Question

Draw the shapes (boundary surfaces) of the following orbitals: (a) \(2 p_{y}\), (b) \(3 d_{z^{2}}\), (c) \(3 d_{x^{2}-y^{2}}\). (Show coordinate axes in your sketches.)

Short answer

The \(2p_y\) orbital is dumbbell-shaped along the \(y\)-axis, \(3d_{z^2}\) features a donut and lobes along \(z\), and \(3d_{x^2-y^2}\) has four lobes in the \(xy\)-plane.

Step-by-step solution

Understanding Orbital Shapes and Notation

Orbitals are regions around an atom where electrons are likely to be found. The notation, such as \(2p_y\), \(3d_{z^2}\), and \(3d_{x^2-y^2}\), tells us the principal quantum number and the type (shape) of the orbital. \(2\) refers to the 2nd energy level; \(p\) and \(d\) refer to specific shapes, and subscripts like \(y\) or \(z^2\) specify orientation.

Drawing the 2p_y Orbital

The \(2p_y\) orbital is a dumbbell-shaped orbital aligned along the \(y\)-axis. Sketch two elongated lobes on opposite sides of the origin along the \(y\)-axis. Make sure to show the coordinate axes and label the \(x\), \(y\), and \(z\) axes.

Drawing the 3d_{z^2} Orbital

The \(3d_{z^2}\) orbital has a unique shape often described as a donut (or ring) surrounding two elongated lobes along the \(z\)-axis. Draw the donuts around the origin and position two lobes along the \(z\)-axis, one lobe above and one below the donut. Include and label the coordinate axes.

Drawing the 3d_{x^2-y^2} Orbital

The \(3d_{x^2-y^2}\) orbital has a clover-like shape defined primarily by its lobes. This orbital has four lobes that lie in the \(xy\)-plane oriented along the \(x\) and \(y\) axes. Sketch four lobes with the coordinate axes intersecting in between them and label the axes as you did in previous sketches.

Key concepts

Orbital Shapes

Atomic orbitals describe where an electron is likely to be found around a nucleus. These orbitals have specific shapes and orientations. For instance, the "p" orbitals are known for their distinctive dumbbell shapes which are oriented along different axes like the x, y, or z-axis, hence the names such as \(2p_x\), \(2p_y\), and \(2p_z\).

The "d" orbitals are even more diverse. There are five types of "d" orbitals featuring different forms. Some d orbitals, such as \(3d_{z^2}\), have a complex shape that includes lobes and a donut-like ring. Others, like \(3d_{x^2-y^2}\), have a four-leaf clover formation positioned primarily on the xy-plane. The unique configuration of each orbital arises from the quantum mechanical principles that govern the movement and positioning of electrons within atoms.

Being aware of these shapes is crucial because they can predict atom behavior in chemical reactions.

Quantum Numbers

Quantum numbers provide a comprehensive description of an electron within an atom, detailing the energy level, shape, and orientation of an electron's orbital, among other properties. In many ways, they act like an address system for electrons.

There are four main types of quantum numbers:

• **Principal Quantum Number (n)**: This dictates the energy level of the orbital. For instance, in a \(2p\) orbital, the number \'2\' represents the second energy level.
• **Angular Momentum Quantum Number (ℓ)**: This determines the shape of the orbital. For \(p\) orbitals, ℓ = 1; for \(d\) orbitals, ℓ = 2.
• **Magnetic Quantum Number (m_ℓ)**: This indicates the specific orientation of the orbital in space relative to the three axes (x, y, and z).
• **Spin Quantum Number (m_s)**: This specifies the electron spin direction within the orbital, which can be either +1/2 or -1/2.

Understanding quantum numbers is key to determining how electrons are distributed in an atom, which directly affects the atom's chemical properties.

Electron Configuration

Electron configuration describes the arrangement of electrons in the orbitals of an atom. It unfolds how electrons are distributed among the atomic orbitals and energy levels. This organization is crucial because it influences the chemical properties and reactivity of the element.

Electrons follow a specific order when filling orbitals, respecting the Aufbau principle which states that lower energy levels fill before higher ones. The typical electron configuration notation uses numbers to represent energy levels and letters like "s", "p", and "d" to represent the orbital types, followed by a superscript showing the number of electrons in those orbitals. For instance, the configuration \(1s^2 2s^2 2p^6\) indicates a filled "s" orbital in the first shell and filled "s" and "p" orbitals in the second shell.

This systematic filling affects how atoms bond and interact with one another, dictating outcomes in chemical reactions.

Coordinate System

The coordinate system is essential when visualizing and sketching atomic orbitals, such as those mentioned in the original exercise (e.g., \(2p_y\), \(3d_{z^2}\), and \(3d_{x^2-y^2}\)). It provides a spatial reference that helps illustrate how these orbitals exist in three-dimensional space.

Typically, we use a Cartesian coordinate system, which consists of the x, y, and z axes:

• The **x-axis** runs horizontally from left to right.
• The **y-axis** runs vertically from bottom to top.
• The **z-axis** extends forward and backward, often depicted coming out of or into a page or screen.

In orbital sketches, different orbitals align along particular axes, such as x, y, or z. By accurately positioning these orbitals using the coordinate system, we can better understand their orientation and relation to each other, aiding in predictions of molecular shape and behavior. Understanding the coordinate system helps simplify what might first seem like complex three-dimensional shapes.