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, the 2p_x orbital has two lobes on the x-axis, the 3d_{xy} orbital has four lobes in the xy-plane, and the 3d_{z^2} orbital has a dumbbell shape along the z-axis with a toroidal region in the xy-plane.

Step-by-step solution

Sketch the 1s Orbital

Draw a sphere centered at the origin of the coordinate system, where the origin represents the nucleus of the atom. The sphere indicates a region of space where there's a high probability of finding an electron in the 1s orbital.

Sketch the 2p_x Orbital

Draw two lobes on either side of the x-axis, with their centers at the origin. The lobes should not touch each other. They represent areas of high electron probability for the 2p_x orbital. Include the x, y, z axes, and remember that the 2p_x orbital is aligned along the x-axis.

Sketch the 3d_{xy} Orbital

Draw two lobes in the shape of a four-leaf clover lying in the xy-plane with the lobes centered at the origin. The lobes should be positioned between the axes, not along them, indicating areas where the electron is likely to be found in the 3d_{xy} orbital.

Sketch the 3d_{z^2} Orbital

Draw a dumbbell shape along the z-axis with a doughnut (torus) around the nucleus in the xy-plane. This shape represents the 3d_{z^2} orbital, where there are two lobes along the z-axis and a ring of high electron density in the xy-plane centered on the nucleus.

Key concepts

s Orbital

Envisioning the s orbital is like imagining a soft cloud surrounding the nucleus of an atom. This cloud represents a three-dimensional area in space where the probability of finding an electron is the greatest. Unlike other orbitals that might have complex shapes, the s orbital is spherical, embodying simplicity in the world of quantum chemistry. It is uniform and doesn't favor any particular direction, which signifies that the electron has an equal probability of being located at any point at a particular distance from the nucleus. This spherical shape is consistent across all s orbitals, such as the 1s orbital, but the size of the sphere increases as you move to higher principal quantum numbers like 2s, 3s, and beyond.

It's crucial to note that each energy level can only have one s orbital, and within it, you can find up to two electrons sharing that space, spinning in opposite directions due to the Pauli exclusion principle.

p Orbitals

Unlike the uniform s orbitals, the p orbitals are more akin to dumbbells, showcasing a directional characteristic that the s orbitals do not possess. For example, a 2p_x orbital showcases a pair of lobes extending in opposite directions along the x-axis, and these lobes are areas where the likelihood of locating an electron is high. Each energy level above n=1 has three p orbitals, dubbed p_x, p_y, and p_z, which are oriented at right angles to each other along the corresponding axes in three-dimensional space.

Orientation and Probability

The orientation of the lobes is related to the angular momentum of the electrons, with a node, or a region of zero probability, at the nucleus. The idea of electron probability is particularly salient in the case of p orbitals where high electron density occurs at a certain distance from the nucleus. This creates a directional probability distribution, explaining the electron's position not just by how far it is from the nucleus but also by pointing out the direction it can be found in relation to the central point of the atom.

d Orbitals

When we escalate to d orbitals, things get even more interesting in terms of shapes and electron probability. d orbitals begin to appear at the third energy level (n=3) and bring five distinct orientations into play: d_xy, d_yz, d_xz, d_x2-y2, and d_z2. These orbitals exhibit more complex shapes, including clover-like forms and even a donut shape mingled with dumbbell-shaped lobes.

Take the 3d_xy orbital, for instance; its clover-shaped lobes lie in the xy-plane and indicate the regions of highest electron probability between the x and y axes, without touching the axes themselves. On the flip side, the 3d_z2 orbital, shows a unique 'dumbbell with a donut' shape. A pair of lobes extends along the z-axis and a torus of electron density encompasses the equatorial plane. These complex patterns signify nuances in electron behavior, as angular and magnetic quantum numbers manifest through space in more sophisticated forms compared to s and p orbitals.

Electron Probability

The heart of quantum chemistry lies in its ability to predict behaviors using probabilities rather than certainties, and this is vividly illustrated in the concept of electron probability. This fundamentally means that until measured, an electron is not in a single, definitive location. Instead, it exists in a state of probability, described by a cloud or shape around the nucleus, where the electron is more or less likely to be found.

These clouds and shapes, which correspond to the s, p, and d orbitals, are not just theoretical constructs but are based on mathematical functions derived from Schrödinger's equation. Each orbital has a characteristic probability distribution or electron density that scientists can calculate. The regions with the highest electron density within these orbitals indicate where it is most probable to find an electron at any given time, aiding chemists in visualizing and understanding electron configurations and chemical bonding patterns.

Quantum Chemistry

Quantum chemistry is the profound field that marries quantum physics with chemistry to explore the behavior of particles at the atomic and molecular levels. It plunges beyond the traditional rules of classical physics, navigating through the subatomic realm where the classical intuitions of how particles should behave no longer hold true. The discipline leverages complex mathematical formulas and principles such as wave functions, perturbation theory, and quantum mechanics to elucidate the arrangements and reactions of electrons within atoms and molecules.

Featuring prominently in quantum chemistry are the energy levels and orbitals, which provide a framework for understanding the spatial distribution and energies of electrons. The visualization and sketching of s, p, and d orbitals are more than an artistic endeavor; they are essential for students and chemists to conceive how electrons inhabit and move around atoms, determining the intrinsic properties of the elements and the way they bond to form the myriad of substances that define our world.