but the most relevant here is beta-minus ( \(\beta^-\)) decay:
The emission includes:

• A beta particle, which is an electron (\(e^-\)).
• An antineutrino.

In beta-minus decay, a neutron in the nucleus changes into a proton:

• The atomic number increases by 1 (\(Z \rightarrow Z + 1\)) while the mass number remains unchanged.
• For this reason, beta decay involves elements located next to each other on the periodic table.

In the decay sequence \(\mathrm{Q} \rightarrow \mathrm{R}\) and \(\mathrm{R} \rightarrow \mathrm{S}\), the atomic number increases, reflecting the nature of beta decay where neutron to proton conversion takes place.

Isotopes

Isotopes are different forms of the same element, sharing the same atomic number but having varying mass numbers. They arise from differences in the number of neutrons within the nucleus:
For example:

• Carbon has isotopes such as \(\text{C-12}\) and \(\text{C-14}\).
• Both have 6 protons, but \(\text{C-12}\) has 6 neutrons compared to 8 neutrons in \(\text{C-14}\).

In the decay process of the exercise, specifying \(\mathrm{P}\) and \(\mathrm{S}\) as isotopes would imply they had the same atomic number. However, due to the differing atomic numbers post-decay steps, this is not the case.
Understanding isotopes is crucial in fields like chemistry and nuclear physics, as isotopic forms can exhibit different characteristics, such as - radioactivity- or absorbance levels.

Isobars

Isobars are nuclides that have the same mass number but different atomic numbers. This means:

• The number of protons and neutrons are combined in such a way as to maintain constant mass.
• They belong to different elements but have the same number of nucleons in total.

In the decay chain, \(\mathrm{Q}, \mathrm{R},\) and \(\mathrm{S}\) all have the mass number \(A - 4\), even though their atomic numbers differ. This validates their classification as isobars.
Isobars play an important role in processes such as nuclear decay where elements have varying stability levels despite having the same overall mass.

Isotones

Isotones are nuclides that have the same number of neutrons but different atomic numbers and mass numbers. This property is - crucial - in the study of nuclear structure and stability.
For two elements to be isotones, they must satisfy the condition:

• Neutron number \(N = A - Z\) remains constant.
• Atomic number \(Z\) can vary.

In the example provided, \(\mathrm{P}\) and \(\mathrm{R}\) are isotones, meaning they have the same neutron count despite their atomic number and mass number changes after decay events. Studying isotones helps scientists understand how nuclear parameters affect stability and transformation.