Question

Sketch a three-dimensional representation for each of the following orbital sets. Label the \(x\) -axis, \(y\) -axis, and \(z\) -axis. (a) \(1 s, 2 s, 2 p_{x}\) (b) \(3 p_{x}, 3 p_{Y}, 3 p_{z}\)

Short answer

Sketch spheres for the 1s and 2s orbitals and dumbbells along their respective axes for the 2px, 3px, 3py, and 3pz orbitals.

Step-by-step solution

Understand the Orbital Types

Recognize that the problem involves sketching atomic orbitals. The spherical orbitals (
s and 2s ) are non-directional, while the p orbitals (2p_x , p_x ,p_y , and p_z ) are dumbbell-shaped and oriented along specific axes.

Sketch the 1s and 2s Orbitals

Both
s and 2s orbitals are spherical, centered on the origin of the axes. The
s orbital is smaller than the 2s orbital, so sketch a small sphere for
s and a larger sphere for 2s. Label the axes with x , y , and z .

Sketch the 2px Orbital

The 2p_x orbital is shaped like a dumbbell oriented along the x -axis. Draw two lobes symmetrically extending from the origin along the x -axis. Ensure the lobes are labeled to denote the orientation along x.

Sketch the 3px, 3py, and 3pz Orbitals

For the 3p_x orbital, draw a larger dumbbell shape along the x -axis. Similarly, draw the p_y orbital with its dumbbell along the y -axis and the p_z orbital along the z -axis. These should resemble the 2p orbitals but be larger, as they are in a higher energy level (n=3).

Key concepts

Spherical Orbitals

Spherical orbitals refer to the atomic orbitals characterized by their spherical shape. These orbitals are classified as "s" orbitals and include the 1s, 2s, 3s, etc., corresponding to their principal quantum numbers.

• 1s Orbital: The 1s orbital is the simplest type of spherical orbital. It surrounds the nucleus symmetrically and is the closest orbital to it. This means it also has the lowest energy level.
• 2s Orbital: Unlike the 1s orbital, the 2s orbital is larger, extending further from the nucleus. While it's still spherical, its larger radius contributes to higher energy compared to the 1s orbital.

These spherical orbitals are not directional, meaning they don't orient along any specific axis. Due to their shape, electrons in these orbitals are equally probable to be found in any direction from the nucleus. In sketches, these are represented as circles encompassing the origin of the axes, with the 2s drawn as a larger circle than the 1s due to its higher energy level. Such sketches help visualize how atomic orbitals differ in size and energy levels as they move further from the nucleus.

Dumbbell-Shaped Orbitals

Dumbbell-shaped orbitals, also known as "p" orbitals, are oriented along specific axes and represent orbitals starting from the second principal quantum level ( =2 and higher). These orbitals include 2p, 3p, etc., and are characterized by their distinctive shape.

• 2p Orbitals: The 2p orbitals consist of three distinct orbitals named 2p_x, 2p_y, and 2p_z. Each of these orbitals is oriented along the respective x, y, and z axes. The dumbbell shape features two lobes, one lobe extending along the positive axis, and the other along the negative axis, with the nucleus located at the node, where there is a zero probability of finding electrons.
• 3p Orbitals: The 3p orbitals follow the same orientation convention as the 2p orbitals but have larger sizes due to their placement at a higher energy level. These orbitals maintain the dumbbell shapes fixed along the axes, but with an increased probability zone for electrons due to their larger size.

When sketching these orbitals, the axis alignment is crucial. Each orbital should be represented with lobes along its specified axis, and when viewed in 3D, they illustrate how electrons are arranged within specific energy levels based on their quantum numbers. Understanding their orientation assists in comprehending how atoms interact to form chemical bonds.

Orbital Sketching

Orbital sketching involves creating three-dimensional visual representations of atomic orbitals. This visual approach helps to illustrate the complex nature of electron clouds and how they are distributed in space around a nucleus.

• Axes Labeling: When sketching orbitals, it is vital to label the axes as x, y, and z to help identify the orientation of the orbital. This labeling is essential for correctly depicting p orbitals, which are directionally dependent.
• Sketching Spherical Orbitals: For s orbitals, begin by drawing circles centered on the origin. Use size to distinguish between different s orbitals; smaller for 1s and larger for 2s.
• Sketching Dumbbell-Shaped Orbitals: Draw each p orbital by creating two lobes extending in opposite directions from a node positioned at the origin. Be attentive to the axis along which each orbital is aligned, such as x for 2p_x or y for 3p_y.

Through orbital sketching, students can better understand how orbitals are not just abstract concepts but actual areas where electrons exist. This understanding provides crucial insight into molecular geometry and electron configurations in chemical elements. Practice with varying sizes and orientations enhances both visual learning and conceptual clarity.