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 \({ }_{\mathrm{s}} \mathrm{O}^{16}\) 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

Option (c) is correct - 0Ca^{40} has magic numbers of protons and neutrons.

Step 1: Analyze Option (a)

In the thorium (4n) decay series, the steps involve thorium isotopes undergoing alpha and beta decay until forming a stable isotope. Bi-200 is not the end nuclide of this series; instead, it would be a stable isotope of lead (Pb). Therefore, this statement is incorrect.

Step 2: Analyze Option (b)

Isobars are nuclides that have the same mass number. N-15 and O-16 have different mass numbers. N-15 has a mass number of 15 and O-16 has a mass number of 16, hence they are not isobars. This statement is incorrect.

Step 3: Analyze Option (c)

A magic number is a number of nucleons (either protons or neutrons) that result in complete nuclear shells. Ca-40 has 20 protons and 20 neutrons. Both 20 and 20 are magic numbers for protons and neutrons, respectively. Thus, this statement is correct.

Step 4: Analyze Option (d)

The formula for the nuclear radius is actually given by \( R = R_0 A^{1/3} \), where \( A \) is the mass number, not \( R = R_0 A^{1/2} \). Thus, this statement is incorrect.

Key Concepts

Isobars

In nuclear physics, isobars are different elements whose atoms have the same mass number. The mass number is the sum of protons and neutrons in the atomic nucleus. Even though isobars have the same mass number, they possess different chemical properties because they have different numbers of protons, which means different elements. To identify whether two nuclides are isobars, simply compare their mass numbers. If two nuclei have identical mass numbers but belong to different elements, they are isobars. For example, the nuclei of

• Argon-40: with 18 protons and 22 neutrons, and
• Ca-40: with 20 protons and 20 neutrons

are isobars, as both have a mass number of 40. Understanding isobars is crucial in nuclear physics because, despite having the same mass number, their other nuclear properties could differ, giving insight into nuclear forces and stability.

Nuclear Decay Series

A nuclear decay series, such as the thorium series, traces the sequence of transformations that a radioactive element goes through to eventually become a stable element. During these transformations, an element can undergo various alpha and beta decays. An alpha decay involves the emission of an alpha particle (two protons and two neutrons), which reduces the atomic mass by four and atomic number by two. A beta decay involves the conversion of a neutron into a proton while emitting an electron, thereby increasing the atomic number by one without changing the mass number. For example, the

• Thorium-232 decay series, which is a 4n series, starts with Thorium-232 and ends with Lead-208, a stable nuclide, after multiple decay steps. Throughout this process, nuclides may change into various elements like Uranium and Radium, before finally stabilizing into Lead.

Understanding decay series is fundamental in nuclear physics to comprehend the natural transmutation of elements and the energy transformations involved.

Nuclear Radius Formula

The nuclear radius formula expresses how the size of an atomic nucleus relates to its mass number, represented by the nuclear radius equation. For a nucleus with a mass number, A, the radius, R, is given by \[R = R_0 A^{1/3}\]In this equation, \( R_0 \) is a constant and usually takes a value around 1.2 to 1.3 femtometers (fm).The use of A raised to the power of 1/3 suggests that the nuclear radius is proportional to the cube root of the mass number. This proportionality signifies that nuclei grow larger relatively slowly as the number of nucleons increases.Understanding the nuclear radius helps in analyzing how nuclear forces keep the nucleons together within the nuclear boundary and provides insight into nuclear densities and structural properties. The incorrect formula provided, \( R = R_0 A^{1/2} \), does not align with empirical observations, thus highlighting the need for precision in mathematical modeling of physical phenomena.