Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement is/ are correct? (a) The number of unpaired electrons in both \(\mathrm{Fe}^{2+}\) and \(\mathrm{Mn}^{2 *}\) are five. (b) In silver atom, 23 electrons have a spin of one type and 24 of the opposite (atomic number of \(\mathrm{Ag}=\) 47). (c) The azimuthal quantum number may have a negative value. (d) The electronic configuration of \(\mathrm{Cr}\) is \([\mathrm{Ar}] 3 \mathrm{~d}^{5} 4 \mathrm{~s}^{1}\) (atomic number of \(\mathrm{Cr}=24\) )

Step-by-Step Solution

Answer

Statements (b) and (d) are correct.

Step 1: Understanding Electric Configurations

To begin with, we need to determine the electronic configurations for the ions and atom given in the options. Given that the atomic number of Fe is 26, the configuration of \(\mathrm{Fe}^{2+}\) is \[\mathrm{[Ar]}\,3\,d^6\]. Manganese's atomic number is 25, so \(\mathrm{Mn}^{2+}\) has the configuration \[\mathrm{[Ar]}\,3\,d^5\]. For silver, \(\mathrm{Ag}\) has a configuration of \[\mathrm{[Kr]}\,4d^{10}\,5s^1\]. Finally, using Cr's atomic number, we check the configuration given.

Step 2: Counting Unpaired Electrons

With the configurations, count the unpaired electrons in \(\mathrm{Fe}^{2+}\) and \(\mathrm{Mn}^{2+}\). \(\mathrm{Fe}^{2+}\,\mathrm{has}\,4\,unpaired\,electrons\) in its \(3d^6\) subshell as \[\uparrow\downarrow\,\uparrow\,\uparrow\,\uparrow\,\uparrow\]. In contrast, \(\mathrm{Mn}^{2+}\,\mathrm{has}\,5\,unpaired\,electrons\) with \[\,\uparrow\,\uparrow\,\uparrow\,\uparrow\,\uparrow\].

Step 3: Evaluating Electron Spins in Silver

For silver, the configuration \([\mathrm{Kr}]\,4d^{10}\,5s^1\) means there are 11 electrons in sub-orbitals with magnetic spin of one kind and 12 with the opposite, indicating a total difference of just one spin not following a symmetry.

Step 4: Azimuthal Quantum Number Properties

Review the definition of the azimuthal (angular momentum) quantum number \(l\), which ranges from zero up to \(n-1\) where \(n\) is the principal quantum number. Important to note is that \(l\) can never be negative.

Step 5: Confirm Electronic Configuration of Chromium

Chromium, with atomic number 24, has an unusual electron configuration \[\mathrm{[Ar]}3d^5\,4s^1\,\] instead of \[\mathrm{[Ar]}3d^4\,4s^2\,\], to minimize energy by half-filling the 3d subshell. This aligns with the given statement.

Key Concepts

Unpaired Electrons

Unpaired electrons in an atom occupy different orbitals within the same sublevel and usually have parallel spins. Understanding the unpaired electrons in ions like \(\mathrm{Fe}^{2+}\) and \(\mathrm{Mn}^{2+}\) involves looking at their electronic configurations. For \(\mathrm{Fe}^{2+}\), the electronic configuration is

• \([\mathrm{Ar}]\,3d^6\)

This means four electrons are unpaired, occupying separate orbitals, indicated with upward and outward spins \(\uparrow\,\uparrow\,\uparrow\,\uparrow\).
In the case of \(\mathrm{Mn}^{2+}\), its electronic configuration is

• \([\mathrm{Ar}]\,3d^5\)

Here, all five electrons in the 3d subshell are unpaired.
Understanding the unpaired electrons helps determine magnetic properties of the element. More unpaired electrons typically mean stronger magnetic effects.

Electron Spin

Electron spin is a fundamental property of electrons that plays a critical role in their magnetic interactions and atomic structure. Each electron spin can be either "up" \((+1/2)\) or "down" \((-1/2)\). In the case of the silver atom, which has an electron configuration:

• \([\mathrm{Kr}]\,4d^{10}\,5s^1\)

The electrons fill their respective orbitals based on the Pauli Exclusion Principle, which mandates no two electrons can have the same set of quantum numbers in an atom.
Understanding the spin in electrons:

• The total number of electrons with one type of spin tells about the atom's net magnetic moment.
• Silver has 23 electrons with one spin and 24 with the opposite spin, excluding any disparity or pairs in the configuration.

This suggests that the difference in spins contributes to the atom's magnetic properties.

Quantum Numbers

Quantum numbers describe the position and energy of electrons within an atom, functioning like addresses. The azimuthal quantum number \(l\) is specifically crucial, defining the shape of an electron's orbital. It follows a simple rule

• \(l = 0, 1, 2, \ldots, n-1\)

where \(n\) is the principal quantum number.
Important aspects of the azimuthal quantum number include:

• It can never have negative values, disputing any notion that it might. Incorrect statements could stem from misunderstandings of quantum mechanics.
• For example, if \(n = 3\), possible values of \(l\) are 0, 1, or 2, corresponding to s, p, and d orbitals respectively.

Understanding quantum numbers provides a framework for predicting and explaining the electron distribution in atoms, and their related chemical properties.