Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

Which set represents the quantum Numbers of 19 th electron in Cr-atom? (a) \(4,0,0,1 / 2\) (b) \(4,1,0,1 / 2\) (c) \(3,2,2,1 / 2\) (d) \(3,2,-2,1 / 2\)

Step-by-Step Solution

Answer

The answer is (c) \(3,2,2,1 / 2\).

Step 1: Determine the Electronic Configuration

First, determine the electronic configuration for Chromium (Cr), which has an atomic number of 24. Chrome has an unusual electronic configuration due to the stability offered by half-filled d-subshells. Its configuration is: \[ \text{Cr: } 1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^1 \] This is atypical as one electron is promoted from the 4s subshell to the 3d subshell to give more stability.

Step 2: Identify the 19th Electron's Position

From the configuration, count to find the position of the 19th electron. \[ 1s^2, 2s^2, 2p^6, 3s^2, 3p^6, \text{3d}^5, 4s^1 \] 1s^2, 2s^2, and 2p^6 together hold the first 10 electrons. 3s^2 adds 2 more electrons (total 12), and 3p^6 makes it 18 electrons. The 19th electron goes into the 3d subshell.

Step 3: Assign Quantum Numbers for the 19th Electron

Each electron in an atom can be represented by a set of four quantum numbers: 1. The principal quantum number \(n\), which corresponds to the shell. For 3d, \(n = 3\). 2. The azimuthal quantum number \(l\) for a d subshell is 2. 3. The magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\). For a 3d electron, possible \(m_l\) values are -2, -1, 0, +1, and +2. 4. The spin quantum number \(m_s\) is either \(+1/2\) or \(-1/2\). Electrons in the same orbital must have opposite spins, so assume the 19th electron has \(+1/2\). Thus, the set of quantum numbers is \(3,2,2,+1/2\).

Step 4: Choose the Correct Option

Compare the obtained set of quantum numbers \(3,2,2,+1/2\) with the given options: (a) \(4,0,0,1 / 2\) (b) \(4,1,0,1 / 2\) (c) \(3,2,2,1 / 2\) (d) \(3,2,-2,1 / 2\) The correct set is option (c) \(3,2,2,1 / 2\).

Key Concepts

Electronic Configuration

Electronic configuration is the arrangement of electrons in an atom's orbitals. Each element has a unique electronic configuration that provides information about how its electrons are distributed among the different energy levels and subshells. Understanding this setup is crucial for predicting an element's chemical properties and behavior.
For Chromium (Cr), which has an atomic number of 24, the electronic configuration is exceptional. Typically, you would expect Chromium to fill its subshells in the order of energy levels. However, due to the stability associated with half-filled and fully filled d-subshells, Chromium's setup is:

• \[ ext{Cr: } 1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^1 \]

This reflects an electron being promoted from the 4s to the 3d subshell, achieving a more stable half-filled state for the 3d subshell. This odd configuration significantly influences its physical and chemical traits.

Chromium Atom

The Chromium atom is an intriguing element, placed under group 6 of the periodic table. What makes it particularly interesting is its unconventional electronic configuration which contrasts with the typical order of filling electron shells. Chromium's atomic number is 24, meaning it has 24 protons in its nucleus and, in a neutral state, 24 electrons.
The electron configuration is not what one would expect purely based on the Aufbau principle, which usually predicts that the 4s subshell fills before the 3d. However, Chromium's configuration is influenced by the electron-electron interactions and the stability provided by a half-filled d subshell. This results in the configuration:

• \[ 1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^1 \]

In chemical reactions, this unique arrangement leads to varied oxidation states and impressive catalytic properties. Chromium can have oxidation states of +2, +3, and +6, which contribute to its versatility in compounds.

Quantum Mechanics

Quantum mechanics is the branch of physics that explains the behavior of particles on the smallest scales—at the level of atoms and subatomic particles. It fundamentally shifts the perspective from classical deterministic views to probabilistic interpretations. In quantum mechanics, electrons are described by quantum numbers and exist in orbitals that are areas around the nucleus where there is a high probability of finding the electron.
Understanding an electron's behavior in an atom requires considering four key quantum numbers:

• The principal quantum number (\( n \)) defines the main energy level.
• The azimuthal (or angular momentum) quantum number (\( l \)) defines the subshell and gives shape to the orbital.
• The magnetic quantum number (\( m_l \)) specifies the orientation of the orbital in space.
• The spin quantum number (\( m_s \)) represents the intrinsic spin direction of the electron.

Together, these numbers form a complete "address" for electrons within an atom, providing a robust framework for understanding complex atomic structures like that of Chromium.

Electron Spin

Electron spin is a fundamental property of electrons, a type of intrinsic angular momentum that is independent of the everyday concept of "spin" as in rotation. In the realm of quantum mechanics, electron spin is quantized, meaning it can only take on certain discrete values, either +1/2 or -1/2. This property is crucial in many of the electron arrangements and chemical properties observed in materials.
In orbitals, electrons are arranged in pairs with opposite spins. This is explained by the Pauli Exclusion Principle, which states no two electrons in an atom can have the same set of four quantum numbers. This means that spin is a key factor in determining the electronic configuration. For the 19th electron in Chromium's 3d subshell, the spin is typically +1/2 if the orbital is initially empty or occupied only by electrons with spins of -1/2.