Question

Sketch the shape and orientation of the following types of orbitals: (a) \(p_{x}\), (b) \(d_{z^{2}}\),(c) \(d_{x^{2}-y^{2}}\).

Short answer

The \(p_{x}\) orbital is a dumbbell-shaped orbital along the x-axis. The \(d_{z^{2}}\) orbital consists of a torus in the xy-plane with two lobes along the z-axis. The \(d_{x^{2}-y^{2}}\) orbital is a cloverleaf with four lobes, each pointing along the diagonal direction of the Cartesian coordinate system, lining up with either x or y-axis.

Step-by-step solution

(a) Sketching and Describing the px Orbital

A \(p_{x}\) orbital is a dumbbell-shaped orbital, where the 'lobe' of the dumbbell lies along the x-axis. That is to say, the electron density is higher and symmetrically distributed along the x-axis. Sketch the \(p_{x}\) orbital by drawing two oval shapes (lobes) on either side of the nucleus (represented by a dot) along the x-axis. These lobes should not touch the nucleus. Label the orbital 'px.'

(b) Sketching and Describing the dz² Orbital

A \(d_{z^{2}}\) orbital consists of a torus or doughnut-shaped region in the xy-plane surrounding the atom's nucleus, with two lobes along the z-axis. To sketch this, draw two oval lobes along the z-axis - one above and one below the nucleus. Both of these lobes should not touch the nucleus. Then, draw a torus shape in the xy-plane, around the nucleus. Label the orbital 'dz².'

(c) Sketching and Describing the d(x²-y²) Orbital

The \(d_{x^{2}-y^{2}}\) orbital is shaped like a cloverleaf with four lobes. Each lobe points along the diagonal direction of the Cartesian coordinate system, lining up with either x or y-axis. To sketch the \(d_{x^{2}-y^{2}}\) orbital, draw four elongated lobes - two pointing in the positive and negative x-axis directions, and the other two pointing in the positive and negative y-axis directions. They should all intersect at the origin (the nucleus). Label the orbital 'd(x²-y²).' By analyzing and sketching these three atomic orbitals, we can better understand their shapes and orientations in 3D space. This knowledge is fundamental in understanding the electronic structure of atoms in chemistry and physics.

Key concepts

px Orbital

The px orbital is a fascinating structure belonging to the family of p orbitals, which includes three types: px, py, and pz. These orbitals have distinct shapes and orientations that contribute to the overall electronic configuration of an atom.
In the case of the px orbital, its unique feature is the dumbbell shape, where two lobes extend symmetrically along the x-axis. This shape represents regions where the probability of finding an electron is high. The lobes are separate and do not meet at the nucleus, which sits at the mid-point between them. It's as if the px orbital is hugging the x-axis, with each lobe reaching out but not quite touching the center, which is the nucleus.
To sketch the px orbital, draw two oval lobes stretching away from each other along the x-axis. These lobes reflect the distribution of electron probability, showing that electrons are more likely to be found the further they are from the nucleus, yet within these specific regions. Through understanding the px orbital's shape and orientation, chemistry students can gain insights into how these orbitals overlap to form bonds.

dz² Orbital

The dz² orbital stands out among the d orbitals due to its unique shape and intriguing design. This orbital is not just about lobes; it incorporates a toroidal or doughnut shape that sets it apart.
The foundational feature of the dz² orbital is its pair of lobes aligned along the z-axis, which is vertical in a typical coordination system. These lobes represent areas where electron density is higher and extend above and below the nucleus. But what makes it particularly interesting is the torus or doughnut-shaped region that sits in the xy-plane, surrounding the nucleus.
When sketching the dz² orbital, begin by drawing two lobes on the z-axis - one pointing upwards and another downwards. Then add the torus encircling the nucleus in the xy-plane. While it may look complex, this shape contributes significantly to the character and bonding abilities of atoms, particularly in transition metals. The dz² orbital illustrates how complex spatial arrangements can arise from simple spherical harmonics.

dx²-y² Orbital

The dx²-y² orbital is another member of the d orbital family, recognizable by its distinct cloverleaf shape. It is named for its relation with the x and y axes but arranged at an angle that is diagonal to these axes.
Its standout feature is the four lobes, each extending along the diagonal directions relative to the x and y axes. This lobe placement results in a shape that looks like a four-leaf clover, hence its nickname. These lobes intersect at the nucleus, resembling a pattern that's centered and symmetrical in the xy-plane.
To sketch the dx²-y² orbital, depict four lobes aligned diagonally with the x and y axes. Each lobe stretches from the central nucleus toward the diagonal, making this configuration particularly suited for forming strong directional bonds, as often seen in complex metallic structures.
Understanding the dx²-y² orbital enriches students' comprehension of atomic structure, illustrating how electrons distribute themselves in complex geometric patterns to balance and stabilize atom's energy.