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 orbitals are as follows: (a) For the \(p_{z}\) orbital, the nodal plane is the \(x y\) plane. (b) For the \(d_{xy}\) orbital, there are two nodal planes: the \(x\)-axis and the \(y\)-axis. (c) For the \(d_{x^2-y^2}\) orbital, there are two nodal planes: the plane where \(x=y\) and the plane where \(x=-y\).

Step-by-step solution

Determine the nodal plane of the \(p_{z}\) orbital

To find the nodal plane of the \(p_{z}\), we are asked to consider where the \(z\) value is equal to zero. This equation \((z=0)\) is satisfied by all the points on the \(x y\) plane. Therefore, the \(x y\) plane is the nodal plane for the \(p_{z}\) orbital.

Find the nodal planes of the \(d_{xy}\) orbital

To find the nodal planes of the \(d_{xy}\) orbital, we need to consider where the product of \(x\) and \(y\) values equals zero. This means either \(x=0\) or \(y=0\). So, there are two nodal planes here: the \(x\)-axis (where \(y=0\)) and the \(y\)-axis (where \(x=0\)).

Identify the nodal planes of the \(d_{x^2-y^2}\) orbital

To find the nodal planes of the \(d_{x^2-y^2}\) orbital, we need to consider where the difference of the squares of \(x\) and \(y\) values is equal to zero. This means either \(x^2=y^2 \Rightarrow x=y\) or \(x^2=y^2 \Rightarrow x=-y\). So, there are two nodal planes here: the plane where \(x=y\) (a diagonal plane from the positive \(x\)-axis to the positive \(y\)-axis) and the plane where \(x=-y\) (a diagonal plane from the positive \(x\)-axis to the negative \(y\)-axis).

Key concepts

Nodal Planes

Nodal planes are regions in an atom where the probability of finding an electron is zero. Imagine these as invisible sheets slicing through an orbital, marking the spots where electron density is non-existent.
These planes are crucial in understanding the shape and directionality of electron orbitals.

• For the p orbitals, nodal planes are easier to spot. Take the \( p_{z} \) orbital, which features a nodal plane along the \( xy \) plane because here, the \( z \) value is zero (\( z = 0 \)).
• In the case of d orbitals, there are often more than one nodal planes. Consider the \( d_{xy} \) orbital, which has two nodal planes: along the \( x \)-axis (\( y=0 \)) and the \( y \)-axis (\( x=0 \)). This happens because we consider regions where either \( x \) or \( y \) is zero causing no electron density in those planes.

Nodal planes offer a visual indicator of electron distribution, shaping our understanding of electron behavior within atoms.

p Orbitals

The p orbitals are a set of three dumbbell-shaped orbitals, namely \( p_x \), \( p_y \), and \( p_z \), each oriented along one of the three principal axes in three-dimensional space. These orbitals can accommodate a total of six electrons, with each orbital holding up to two electrons of opposite spin.
In visual terms, each p orbital consists of two lobes extending opposite directions from the nucleus, with a nodal plane passing through the nucleus.

• For instance, the \( p_x \) orbital is oriented along the x-axis, and its nodal plane is the \( yz \) plane, where x equals zero.
• The \( p_z \) orbital, meanwhile, aligns along the z-axis, with a nodal plane on the \( xy \) plane, given that \( z = 0 \) in this region.

Understanding the orientation and nodal structure of p orbitals is essential for predicting the shape of molecules and the types of chemical bonds they can form.

d Orbitals

d orbitals are more complex than s and p orbitals. They consist of five distinct shapes within each main energy level, accommodating up to 10 electrons. Each d orbital consists of four lobes pointing in various directions and features two nodal planes.
These characteristics make them fascinating and important in various chemical processes, particularly in transition metals.

• The \( d_{xy} \) orbital, for instance, has nodal planes on both the x and y axes, indicating no electron density there, helping define its four-lobed shape lying between these axes.
• The \( d_{x^2-y^2} \) orbital is characterized by nodal planes along the diagonals of the x and y axes, specifically where \( x = y \) or \( x = -y \), giving rise to its distinctive cross or cloverleaf shape.

Comprehending the nodal planes and lobes in d orbitals aids in the analysis of bonding and electron arrangements, particularly when assessing the electron configuration of transition elements and their compounds.