Question

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

Short answer

a) The s orbital is a symmetric spherical shape centered around the nucleus. b) The p_z orbital is a dumbbell shape oriented along the z-axis, with two equally sized lobes and a nodal plane at the nucleus. c) The d_xy orbital is a cloverleaf shape with four lobes in the xy plane, symmetric around the nucleus, with nodal plane at the nucleus and two nodal lines on the x and y axes.

Step-by-step solution

a) Sketching the s orbital

The s orbital is a spherical orbital that represents the lowest energy state in an atom. It is also the simplest type of orbital to sketch, as it is symmetric about the nucleus. 1. Draw a small circle to represent the nucleus of the atom. 2. Draw a larger circle around the nucleus, with its center at the nucleus. This circle represents the boundary of the s orbital, where the electron is most likely to be found. 3. Shade in the area between the nucleus and the outer circle to show the distribution of the electron in the s orbital. The shading should be uniform since the electron density is the same in all directions. The result is a simple, symmetric spherical shape around the nucleus, which represents the s orbital.

b) Sketching the p_z (p orbital aligned to the z-axis)

The p orbitals are more complex than the s orbitals, and they have a dumbbell shape. There are three different p orbitals (pₓ, pᵧ, p_z), and they are oriented along the x, y, and z axes, respectively. In this case, we will sketch the p_z orbital, which is aligned along the z-axis. 1. Draw a small circle to represent the nucleus of the atom. 2. Draw the x, y, and z axes such that the nucleus is at the origin. Be sure to label the z-axis since we are focusing on the p_z orbital. 3. Draw a dumbbell shape along the z-axis with two lobes, one on the positive side and one on the negative side. The lobes should be elongated and symmetrical with respect to the nucleus. 4. Shade in the lobes to represent the electron density within the p_z orbital. The shading should be uniform within each lobe, with a nodal plane (an area of zero electron density) at the nucleus, where the two lobes meet. The final sketch should show a dumbbell-shaped p_z orbital oriented along the z-axis, with two equally sized lobes.

c) Sketching the d_xy orbital

The d orbitals are even more complex than the p orbitals and have various shapes. There are five different d orbitals, and in this case, we will sketch the d_xy orbital. The d_xy orbital lies in the xy plane and has a cloverleaf shape with four lobes. 1. Draw a small circle to represent the nucleus of the atom. 2. Draw the x and y axes such that the nucleus is at the origin. 3. Draw the d_xy orbital as four lobes in the xy plane, with each lobe located in a different quadrant between the x and y axes and symmetric around the nucleus. 4. Shade in the lobes to represent the electron density within the d_xy orbital. The shading should be uniform within each lobe, with a nodal plane (an area of zero electron density) at the nucleus, and two nodal lines on the x and y axes. The final sketch should show a four-lobed, cloverleaf-shaped d_xy orbital oriented in the xy plane.

Key concepts

S Orbital

The s orbital represents the fundamental level where electrons can reside in an atom. It sports a simple spherical shape, reflecting the symmetrical distribution of electron density around an atom's nucleus. Imagine a perfectly even cloud of negative charge embracing the positive nucleus, where the probability of finding an electron is equal at all points on the sphere's surface. Despite the uniform appearance, the electron is not still but darts around within this spherical space in a dynamic fashion. The s orbital is unique in having no nodal planes – areas where the probability of finding an electron is zero – showcasing its simplicity in structure compared to more complex orbitals.

P Orbital

On the more complex end, the p orbitals diverge into three orientations, each aligning with a different axis: x, y, and z. Unlike the s orbital, p orbitals have a distinctive dumbbell shape, with a node—a region of zero electron density—at the nucleus. Focusing on the p_z orbital, it aligns with the z-axis, protruding electron density in two opposing lobes. One extends along the positive z-axis and the other in the negative direction, resembling a pair of weights held in balance on a weight scale, with the atom's nucleus at the center fulcrum. This nodal plane is significant as it divides the orbital into two, a stark contrast to the node-free s orbital.

D Orbital

D orbitals introduce an intricate dance of electron density and shape. Among the five d orbitals, the d_xy orbital is especially fascinating with its cloverleaf pattern. Positioned within the xy plane and slicing through four quadrants, the orbitals feature four lobes where the likelihood of finding an electron peaks. Moreover, d orbitals introduce not just nodal planes but nodal lines as well—the d_xy orbital has two, coinciding with the x and y axes. This arrangement results in a striking visual of electron density peaks and troughs, offering a complex environment for electron habitation.

Electron Density

The term electron density describes the probability of finding an electron in a given space around the nucleus. It’s the backbone of orbital shapes, as these regions are characterized by varying electron densities. In s orbitals, the density is uniform, manifesting in a spherical configuration, but in p and d orbitals, the density is spatially variable, leading to more elaborate shapes like dumbbells and clovers. Understanding electron density is crucial—it's a map that predicts the electron's whereabouts, determining atomic bonding and property manifestations.

Nodal Planes

The concept of nodal planes is linked with the quantum mechanics of electron orbitals. These are regions where there is zero probability of finding an electron, acting as boundaries of electron presence. While the s orbital is devoid of nodal planes, p orbitals have one, and d orbitals are accompanied by an intricate combination of planes and lines. The presence of nodal planes affects the chemical bonding potential and reactivity of an element, as they stand as zones bereft of electron activity within otherwise electron-rich spaces.