Problem 92 - Chemistry The Central Science

Question

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.22 and 6.23 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?

Step-by-Step Solution

Answer

\(p_z\) orbital nodal plane: x-y plane \(d_{xy}\) orbital nodal planes: planes containing x-axis and y-axis \(d_{x^2-y^2}\) orbital nodal planes: planes containing lines y = x and y = -x

Step 1: (a) Nodal plane of \(p_z\) orbital

The \(p_{z}\) orbital has a dumbbell shape, with two lobes along the z-axis. The point where the electron probability density is zero is the x-y plane. All points in the x-y plane satisfy the condition for zero electron density, thus the nodal plane of the \(p_z\) orbital is the x-y plane.

Step 2: (b) Nodal planes of \(d_{xy}\) orbital

The \(d_{xy}\) orbital has four lobes, and they lie in the x-y plane between the x and y axes. The electron probability density is zero along the x and y axes. Thus, the two nodal planes of the \(d_{xy}\) orbital are the planes perpendicular to the x-y plane, containing the x-axis and y-axis respectively.

Step 3: (c) Nodal planes of \(d_{x^2-y^2}\) orbital

The \(d_{x^2-y^2}\) orbital has four lobes, with two along the x-axis and two along the y-axis. The electron probability density is zero along the lines y = x and y = -x. Thus, the two nodal planes of the \(d_{x^2-y^2}\) orbital are the planes perpendicular to the x-y plane, containing the lines y = x and y = -x respectively.

Key Concepts

p Orbitals

The p orbitals are fascinating components of atomic structure due to their unique shapes and nodal plane characteristics. They are denoted as \(p_x\), \(p_y\), and \(p_z\), each aligned along the respective coordinate axes.
The shape of a p orbital is often compared to a dumbbell or two teardrop shapes joined at their tips. This gives rise to their distinctive lobe structure, where the electron density is concentrated in two regions.
Each p orbital possesses a nodal plane, which is a plane where the probability of finding an electron is zero:

• For the \(p_x\) orbital, the nodal plane is the yz-plane.
• For the \(p_y\) orbital, it is the xz-plane.
• For the \(p_z\) orbital, it is the xy-plane.

This arrangement contributes to our understanding of more complex molecular shapes and bonding.

d Orbitals

D orbitals add a layer of complexity and intricacy compared to p orbitals, present in transition metals and crucial for their chemical behavior. These orbitals are denoted as \(d_{xy}\), \(d_{yz}\), \(d_{zx}\), \(d_{x^2-y^2}\), and \(d_{z^2}\).
Each d orbital has a peculiar shape that can influence chemical bonding and properties. The \(d_{xy}\), \(d_{yz}\), and \(d_{zx}\) orbitals each have four lobes located between the coordinate axes.

• The \(d_{xy}\) orbital has nodal planes along the x and y axes.
• The \(d_{x^2-y^2}\) orbital features nodal planes along lines \(y = x\) and \(y = -x\).
• Other d orbitals may include distinctive donut-shaped lobes, specifically the \(d_{z^2}\) orbital.

D orbitals are integral in complex formation and in determining the geometries of transition metal compounds.

Electron Density

Electron density is a critical concept in quantum chemistry describing where electrons are likely to be found around an atom. It is influenced by the shape of orbitals.

• High electron density regions correspond to areas within an orbital where it is highly probable to find electrons.
• Nodal planes are where the electron density is exactly zero, meaning no electrons will be found in these planes.

The significance of electron density extends to how atoms interact, form bonds, and ultimately structure molecules and solids.
Understanding electron density distribution helps predict molecule reactivity, polarity, and physical properties like melting and boiling points.

Orbital Shapes

Orbital shapes are visual representations of where electrons reside in an atom, and they vary among s, p, d, and f orbitals.
It's crucial to comprehend solid shapes as they dictate electron configurations and the nature of chemical bonds.
Different orbitals have different numbers of nodal planes, which are indicative of the phases in which electrons can or cannot exist.

• S orbitals are spherical and have no nodal planes.
• P orbitals have a dumbbell shape with one nodal plane.
• D orbitals are more complex, with multiple nodal planes and lobes.

Recognizing these shapes aids in visualizing molecular geometries and provides insight into the behaviors and interactions of molecules during chemical reactions.