Problem 162 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the ground state of \(\mathrm{Cr}\) atom \((\mathrm{Z}=24)\). The numbers of electrons with the azimuthal quantum numbers, \(l=1\) and 2 are, respectively \(\quad\) [2004] (a) 12 and 4 (b) 12 and 5 (c) 16 and 4 (d) 16 and 5

Step-by-Step Solution

Answer

Option (b) 12 and 5 is correct.

Step 1: Understanding Electron Configuration

The electron configuration of an atom provides the distribution of electrons in an atom's orbitals. The atomic number of Chromium (Cr) is 24, indicating it has 24 electrons.

Step 2: Write Chromium's Electron Configuration

The electron configuration for Cr, which has some unique arrangements due to its half-filled stability is: \[ [ ext{Ar}] \, 3d^5 \, 4s^1 \] Creating a stable half-filled d subshell.

Step 3: Identify Azimuthal Quantum Number l=1 Electrons

The azimuthal quantum number \( l=1 \) corresponds to the p subshell. Chromium has electrons filled in the 2p and 3p orbitals:\[ 2p^6 \text{ and } 3p^6 \] Thus, the total number of \( l=1 \) electrons is \( 6+6 = 12 \).

Step 4: Identify Azimuthal Quantum Number l=2 Electrons

The azimuthal quantum number \( l=2 \) corresponds to the d subshell. In Cr, the \(3d\) subshell is filled with: \[ 3d^5 \] Thus, the total number of \( l=2 \) electrons is \(5\).

Step 5: Verify the Results with Options

From the calculation, \( l=1 \) electrons total 12, and \( l=2 \) electrons total 5. Comparing this with the given options, choice (b) 12 and 5 matches these numbers.

Key Concepts

Azimuthal Quantum Numbers

The azimuthal quantum number, denoted by \( l \), is crucial for understanding electron configurations. It's a part of the quantum mechanical model of the atom that defines the shape of the electron's orbital. There are four quantum numbers that describe the state of an electron, and the azimuthal quantum number is one of them.

In simpler terms, \( l \) indicates the type of subshell an electron occupies:

• \( l=0 \): s subshell (spherical shape)
• \( l=1 \): p subshell (dumbbell shape)
• \( l=2 \): d subshell (cloverleaf shape)
• \( l=3 \): f subshell (complex shapes)

These subshells help determine the arrangement of electrons in atoms. By using the quantum number \( l \), you can assign electrons to specific subshells according to their energy levels, which contributes to the unique properties of elements.

Chromium Electron Configuration

The electron configuration of Chromium (Cr) is particularly interesting because it doesn't follow the typical filling order we might expect. Chromium, with an atomic number of 24, is expected to fill electrons as \( [\text{Ar}] \, 3d^4 \, 4s^2 \) based on the standard order. However, the actual configuration is \( [\text{Ar}] \, 3d^5 \, 4s^1 \).

This is because chromium prefers to have a half-filled \( 3d \) subshell and a single \( 4s \) electron. A half-filled \( 3d \) subshell provides stability and a reduced energy state, leading to this adjustment. This behavior is an example of an electron configuration anomaly.

Such anomalies illustrate the balance between electron-electron interactions and overall energy stability, making chromium's configuration a fascinating study in quantum chemistry.

D Subshell Electrons

In the context of quantum numbers, the \( l=2 \) azimuthal quantum number corresponds to the d subshell. The d subshell can hold a maximum of 10 electrons across its 5 orbitals (with 2 electrons per orbital).

The distribution of electrons within these orbitals depends on energy levels and external conditions. For chromium, this means having \( 3d^5 \) electrons, as its configuration seeks stability by half-filling the d subshell. This arrangement allows for minimized repulsion between electrons while providing a structurally balanced electron shell.

Understanding d subshell electrons is critical for predicting an element's chemical reactivity and bonding behavior. Elements with partly filled d subshells, such as transition metals, often display unique properties, such as variable oxidation states and strong metallic bonds.