Question

Sketch the following orbitals (including the \(x, y\), and \(z\) axes): \(1 s, 2 p_{x}, 3 d_{x y}, 3 d_{z} 2\).

Short answer

The 1s orbital is a sphere centered at the origin; the 2p_x orbital is a dumbbell along the x-axis; the 3d_{xy} orbital is clover-shaped in the xy-plane; the 3d_{z^2} orbital has a dumbbell along the z-axis with a doughnut-shaped ring in the xy-plane.

Step-by-step solution

Sketch the 1s Orbital

The 1s orbital is a spherical shape representing the space where there's a high probability of finding an electron. Sketch a circle or sphere centered at the origin of the coordinate system where your axes intersect.

Sketch the 2p_x Orbital

The 2p_x orbital has a dumbbell shape and is oriented along the x-axis. Sketch two symmetrical lobes lying along the x-axis, with the node, where the probability of finding an electron is zero, at the origin.

Sketch the 3d_{xy} Orbital

The 3d_{xy} orbital is shaped like a four-leafed clover lying on the xy-plane. Draw two lobes above the xy-plane and two lobes below the xy-plane, with the lobes centered on the lines x=y and x=-y, intersecting at the origin.

Sketch the 3d_{z^2} Orbital

The 3d_{z^2} orbital has a distinct shape with a dumbbell along the z-axis and a ring around the xy-plane. Draw the dumbbell with its lobes along the z-axis and the doughnut-shaped ring in the xy-plane around the nucleus at the origin.

Key concepts

Orbitals Sketching

Understanding atomic orbitals begins with knowing how to sketch them. These sketches are visual tools that help us comprehend where electrons are likely to be found around an atom. A fundamental knowledge of the coordinate system is beneficial, as orbitals are often described in relation to the Cartesian axes ( x, y, z). When sketching, keep in mind that each orbital shape represents a three-dimensional region with a particular electron probability distribution. For example, an 's' orbital appears spherical in sketches, while 'p' orbitals have a dumbbell form. The 'd' orbitals may resemble a four-leaf clover or have more complex shapes. These sketches are not exact depictions but conceptual aids for visualizing an orbital's geometry.

1s Orbital

The 1s orbital is the simplest to visualize. It represents the electron cloud for an electron in the lowest energy state of an atom. If we could measure where an electron is over time, we'd find it within this spherical space most frequently. In essence, the size of the sphere in a 1s orbital sketch corresponds to the effective 'boundary' within which an electron's presence is most probable. The nucleus—or the atom's center—coincides with the center of this sphere.

2px Orbital

Moving on to the 2px orbital, we see a shift from spherically symmetrical shapes to more directional ones. Specifically, the 2px orbital has a dumbbell shape, with two lobes on either side of the atomic nucleus, extending along the x-axis. The space between the lobes, at the nucleus, is known as a nodal plane where the probability of finding an electron drops to zero. These orbitals reflect the higher energy and complexity of the electron behavior as we move away from the nucleus.

3d Orbitals

The 3d orbitals are more complex. These orbitals are characterized by their distinctive shapes and orientations. For instance, the 3dxy orbital looks like a four-leaved clover lying flat on the xy-plane. The leaves of this 'clover' point toward the corners of a square centered on the nucleus. For the 3dz^2 orbital, the shape is like a dumbbell aligned with the z-axis and a donut-shaped ring lying in the xy-plane. These orbitals indicate regions where there's a significant chance of finding an electron in atoms with more than ten electrons.

Electron Probability

Electron probability, or electron density, is at the heart of how we understand atomic orbitals. It quantifies the likelihood of finding an electron in a specific region around the nucleus. Orbitals are drawn as surfaces that encompass a high probability region—often 90% or more of where an electron could be. More technically, the shapes of orbitals are determined by the solutions to the Schrödinger equation for electrons in atoms, which give us the electron probability distributions.

Atomic Orbital Shapes

Lastly, the fascination with atomic orbital shapes is due to their determining role in an atom's chemistry. Each type of orbital—s, p, d, and f—has a unique orientation and shape, influencing how atoms bond and interact. The 's' orbitals are spherical; 'p' orbitals are dumbbell-shaped with nodal planes; 'd' orbitals come in four-leaf clover shapes or with rings and lobes, and 'f' orbitals are even more complex. The spatial orientation of these orbitals is critical when it comes to the formation of chemical bonds and the properties of molecules.