Introduction
In this article, we will be looking at six questions that are commonly asked about orbitals. At the same time, those who have no idea what orbitals are can also understand what orbital theory entails through this article. The questions are as follows (in no particular order):
- What are orbitals?
- What types of orbitals are there?
- How do electrons fill the orbitals?
- What are the quantum numbers?
- What are bonding and antibonding orbitals?
- What are degenerate orbitals and how are they important?
Finally, we will also be discussing a sub-question: how many orbitals are there in n = 3, 4 or 5? The letter n, as described here, may not make sense to us as of now, but we will see what they mean later. (This sub-question also counts as 1 common question, so the title gives 6, not 5, common questions about orbitals).
So without further ado, let us begin.
Question 1: What are orbitals?
A lot of people can’t seem to get what orbitals are, exactly, because they believe that orbitals are ‘solid shapes’. In fact, orbitals represent a 3D area where an electron is highly likely to be found within (it is also a wave function, although we will deal with that later). The shape of the simplest orbital, the 1s orbital, is shown below (Fig. 1).
Fig. 1: Shape of an 1s orbital.
Why do we need to pinpoint an approximate area where an electron is likely to be present? There are a few reasons to explain this, but perhaps the simplest (for me) is Heisenberg’s Uncertainty Principle, which states that it is impossible to pinpoint the exact position and momentum of a particle at the same time.
This means that it is not possible for us to completely accurately determine the position of an electron. Thus, only an approximate area can be determined, and this is given by the shape of the orbital. Also, do take note that at most two electrons are present in any orbital (as given by Pauli’s exclusion principle, which will be explained later).
Secondly, we will also explain the definitions of shell and subshell. Basically, in every shell there are a few orbitals (we will be explaining this fully later). In every shell, there are different subshells, which in turn contain several orbitals. For example, the 2 shell contains 1 2p subshell which in turn contains 3 2p orbitals.
This means that ‘2p’ can refer to either the subshell or the orbital. Do take note of this when mentioning the ‘names’ or types of subshells and orbitals. The number of orbitals in each subshell is depicted in the diagram below (Fig. 2).
Fig. 2: Table depicting number of orbitals.
Question 2: What types of orbitals are there?
In general, there are four types of orbitals, the s, p, d and f orbitals. These letters refer to the words sharp, principal, diffuse and fundamental. But, we won’t see orbitals being described as s, p, d or f — there is usually a number in front of it. For example, say, 1s or 3d.
What do these letters and numbers mean? For the letters, they are used to refer to a specific type of orbital, such as 1s or 2p. The numbers refer to the shell of the orbital. For example, in the 1st shell there is the 1s orbital, while in the 2nd shell there are the 2s and 2p orbitals.
So, what is the significance of such numbering? In fact, the numbering also refers to a separate value, the quantum number l. The lettering is used to refer to the quantum number n. However, we shall discuss these topics only later.
Question 3: How do electrons fill the orbitals?
The electrons fill the orbitals by a few principles — these are Aufbau’s principle, Hund’s rule, and Pauli’s exclusion principle. We discuss Aufbau's principle first. It is also known as the ‘building-up’ principle (direct translation), because it dictates that subshells of lower energies are filled before those of higher energies.
Next, we have Hund’s rule of maximum multiplicity. It states that every orbital in a subshell (which can be filled by two electrons) is filled singly before being filled doubly. Note that this applies for every atomic orbital of the same energy (i.e. of the same subshell), as per Aufbau’s principle.
Finally, we have Pauli’s exclusion principle (Fig. 3). As we will learn in Question 4, every electron has either an up spin or a down spin. In every orbital, only two electrons, one with up spin and the other with down spin, are permitted. Any state where both electrons in the orbital are of the same spin is forbidden.
Fig. 3: Forbidden state of Pauli’s exclusion principle.
Question 4: What are the quantum numbers?
Before we look at the quantum numbers themselves, let us first think about why they exist in the first place. Simply, the quantum numbers represent solutions to Schrodinger’s equation for a hydrogen atom. Schrodinger's equation itself is used to describe the atomic structure of the hydrogen atom.
