Question

For orbitals that are symmetric but not spherical, the contour representations (as in Figures \(6.23\) and 6.24) 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?

Short answer

The nodal planes for the given orbitals are: (a) the \(p_{z}\) orbital has the \(xy\) plane as its nodal plane; (b) the \(d_{xy}\) orbital has two nodal planes – the \(yz\) plane and the \(xz\) plane; (c) the \(d_{x^2-y^2}\) orbital has two nodal planes along the lines \(y=x\) and \(y=-x\) on the \(xy\) plane.

Step-by-step solution

a) Nodal Plane of the \(p_{z}\) Orbital

The \(p_{z}\) orbital is symmetric about the z-axis. Thus, the nodal plane should be one where \(z=0\). This equation is satisfied by all points on the \(xy\) plane. Therefore, the nodal plane of the \(p_{z}\) orbital is the \(xy\) plane.

b) Nodal Planes of the \(d_{xy}\) Orbital

The \(d_{xy}\) orbital has two nodal planes. Since it involves both x and y coordinates, it has nodes where either of them is zero. There are two equations to consider: 1. \(x=0\): This equation is satisfied by all points on the \(yz\) plane. 2. \(y=0\): This equation is satisfied by all points on the \(xz\) plane. Therefore, the two nodal planes of the \(d_{xy}\) orbital are the \(yz\) plane and the \(xz\) plane.

c) Nodal Planes of the \(d_{x^2-y^2}\) Orbital

The \(d_{x^2-y^2}\) orbital also has two nodal planes. There are two equations to consider for this orbital: 1. \(x^2=y^2\): This equation is satisfied by all points on the line \(y=x\), which is the diagonal of the \(xy\) plane. 2. \(x^2=y^2\): This equation is also satisfied by all points on the line \(y=-x\), which is the other diagonal of the \(xy\) plane. Therefore, the two nodal planes of the \(d_{x^2-y^2}\) orbital are the lines \(y=x\) and \(y=-x\) on the \(xy\) plane.

Key concepts

Nodal Planes

Nodal planes are fascinating features of atomic orbitals that help us understand electron distributions. These planes are defined as regions where the probability of finding an electron is zero. In simpler terms, nodal planes have no electron density; they are like 'no-go' zones for electrons. Let's dive into a few examples to understand this concept better.

Take, for example, the well-known \(p\) orbitals: the \(p_{x}\), \(p_{y}\), and \(p_{z}\). Each of these orbitals has a unique nodal plane due to their orientation.

• The \(p_{x}\) orbital has a nodal plane at \(x=0\), which corresponds to the \(yz\) plane.
• Similarly, the \(p_{y}\) orbital has a nodal plane at \(y=0\), corresponding to the \(xz\) plane.
• For the \(p_{z}\) orbital, the nodal plane is at \(z=0\), lying along the \(xy\) plane.

Visualizing nodal planes helps us predict where electron density will be absent around the nucleus, which is useful when considering molecular geometry and chemical bonding.

P Orbitals

\(p\) orbitals are among the simplest atomic orbitals, yet they have intriguing properties. Each principal energy level above \(n=1\) contains three \(p\) orbitals. These orbitals are dumbbell-shaped and oriented along the x, y, and z-axis respectively, thus named \(p_{x}\), \(p_{y}\), and \(p_{z}\).

A distinctive feature of the \(p\) orbitals is their angular node, also known as a nodal plane. These arise due to the orbital's wave-like nature, reinforcing the fundamental quantum mechanical behavior. For instance:

• The \(p_{x}\) orbital has a nodal plane on the \(yz\) plane.
• The \(p_{y}\) orbital holds its nodal plane on the \(xz\) plane.
• The \(p_{z}\) orbital's nodal plane rests on the \(xy\) plane.

Due to the angular shape and nodal configuration, \(p\) orbitals play a critical role in the formation of pi bonds in molecules, essential for chemical reactivity.

D Orbitals

\(d\) orbitals are more complex than \(s\) and \(p\) orbitals, adding depth to the atomic structure starting from the third period of the periodic table. There are five \(d\) orbitals (\(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{z^2}\), and \(d_{x^2-y^2}\)), each with unique shapes and nodal planes.

The nodal planes in \(d\) orbitals are especially interesting as they have two nodal planes within each orbital. For example:

• The \(d_{xy}\) orbital has nodal planes at \(x=0\) and \(y=0\), which correspond to the \(yz\) and \(xz\) planes, respectively.
• Meanwhile, the \(d_{x^2-y^2}\) orbital has nodal features along the lines \(y=x\) and \(y=-x\) on the \(xy\) plane.

The presence of more complex nodal planes can explain the unique variety in shapes and orientations of \(d\) orbitals compared to the simpler \(s\) and \(p\) orbitals. This complexity is integral in advanced topics like transition metal chemistry and crystal field theory, which influence electronic configurations and the resulting chemical properties of elements and compounds.