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: (a) 12 and 4 (b) 12 and 5 (c) 16 and 4 (d) 16 and 5

Short answer

The correct answer is (b) 12 and 5.

Step-by-step solution

Understand the Electronic Configuration

The electronic configuration of chromium () is given by \[ \text{{Cr: }} [\text{{Ar}}] 3d^5 4s^1 \]Chromium has 24 electrons to fill the orbitals. It follows the principle where electrons prefer to stay unpaired in order to minimize electron-electron repulsions, thus adopting the configuration \([\text{{Ar}}] 3d^5 4s^1\) instead of the expected \([\text{{Ar}}] 3d^4 4s^2\).

Identify Electrons with Azimuthal Quantum Number l=1

The azimuthal quantum number \( l = 1 \) corresponds to the \( p \)-orbitals. In the electronic configuration of chromium, we identify electrons in the \( p \)-orbitals:- From \([\text{{Ar}}]\), the \( p \)-electrons come from the 2p and 3p orbitals, each having 6 electrons. Thus, there are \( 6 + 6 = 12 \) electrons with \( l = 1 \).

Identify Electrons with Azimuthal Quantum Number l=2

The azimuthal quantum number \( l = 2 \) corresponds to the \( d \)-orbitals. In the electronic configuration of chromium, the electrons in the 3d orbital are the ones that contribute:- The 3d sublevel contains 5 electrons as seen in \( 3d^5 \).Therefore, there are 5 electrons with \( l = 2 \).

Verify and Compare with Options

We calculated there are 12 electrons with \( l = 1 \) and 5 electrons with \( l = 2 \). Comparing this result with the given options, the correct answer is (b) 12 and 5.

Key concepts

Understanding Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( l \), plays a critical role in defining the shape of an atomic orbital and is crucial in determining an electron's angular momentum. It is an integer value that can range from 0 to \( n-1 \), where \( n \) is the principal quantum number representing the energy level of an electron. Each value of \( l \) corresponds to a specific type of orbital:

• \( l = 0 \) corresponds to s-orbitals, spherical in shape.
• \( l = 1 \) represents p-orbitals, which have a dumbbell shape.
• \( l = 2 \) designates d-orbitals, more complex in shape with four lobes.
• Greater values of \( l \) correspond to f-orbitals and beyond, which are even more complex.

In the context of the chromium atom, focusing on the azimuthal quantum numbers \( l = 1 \) and \( l = 2 \) allows us to analyze which electrons are occupying the p- and d-orbitals, which are pivotal in determining the chromium's electronic configuration.

The Role of P-Orbitals

P-orbitals are a fascinating aspect of atomic structure. Each principal energy level starting from the second level (\( n = 2 \)) has three p-orbitals: \( p_x, p_y, \) and \( p_z \), each oriented along a different axis. These orbitals are not just about three-dimensional shapes; they also influence chemical bonding and molecular geometry. Each p-orbital can hold up to two electrons, leading to a total of six electrons if all p-orbitals are filled.
In case of chromium’s electron configuration, the full p-orbitals in the 2p and 3p levels (\( [ ext{Ar}] \) contributes with 6 electrons each) cumulatively account for the 12 electrons:

• The configuration \( 2p^6 \) and \( 3p^6 \) jointly results in these 12 electrons, all possessing an azimuthal quantum number \( l = 1 \).

This understanding plays a pivotal role when determining how an element like chromium bonds and what properties it exhibits.

D-Orbitals and their Complexities

D-orbitals add significant complexity and richness to electron configurations, particularly in transition metals such as chromium. D-orbitals, associated with \( l = 2 \), emerge starting from the n=3 principal quantum level and possess a unique set of spatial orientations. Unlike simpler s- and p-orbitals, d-orbitals come in five distinct shapes—\( d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, \) and \( d_{z^2} \). Each can house two electrons, allowing isotopic sublevels like 3d to hold up to ten electrons.
In chromium's ground state, the 3d orbital contains five electrons as indicated by the configuration \( [ ext{Ar}] 3d^5 \). By occupying half of the d-orbitals, electrons assume a configuration minimizing electron repulsion, adding to the atom’s stability.

• These five electrons distinctly have an azimuthal quantum number \( l = 2 \).

This arrangement plays a crucial role in the distinctive chemical properties of chromium and other transition metals.