Problem 4 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following statement is correct? (a) the end nuclide formed in thorium (4n) series is \({ }_{83} \mathrm{Bi}^{200}\) (b) \({ }_{7} \mathrm{~N}^{15}\) and \({ }_{8} \mathrm{O}^{16}\) are isobars (c) \({ }_{20} \mathrm{Ca}^{40}\) has magic number of protons and magic number of neutrons (d) The radius (R) of a nuclide of mass number \(\mathrm{A}\) is given by the equation \(\mathrm{R}=\mathrm{R}_{0}(\mathrm{~A})^{1 / 2}\left(\mathrm{R}_{0}=\right.\) constant \()\)

Step-by-Step Solution

Answer

Statement (c) is correct.

Step 1: Understanding the Thorium (4n) Series

The thorium (4n) series is one of the four major natural radioactive decay series. It begins with thorium-232 and ends with a stable isotope of lead, which has atomic number 82, not bismuth as stated in option (a). Therefore, statement (a) is incorrect.

Step 2: Analyzing Isobars

Isobars are nuclides with the same mass number but different atomic numbers. The mass numbers for \({ }_{7} \mathrm{~N}^{15}\) and \({ }_{8} \mathrm{O}^{16}\) are 15 and 16 respectively, which are different. Thus, statement (b) is incorrect.

Step 3: Checking for Magic Numbers

Magic numbers are numbers of protons or neutrons that result in a complete nuclear shell. Calcium-40 \({ }_{20} \mathrm{Ca}^{40}\) has 20 protons and 20 neutrons. Both numbers are indeed magic numbers (20 is a magic number), making statement (c) correct.

Step 4: Evaluating the Radius Equation

The radius \(R\) of a nuclide is proportional to the cube root of its mass number A, given by \(R = R_0 A^{1/3}\), not \(A^{1/2}\) as suggested in statement (d). Hence, statement (d) is incorrect.

Key Concepts

Radioactive Decay Series

In the world of nuclear chemistry, a radioactive decay series is like a roadmap. It charts the transformation routes of unstable isotopes as they decay into stable isotopes. This process is crucial as it helps us understand how elements change and eventually stabilize.

The thorium series, for instance, is one of the four well-known decay chains. This series starts with thorium-232. Through a series of alpha and beta decays, it eventually transitions into a stable form of lead (Pb-208). The complete chain includes several intermediate isotopes, each step edging closer to stability.

Understanding these decay series is essential for applications ranging from energy generation in nuclear reactors to archaeological dating using isotopic compositions. Each step in the decay involves releasing radiation, which is why handling such materials requires care.

Isobars

Isobars might sound complex, but they’re simply nuclides that share the same total number of nucleons (mass number) but differ in their proton or neutron numbers.

For example, if we take nitrogen-15 \(^{15}_{7}N\) and oxygen-16 \(^{16}_{8}O\), it's easy to see how they don't fit the definition since they don’t possess the same mass number. This distinction is essential because isobars undergo different nuclear reactions and have unique applications.

Studying isobars allows scientists to investigate how changes in neutron count can affect an element's properties without altering its mass, contributing to our broader understanding of nuclear stability and reactions.

Magic Numbers

Magic numbers are special. In nuclear chemistry, these numbers refer to the number of protons or neutrons in a nucleus that creates complete energy levels or shells.

The presence of magic numbers is vital because a nucleon count matching these numbers translates to a more stable and less reactive nucleus. Numbers such as 2, 8, 20, 28, 50, 82, and 126 are considered magic. This is why calcium-40 \(^{40}_{20}Ca\) is fascinating; it possesses both 20 protons and 20 neutrons, aligning with magic numbers and resulting in enhanced nuclear stability.

Magic numbers greatly guide the theoretical predictions of nuclear models, helping scientists predict the stability and existence of superheavy elements.

Nuclear Shell Model

The nuclear shell model, akin to the electron shell model in atomic theory, describes how protons and neutrons are organized within an atomic nucleus.

In this model, nucleons (protons and neutrons) fill distinct energy levels or shells, very much like electrons do around an atom. The significance of this model is in its explanation of the reinforcement effects brought by magic numbers, where nucleus configurations reach enhanced stability.

The shell model allows for a deeper understanding of nuclear structure and facilitates predictions on nuclear behavior, stability, and reactions. It explains why isotopes like calcium-40 are stable: both protons and neutrons fill their shells in an optimal compact packed form.

Understanding the nuclear shell model is essential as it offers insights into the forces at play inside a nucleus, enriching our overall comprehension of atomic theory.