Question

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

Short answer

The \(p_x\) orbital has a dumbbell shape aligned along the x-axis. The \(d_{z^2}\) orbital has a dumbbell shape along the z-axis with a donut-shaped region in the xy plane surrounding the nucleus. The \(d_{x^2-y^2}\) orbital consists of four lobes, each located in one of the quadrants of the xy plane, with the lobes aligned along the x and y-axes.

Step-by-step solution

Describing the px orbital

The \(p_x\) orbital is one of the three \(p\) orbitals, which have a dumbbell shape. The \(p_x\) orbital lies along the x-axis, with the nucleus at the origin of our coordinates. The two lobes of the \(p_x\) orbital are on opposite sides of the nucleus, aligned along the x-axis.

Sketching the px orbital

To sketch the \(p_x\) orbital, first draw the x-axis and place the nucleus at the origin (0, 0). Now, draw a dumbbell shape with two lobes, one on each side of the nucleus, such that they are aligned along the x-axis.

Describing the dz^2 orbital

The \(d_{z^2}\) orbital is one of the five \(d\) orbitals, which have more complex shapes than the \(p\) orbitals. The \(d_{z^2}\) orbital has a shape resembling a dumbbell along the z-axis, but with an additional donut-shaped region in the xy plane around the nucleus.

Sketching the dz^2 orbital

To sketch the \(d_{z^2}\) orbital, first draw the x, y, and z-axes, with the nucleus at the origin (0, 0, 0). Then, draw a dumbbell shape aligned along the z-axis, with one lobe above and one lobe below the nucleus. Finally, add a donut-shaped region in the xy plane, surrounding the nucleus.

Describing the dx^2-y^2 orbital

The \(d_{x^2-y^2}\) orbital is another one of the five \(d\) orbitals. The shape of the \(d_{x^2-y^2}\) orbital consists of four lobes, each located in one of the quadrants of the xy plane, with the lobes aligned along the x and y-axes.

Sketching the dx^2-y^2 orbital

To sketch the \(d_{x^2-y^2}\) orbital, first draw the x, y, and z-axes, again with the nucleus at the origin (0, 0, 0). Then, draw four lobes in the xy plane, one in each quadrant. Make sure that the lobes are symmetrical and aligned along the x and y-axes.

Key concepts

Orbital Shapes

Understanding the shape of atomic orbitals is fundamental in grasping the basics of quantum chemistry and how electrons behave in atoms. Orbitals are regions in space where there is a high probability of finding an electron. Each type of orbital has a distinctive shape that indicates the spatial distribution of electron density.

The simplest type is the s orbital, which is spherical. As we move to more complex orbitals, such as p and d orbitals, the shapes become more elaborate. The shapes of these orbitals are not just for theoretical interest; they have direct consequences on the chemical behavior of atoms, including the angles at which atoms bond and the types of bonds that can form.

P Orbitals

P orbitals are one of the first steps beyond the simplest s orbitals in understanding electron configuration. They have a peculiar dumbbell shape, with each lobe representing the area where an electron is likely to be found. There are three p orbitals in every energy level above the first, denoted as p_x, p_y, and p_z, oriented along the three axes of 3D space.

The p_x orbital, for instance, extends along the x-axis on both sides of the nucleus, having two lobes with a node at the nucleus where the probability of finding an electron is zero. These p orbitals can hold a maximum of six electrons in total, and their orientation in space influences the shape and physical properties of molecules.

D Orbitals

D orbitals, present from the third energy level onwards, introduce even more complexity to the understanding of electron distribution within an atom. There are five d orbitals, labeled as d_z², d_x2-y2, d_xy, d_xz, and d_yz. These labels give us an indication of the orbital shapes and the axes along which they are aligned.

The d_z² orbital is characterized by a clover shape along the z-axis with a doughnut-shaped region in the xy plane. In contrast, the d_x2-y2 orbital has four lobes that lie in the xy plane and extend along the x and y axes, forming a shape similar to a four-leaf clover. These complex shapes allow for a variety of bonding patterns, which are crucial in fields such as coordination chemistry.