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 \(\mathrm{R}=\mathrm{R}_{0}(\mathrm{~A})^{1 / 2}\left(\mathrm{R}_{0}=\mathrm{constant}\right)\)

Step-by-Step Solution

Answer

Statement (c) is correct.

Step 1: Analyze Statement (a)

The thorium series, also known as the 4n series, ultimately decays to lead-208 (\({ }_{82}\mathrm{Pb}^{208}\)). It doesn't form \({ }_{83}\mathrm{Bi}^{200}\) as its end product, so this statement is false.

Step 2: Analyze Statement (b)

Isobars are atoms with different numbers of protons but the same mass number. \({ }_{7} \mathrm{N}^{15}\) and \({ }_{8} \mathrm{O}^{16}\) have different mass numbers (15 and 16, respectively), so they are not isobars. Thus, this statement is incorrect.

Step 3: Analyze Statement (c)

Magic numbers are specific numbers of nucleons (either protons or neutrons) that make the nuclei more stable. The magic numbers are 2, 8, 20, 28, 50, 82, 126. \({ }_{20} \mathrm{Ca}^{40}\) has 20 protons, a magic number, and 20 neutrons, also a magic number. Therefore, this statement is true.

Step 4: Analyze Statement (d)

The radius \(R\) of a nucleus with mass number \(A\) is proportional to \(A^{1/3}\), according to the formula \(R=R_{0}(A)^{1/3}\). Therefore, the given formula \(R = R_{0}(A)^{1/2}\) is incorrect, making this statement false.

Key Concepts

Thorium Series

The thorium series is one of the several radioactive decay series found in nature. It is also referred to as the 4n series because all its parent and daughter nuclei have mass numbers that are multiples of 4. This series starts with thorium-232

• Thorium-232 is a naturally occurring radioactive element.
• It gradually decays, undergoing a series of transformations, each time releasing alpha or beta particles.

Over time, it results in different daughter isotopes. The series passes through various other isotopes such as radium, radon, and others, ultimately ending with lead-208 (\({ }_{82}\mathrm{Pb}^{208}\)). This endpoint is a stable, non-radioactive isotope.
This means that over time, any amount of thorium-232 will eventually become lead-208, releasing radiation in the process. Understanding this series is important in nuclear chemistry as it helps scientists predict the behavior and lifespan of certain radioactive materials.

Magic Numbers

Magic numbers are unique numbers of protons or neutrons that allow nuclei to be more stable than their neighbors. These are numbers at which the nuclei form complete shells.

• Some known magic numbers are 2, 8, 20, 28, 50, 82, and 126.
• The complete shells contribute to greater nuclear stability.

An example of this is the calcium-40 nucleus (\({ }_{20} \mathrm{Ca}^{40}\)), which contains 20 protons and 20 neutrons. Both 20 protons and 20 neutrons are magic numbers
This gives the calcium-40 nucleus exceptionally stable characteristics. Scientists are particularly interested in these magical numbers because they provide insights into the structural design of atomic nuclei. Such knowledge is crucial in nuclear physics, helping to explain why some isotopes exist in nature while others don't. This can also guide the creation of new isotopes in laboratory settings.

Isobars

Isobars are atoms that have different atomic numbers but the same mass number.

• The mass number is the sum of protons and neutrons in an atom's nucleus.
• This means isobars have the same total number of nucleons.

However, they differ in the number of protons, meaning they are different elements and have different chemical properties.
An easy way to not confuse isobars with isotopes is to remember that isobars differ in chemical identity. For instance, \({ }_{7} \mathrm{~N}^{15}\) and \({ }_{8} \mathrm{O}^{16}\) are not isobars since their mass numbers are different.
In contrast, carbon-14 and nitrogen-14 are examples of isobars because they both have a mass number of 14. Understanding isobars helps in various applications, such as the study of nuclear reactions and radioactive decay.