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

The main types of orbitals we will focus on are the s, p, and d orbitals. The s orbitals are spherical, while the p orbitals are dumbbell-shaped and occur in three orientations corresponding to the x, y, and z axes. The d orbitals have complex shapes and a total of five orientations.

The s orbital is spherical in shape. Start by drawing a circle, which represents the shape of the s orbital where the electron can be found surrounding the nucleus. Arriving at the representation of the s orbital, an example for the shape of the 1s orbital would be: \[\text{1s orbital:} \circ\]

The pz orbital is a type of p orbital oriented along the z-axis. It has a dumbbell shape. Start by drawing a dumbbell shape where the two lobes are centered along the z-axis. The lobes represent regions where electrons can be found. Draw a small circle (representing the nucleus) between the two lobes along the z-axis. The final representation of the pz orbital would look like: \[\text{pz orbital:} \bullet \longleftrightarrow \bullet\]

The dxy orbital is one of the d orbitals, which are more complex in shape than the s and p orbitals. Draw the dxy orbital as four lobes located in the xy-plane, between the axes rather than on them. Each of the four lobes are oriented between x and y axes. Draw a small circle in the center to represent the nucleus. The final representation of the dxy orbital would look like: \[\text{dxy orbital:} \begin{array}{|c|} \hline \diagup \hspace{5mm}\\\hspace{5mm} \bullet \hspace{5mm} \\ \diagdown \hspace{5mm}\\ \hline \end{array} \hspace{5mm} \begin{array}{|c|} \hline \diagdown \hspace{5mm}\\\hspace{5mm} \bullet \hspace{5mm} \\ \diagup \hspace{5mm}\\ \hline \end{array} \] In summary, the s orbital is spherical, the pz orbital has a dumbbell shape along the z-axis, and the dxy orbital is composed of four lobes in the xy-plane situated between the x and y axes.