In this explainer, we will learn how to recognize atomic orbitals based on their shape and describe their relationship with quantum numbers.
We used to think that electrons were nothing more than simple point particles. The quantum physicists caused a paradigm shift when they showed that we cannot measure the position () and the momentum () of a particle with absolute precision. We do not tend to describe electrons as point particles that are located at a single point in space anymore. Instead, we describe electrons with mathematical expressions that describe the most likely location of an electron in an atom. Such mathematical expressions are known as atomic orbitals.
Definition: Atomic orbital
Atomic orbitals are three-dimensional mathematical expressions that describe the most likely location of an electron in an atom.
There are lots of different types of atomic orbitals but the four most relevant to current chemistry are the s, p, d, and f orbitals. The shapes and orientations of these atomic orbitals are all different. Each orbital can be labeled by a number that indicates the energy level it is associated with and therefore its energy. For example, the 1s orbital is of lower energy than the 4f orbital. The 1s orbital has a relatively low energy because it has a principal quantum number of one () and the 4f orbital has a higher energy because it has a principal quantum number of four ().
The shape and size of any atomic orbital are determined from its quantum numbers. The principal quantum number () determines the size of the atomic orbital and the subsidiary quantum () number determines the shape of the atomic orbital. The magnetic quantum number () determines the direction of the atomic orbital.
An s orbital has a spherical shape. The size of an s orbital depends on the value of its principal quantum number. s orbitals are relatively small when they have a low principal quantum number and they are wide when they have a high principal quantum number. s orbitals have a subsidiary quantum number that is equal to a value of zero (). The figure below shows the appearance of the 1s, 2s, and 3s atomic orbitals.
The structure of the 1s, 2s, and 3s atomic orbitals was determined from their electron density functions. This is explained in the following figure. The electron density functions are shown as graphs that are linked to corresponding atomic orbital diagrams. The shapes of the atomic orbitals are determined as the electron density functions are mapped onto three-dimensional space. This same procedure can be used to determine the shape of any other type of atomic orbital. The shape of any one atomic orbital is determined as its electron density graph is mapped onto three-dimensional space.
The following figures also show that the electron density functions sometimes have a value of zero. These parts of electron density graphs correspond to areas of zero electron density. They basically correspond to areas in the three-dimensional orbital diagrams where there is no electron density whatsoever. These areas are called nodes. Nodes are areas of atomic orbitals that are completely empty. The figure shows that the 2s atomic orbital has one node and the 3s orbital has two nodes. The 2s orbital has one area of empty space in between areas of high electron density. The 3s orbital has two areas of empty space in between areas of high electron density.
Example 1: Identifying the Probability Distribution Function of the 1s Atomic Orbital
The graph shows the probability of finding an electron at a distance from the nucleus for the first three s orbitals of an atom of hydrogen. Which plot corresponds to the 1s orbital?
The electron density function of the 1s orbital has one maximum value. The electron density functions of the 2s and 3s orbitals have two and three maximum values respectively. The figure shows the 1s, 2s, and 3s electron density functions and we have to determine which is which. The blue line must be for the 1s electron density function because it has one maximum value. We can use this line of reasoning to determine that option A is the correct answer. Furthermore, plot A contains a maximum value at a distance closest to the nucleus. Since 1s orbitals are located closest to the nucleus, then we can reaffirm that the correct answer is option A.
The other s-type atomic orbitals have a similar spherical appearance. They are all made up of concentric spheres of electron density that are separated from each other by (radial) nodes. It is relatively simple to determine how many electrons there are in any one s-type subshell because all s-type subshells contain a single atomic orbital and a single atomic orbital can only hold two electrons. We can use this line of reasoning to state that all s-type subshells can hold two electrons. The 1s subshell can hold two electrons. The 2s subshell can hold two electrons. The 3s subshell can hold two electrons, and so on. The following figure shows that the s-type subshells have one atomic orbital whereas the p-type subshells have three atomic orbitals.
Example 2: Identifying Atomic Orbitals from a Simple Figure of a Sphere
Which of the following atomic orbitals is shown in the image?
The low-energy and high-energy s-type atomic orbitals are relatively easy to identify because they have a characteristic spherical shape. The s-type atomic orbitals contain one sphere of electron density or several concentric spherical layers of electron density. The other types of atomic orbitals have much more complex three-dimensional shapes. The figure shows an orbital that must be an s-type atomic orbital because it has a relatively simple spherical shape. We can use this line of reasoning to determine that option D is the correct answer for this question.
The p-type atomic orbitals have more complex shapes. All of the p-type subshells can hold a total of six electrons because they are made up of three atomic orbitals and each atomic orbital can hold two electrons. The following figure shows the shape of the , , and atomic orbitals that make up the 2p subshell. Each one of the 2p orbitals is made up of two lobes of electron density, with a node at the point where they meet. We can describe the orientation of each p orbital as being along the -, -, or -axis in space. Each p orbital is therefore perpendicular to the others.
p orbitals have a subsidiary quantum number that is equal to a value of 1, so . The orientation of each orbital can be described by the magnetic quantum number (). The magnetic quantum number can have any integer value from to . As the value for for a p orbital is 1, then the values for are , 0, and , corresponding to the , , and orbitals.
