Question

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

Short answer

(a) The px orbital has a dumbbell shape, with one lobe on the positive x side and the other on the negative x side of a Cartesian coordinate system. (b) The d₂² orbital has a "four-leaf clover" shape, with lobes placed diagonally between the x, y, and z axes, specifically in the positive-x positive-y, negative-x positive-y, negative-x negative-y, and positive-x negative-y regions. (c) The dx²-y² orbital has a "four-leaf clover" shape similar to d₂², but with lobes aligned along the x and y axes, with one lobe in the positive-x, negative-x, positive-y, and negative-y regions.

Step-by-step solution

(a) Sketching px Orbital

: To sketch the px orbital, we need to recall that p orbitals have a dumbbell shape. The px orbital is aligned along the x-axis. 1. Draw the x, y, and z-axis on a Cartesian coordinate system. 2. Now sketch a dumbbell shape centered on the x-axis, with one lobe on the positive x side and the other on the negative x side. 3. Label the orbital as px.

(b) Sketching d₂² Orbital

: To sketch the d₂² orbital, recall that d orbitals have a complex "four-leaf clover" shape. The orientation of d₂² will lie along the x, y, and z axes. 1. Draw the x, y, and z-axis on a Cartesian coordinate system. 2. Visualize the d₂² clover-like shape with four lobes placed diagonally between the x, y, and z axes. One lobe should be in the positive-x positive-y region, one in the negative-x positive-y region, one in the negative-x negative-y region, and one in the positive-x negative-y region. 3. Label the orbital as d₂².

(c) Sketching dx²-y² Orbital

: To draw the dx²-y² orbital, which is also a type of d orbital, recall that d orbitals have a complex "four-leaf clover" shape. The shape of the dx²-y² orbital is similar to that of d₂², but with its lobes aligned along the x and y axes. 1. Draw the x, y, and z-axis on a Cartesian coordinate system. 2. Visualize the dx²-y² clover-like shape by placing four lobes around the axis with two lobes aligned along the x-axis and the other two lobes aligned along the y-axis. One lobe should be in the positive-x region, one in the negative-x region, one in the positive-y region, and one in the negative-y region. 3. Label the orbital as dx²-y².

Key concepts

p orbitals

p orbitals are one of the most familiar types of atomic orbitals. They are known for their distinctive dumbbell shape. When thinking about p orbitals, imagine a pair of elongated balloons tied along their midpoints. Each p orbital is oriented along one of the Cartesian coordinate axes: x, y, or z. Thus, we have three types: \(p_x\), \(p_y\), and \(p_z\).
These orbitals are essential because they account for the unique shapes of molecules and their chemical bonding behaviors. The dumbbell lobes of a p orbital create regions where electron density is highest, and this arrangement is crucial for the interactions with other atoms.
The symbol \(p_x\) denotes that the orbital is aligned along the x-axis. This orientation signifies that the two lobes of the p orbital extend equally in the positive and negative x directions. This alignment is key in helping chemists predict and explain how atoms bond and interact.

• The shape: Dumbbell
• Primary axes: x, y, z
• Number of orientations: Three – \(p_x\), \(p_y\), \(p_z\)

d orbitals

d orbitals are more complex than p orbitals. Instead of the simpler shapes like the p orbitals, d orbitals have four-lobed structures often compared to a cloverleaf. This cloverleaf design makes them crucial for describing compounds with metals and for understanding the intricate geometries of transition metal complexes.
There are five d orbitals in total, labeled \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{x^2-y^2}\), and \(d_{z^2}\). Each of these has distinct orientations and shapes, which influence how they interact with other orbitals and atoms. For instance, the \(d_{x^2-y^2}\) orbital has its lobes lying along the axes rather than between them. This peculiar orientation is crucial when considering orbital overlap in chemistry.
In contrast, the \(d_{2}^2\) designation in the original exercise appears to be a typographical error or a non-standard notation, as it doesn't align with the conventional labels. Thus, care should be taken to accurately identify and sketch these orbitals.

• Shape: Cloverleaf
• Number of orientations: Five
• Significance: Important for metal atom bonding

orbital orientation

The orientation of an atomic orbital refers to the direction in which the lobes of the orbital extend in space. This spatial orientation is significant for both the shape and function of the orbital, affecting how it overlaps with orbitals from other atoms during bonding.
p orbitals, with their three orientations (\(p_x\), \(p_y\), \(p_z\)), align along the primary axes of a Cartesian coordinate system. This specific alignment helps in predicting molecular geometry and bonding patterns. As a result, the orientation dictates the directionality of electron density.
d orbitals possess different orientations compared to p orbitals. They are more varied and numerous, with five distinct orientations that include alignments between axes or along them. This variation is crucial for explaining aspects of transition metal chemistry, where electronic configuration plays a vital role.
Understanding orbital orientation is fundamental since it helps chemists visualize and forecast how atomic orbitals will influence molecular shape and bonding. Recognizing these orientations helps demystify complex structures into manageable representations.

• Importance: Determines spatial relationships
• p Orbital orientations: x, y, z
• d Orbital orientations: xy, xz, yz, \(x^2-y^2\), \(z^2\)