Problem 146 - A Complete Resource Book in Chemistry for JEE Main

Question

Calculate the number of nodal planes in the \(\mathrm{d}_{x y}\) orbital.

Step-by-Step Solution

Answer

The \\(d_{xy}\\) orbital has 2 nodal planes.

Step 1: Identify the characteristic of \\(d_{xy}\\) orbital

The \(d_{xy}\) orbital is part of the \(d\) subshell, which has five orbitals: \(d_{xy}\), \(d_{yz}\), \(d_{zx}\), \(d_{x^2-y^2}\), and \(d_{z^2}\). The \(d_{xy}\) orbital is oriented in the \xy-plane, and its lobes are between the axes.

Step 2: Understand the concept of nodal planes

A nodal plane is a plane where the probability of finding an electron is zero. For \(d\) orbitals, the number of nodal planes is determined by the orientation and shape of the orbital.

Step 3: Determine the number of nodal planes

The \(d_{xy}\) orbital lies between the \(x\)- and \(y\)-axes, consisting of two nodal planes. These are the \(xz\)-plane and the \(yz\)-plane, where the probability density of finding an electron is zero.

Key Concepts

d-xy orbital

The \(d_{xy}\) orbital is one of the five atomic orbitals found in the \(d\) subshell, representing a fascinating aspect of quantum chemistry. Unlike the other \(d\) orbitals, the \(d_{xy}\) orbital is distinctly positioned in the \(xy\)-plane. This means its lobes, which resemble the shape of a four-leaf clover, exist between the \(x\)- and \(y\)-axes.
This unique orientation differentiates it from the other four \(d\) orbitals: \(d_{yz}\), \(d_{zx}\), \(d_{x^2-y^2}\), and \(d_{z^2}\).
The \(d_{xy}\) orbital's spatial arrangement allows it to interact specifically with other orbitals, influencing how atoms form bonds with one another. Each orbital within the \(d\) subshell is crucial for understanding complex chemical bonding, including transition metals, which often exhibit these orbitals' intriguing properties.The connection between the orbital shape and its placement between the axes makes the \(d_{xy}\) orbital particularly important when analyzing molecular geometries and electron distribution in molecules.

nodal planes

Nodal planes play a pivotal role in quantum mechanics and electron behavior in atomic orbitals. Essentially, a nodal plane is a hypothetical plane within an orbital where the probability of finding an electron at any point is exactly zero.
This absence of electron density in nodal planes contributes to an orbital's shape and characteristics.
In the case of the \(d_{xy}\) orbital, you can visualize two nodal planes: the \(xz\)-plane and the \(yz\)-plane. The electron's probability density becomes zero in these areas. Thus, it helps explain why the \(d_{xy}\) orbital is oriented in its particular way, with its lobes avoiding these planes.
Nodal planes are a direct consequence of the wave function associated with an electron in an orbital, which is an essential part of quantum mechanical theory. By analyzing nodal planes, chemists can gain insights into electron configuration and chemical bonding patterns.

probability density

Probability density is at the heart of understanding electron positions within an atom. In quantum mechanics, it describes how likely you are to find an electron in a specific location around the nucleus. The concept is rooted in the wave-particle duality of electrons, where electrons exhibit properties of both particles and waves.
Mathematically, the probability density is derived from the square of the electron's wave function. This concept dictates the shape of orbitals and helps explain electron cloud distributions. For the \(d_{xy}\) orbital, the probability density is concentrated in the areas between the nodal planes, specifically in regions between the \(x\)- and \(y\)-axes.
Understanding probability density is crucial for visualizing an electron's behavior in atoms and molecules. It provides an elegant explanation for various observable phenomena, such as the appearance of atomic spectra and the chemical behavior of atoms during reactions.

quantum mechanics

Quantum mechanics is the fundamental theory used to explain the behavior and interaction of particles at the atomic and subatomic levels. It reveals a world where particles, like electrons, don't adhere to classical physical laws but instead exist in multiple states, as described by probabilistic wave functions.
In quantum mechanics, the concept of orbitals arises, introducing the notion of electrons existing in specific probability zones around an atom, rather than predictable paths.
This quantum description is why electron behavior in orbitals is predicted using probability densities. Quantum mechanics introduces revolutionary ideas like wave-particle duality, uncertainty principles, and entanglement, redefining how scientists understand matter and chemical reactions.
It provides the mathematical formulas and conceptual frameworks needed to analyze electron configurations, bonding patterns, and energy levels in atoms. The insights gained from quantum mechanics reverberate across physics and chemistry, offering explanations for phenomena that classical mechanics could not, revolutionizing science and technology.