However, being chemists, we are probably not interested in the mathematical portion of orbitals. All we need to know is that Schrodinger's equation can be solved precisely only for a one-electron system, for example the hydrogen atom. To describe the single electron of this one-electron system sufficiently, we need to use the quantum numbers.
So, what are the four quantum numbers? They are n, l, m and s, which stand for the principal, azimuthal, magnetic and spin quantum numbers, respectively. The principal quantum number n refers to the shell of the electron. For example, n = 1 indicates the 1st shell, n = 2 indicates the 2nd shell, and so on (Fig. 4). Thus n must be larger than zero.
Fig. 4: Table for quantum number n.
As for the azimuthal quantum number l (Fig. 5), it refers to whether the subshell is s, p, d or f. Alternatively, it is even possible for a hypothetical g orbital to exist. This case is different from the previous one, with l = 0 indicating the s subshell, l = 1 indicating the p subshell, and so on. After the f subshell, an alphabetical order, i.e. g, h, i, is followed.
Fig. 5: Table for quantum number l.
m refers to the magnetic quantum number. It indicates the type of orbital within the subshell. As we already know, each subshell may contain more than 1 orbital. For s subshells, only 1 orbital is present, and it is given the m value m = 0. For p subshells, 3 orbitals are present giving values of m = -1, m = 0 and m = 1. The values of m range from any negative to any positive integer, including m = 0.
Finally, the spin quantum number is s and indicates the spin of each electron within the orbital itself. This number can only take two values, either -½ or ½, indicating the ‘up’ spin and the ‘down’ spin, as we have already noted in Question 3, by Pauli’s exclusion principle.
Question 4.5: How many orbitals are there in n = 3, 4 or 5?
This subquestion links to the previous one, because n in this case refers to the quantum number n while l refers to the quantum number l. We have discussed what n and l mean in the previous chapter. Now, this subquestion should be an easy one to solve.
Since n = 3 (for example), this means that the orbitals are all in the second subshell, i.e. all have the ‘3’ number in front of them, as in 3s, 3p, or 3d. The general formula to calculate the number of orbitals in any given subshell is n2. This means that if n = 3, there will be 32 = 9 orbitals.
We can also prove this by using knowledge attained from Question 2. For n = 3, there are the 3s, 3p and 3d subshells. As we have learnt, each of these subshells contain 1, 3 and 5 orbitals respectively. Adding these together by 1 + 3 + 5 leads us to the same final value of 9.
Question 5: What are degenerate orbitals and how are they important?
Degenerate orbitals are very simple. They refer to two or more orbitals with the same energy level. There is nothing much to discuss about this, and degenerate orbitals usually refer to molecular orbitals (which will be explained in a separate post), rather than atomic orbitals.
Degenerate atomic orbitals are always present in all subshells, other than the s subshell, which only contains a single orbital (and thus no other orbitals of equal energy are present). For p, d and f subshells they always contain more than 1 orbital, and as such degenerate orbitals may be present.
Degenerate molecular orbitals have been discussed in the previous article. Since lower energy orbitals are filled first (this applies not just for atomic orbitals, as we have discussed for most of this article, but also molecular orbitals), it creates Huckel’s rule and Mobius aromaticity, as we have explained.
Question 6: What are bonding and antibonding orbitals?
Orbitals can interact with each other when bonds are formed, as per molecular orbital (MO) theory. Because of wave-particle duality, mathematically it is convenient to consider atomic orbitals as waves. And this leads to the interactions between atomic orbitals that results in bonding and antibonding molecular orbitals.
Because electrons are waves, it is possible for the waves to cancel out each other, or add on to each other to produce a wave of larger amplitude (i.e. the peaks are higher). When the waves cancel out each other, it is known as destructive interference, producing antibonding molecular orbitals. When the waves add on to each other, it is known as constructive interference, producing bonding molecular orbitals.
This is all we will state for this question, because to fully understand the concept of bonding and antibonding molecular orbitals, we will need to look at MO theory itself, which we have not yet discussed. It will, however, be explained in the next article.