Example 3: Identifying an Atomic Orbital from Its Quantum Numbers
What name is given to an atomic orbital with the quantum numbers , , and ?
Subshell notation terms are written to indicate the values of the principal quantum number () and the subsidiary quantum number (). The subshell notation terms are made up of one number and one letter. The number is used to represent the principal quantum number and the letter is used to represent the subsidiary quantum number.
The letter s represents the situation where the subsidiary quantum number is equal to zero (), and the letter p represents the situation where the subsidiary quantum number is equal to one ().
The 2p subshell term is made up of the number 2 and the letter p. This means that the 2p subshell has a principal quantum number of two () and a subsidiary quantum number of one (). We can use these statements to determine that option B is the correct answer for this question.
The d-type atomic orbitals cannot all be described the same way, because one of the d-type atomic orbitals () always looks different from the other d-type atomic orbitals of that subshell. Each d-type subshell contains five atomic orbitals. The d subshells can all hold up to ten electrons because they are made up of five atomic orbitals. Four of the d orbitals can be described as looking like a four-leaf clover and the fifth d orbital can be described as looking like a donut with lobes on either side. The following image shows the five atomic orbitals that make up the 3d subshell.
The aufbau principle states that electrons fill the lowest-energy atomic orbitals before they can fill any higher-energy atomic orbitals. The following figure represents the aufbau principle.
The figure shows that the electrons of any element fill the lowest-energy atomic orbitals before they fill the higher-energy atomic orbitals. Hydrogen atoms have a single electron and lithium atoms have three electrons. The highest-energy (and only) electron of a hydrogen atom must be found in the lowest-energy 1s atomic orbital. The highest-energy electron of a lithium atom must be found in the 2s atomic orbital because its other two electrons can fill the lowest-energy 1s atomic orbital.
Definition: Aufbau Principle
The aufbau principle states that electrons fill the lowest-energy atomic orbitals before they fill higher-energy atomic orbitals.
The aufbau principle can effectively be mapped onto the periodic table to explain why periodic table blocks have similar chemical properties. The mapping of the aufbau principle shows that periodic table blocks are made up of chemical elements that have their highest energy electrons in the same type of atomic orbitals. The group one and group two elements have similar chemical properties because their highest energy electrons fill s-type atomic orbitals. The halogen elements have similar chemical properties because their highest energy electrons fill p-type atomic orbitals.
Example 4: Identifying the Highest Occupied Orbital in an Atom of Boron
What is the highest occupied orbital in an atom of boron?
The electrons of any one chemical element will fill atomic orbitals according to the aufbau principle. Electrons will first fill the low-energy atomic orbitals and then they will fill the higher-energy atomic orbitals.
Boron atoms have a total of five electrons. Four of the electrons in each boron atom have to fill the lowest-energy 1s and 2s subshells. The other electron must be in the 2p subshell because the 2p subshell has the third-lowest energy of any atomic orbital. We can use this line of reasoning to determine that option B is the correct answer for this question.
The number and type of valence electrons in any atom can be determined from its electron configuration. For example, carbon and oxygen have the and electron configurations and they have four and six valence electrons. The following table shows the number of valence electrons for the first ten elements of the periodic table.
|Atomic Number||Element Symbol||Electron Configuration||Number of Valence Electrons|
Valence electron atomic orbitals can be combined together to make molecular orbitals. The s-type valence electrons of one atom can be combined with the s-type valence electrons of a second atom to make one bonding molecular orbital and one antibonding molecular orbital.
Bonding molecular orbitals strengthen a chemical bond between two atoms because they lower the energy of the molecule relative to the separated atoms. Antibonding molecular orbitals weaken a chemical bond between two atoms because they raise the energy of the molecule relative to the separated atoms.
Definition: Molecular Orbital
Molecular orbitals are mathematical functions that describe the location and wave-like behavior of an electron in a molecule.
p-type orbitals can also interact with each other and produce molecular orbitals. Two atomic orbitals can overlap with each other and produce one bonding molecular orbital and a second antibonding molecular orbital.
Example 5: Determining Which Atomic Orbitals Have Combined Together to Make a Molecular Orbital
Molecular bonds, the chemical bonds in molecules, are formed from the overlap of atomic orbitals. Which two atomic orbitals’ overlap is likely to have formed the molecular orbital shown?
- f orbitals
- p orbitals
- d orbitals
- s orbitals
The figure shows a molecular orbital. The molecular orbital resembles the bonding orbital that is formed through the end-to-end overlap of two adjacent atomic orbitals. We can use these statements to conclude that the molecular orbital was formed through the end-to-end overlap of two p-type atomic orbitals. This suggests that option B is the correct answer for this question.
Let us summarize what we have learned in this explainer.
- Atomic orbitals are three-dimensional mathematical expressions that describe the most likely location of an electron in an atom.
- Atomic orbitals have different shapes.
- The s-type atomic orbitals are shaped like spheres and the p-type atomic orbitals are shaped like a dumbbell.
- The appearance of an atomic orbital is determined from its quantum numbers.
- The electrons of any one chemical element fill low-energy atomic orbitals before they fill higher-energy atomic orbitals.
- Atomic orbitals can be combined together to make molecular orbitals.