Question

For non-spherically symmetric orbitals, 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

(a) The nodal plane of the \(p_{z}\) orbital is the \(xy\) plane. (b) The two nodal planes of the \(d_{xy}\) orbital are the \(xz\) plane and the \(yz\) plane. (c) The two nodal planes of the \(d_{x^2-y^2}\) orbital are the planes where \(x=y\) and \(x=-y\).

Step-by-step solution

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

The \(p_{z}\) orbital is where the probability of finding an electron will be zero when \(z=0\). This condition is satisfied by all points on the \(xy\) plane. Therefore, the nodal plane of the \(p_{z}\) orbital is the \(xy\) plane.

(b) Determine the two nodal planes of the \(d_{xy}\) orbital.

The \(d_{xy}\) orbital has nodes when \(x=0\) or \(y=0\). These conditions are satisfied by all points on the \(xz\) and \(yz\) planes. Therefore, the two nodal planes of the \(d_{xy}\) orbital are the \(xz\) plane and the \(yz\) plane.

(c) Determine the two nodal planes of the \(d_{x^2-y^2}\) orbital.

The \(d_{x^2-y^2}\) orbital has nodes when \(x=y\) or \(x=-y\). These conditions represent all points on two diagonal planes that pass through the origin. Thus, the two nodal planes of the \(d_{x^2-y^2}\) orbital are the planes where \(x=y\) and \(x=-y\).

Key concepts

p Orbitals

Let's talk about the intriguing p orbitals. These are one of the first steps away from the simple spherical s orbitals that students typically learn about first. P orbitals have a distinctive dumbbell shape and are oriented along the x, y, and z axes in three-dimensional space. One of the most fascinating aspects is how the electron density map of a p orbital pinpoints regions where you'll find electrons most of the time.

Now, imagine slicing through a p orbital: at certain planes, the electron density drops to zero. These are known as nodal planes. For instance, the px orbital has a nodal plane along the yz plane. In simple terms, if you stand on the yz plane and look towards the px orbital, you'd see an absence of electron 'cloud' directly in front of you. Understanding these nodal planes helps in visualizing the electron distribution and predicting chemical behavior of atoms in molecules.

d Orbitals

Moving on to d orbitals, these take us into even more complex territory. There are five of them, and they possess a variety of shapes, including cloverleaf and donut-like forms. One d orbital looks like a four-leaf clover lying on the xy plane, known as the dxy orbital. Another, the dx2-y2 orbital, has its leaves along the axes.

The nodal planes of d orbitals are where things get extra interesting. Contrary to the singular nodal plane in p orbitals, some d orbitals have two nodal planes. For example, the dxy orbital has nodal planes along the xz and yz planes—picture a cross slicing right through the middle of the clover. The dx2-y2 orbital's nodal planes slash diagonally across the atom, aligning with the x=y and x=-y lines. Students can better grasp chemical bonding and transitions involving d orbital electrons with a strong mental image of these nodal planes.

Electron Density

Electron density is a concept that often gets oversimplified to 'where the electron is.' However, it's much more nuanced—it represents the probability of finding an electron in a given location around the nucleus of an atom. Think of it like a weather forecast map that shows where it's likely to rain, except instead of rain, it's the electron 'cloud' that's more or less dense in different regions.

When we discuss orbitals and their nodal planes, we are fundamentally talking about understanding the electron density distribution. For p and d orbitals, the electron density isn't uniform, and these nodal planes are spots where the electron density is zero. It’s critical to understand that areas of high electron density often correspond to where bonds may form, while nodal planes can indicate regions of chemical inactivity.

Molecular Orbitals

Molecular orbitals are where the quantum mechanics hits home for chemists. These are not just confined to single atoms; instead, molecular orbitals stretch over multiple atoms in a molecule, creating regions where electrons are likely to be found. They arise from the combination of atomic orbitals when atoms bond together to form molecules.

Molecular orbitals can also have nodal planes, just like atomic orbitals. However, their nodal planes have to do with the molecular geometry and how different atomic orbitals intersect. Sometimes they align symmetrically to create bonding orbitals with high electron density between the nuclei, and other times asymmetrically to create antibonding orbitals with a nodal plane—meaning low electron density—between the nuclei. Visualizing molecular orbitals can be complex, but breaking them down into the contributions of individual atomic orbitals can simplify this massively, and this is where understanding nodal planes is so vital.