Energy Levels and Electron Configurations

Recall that the quantum mechanical model of the atom yields wave functions and corresponding energies called orbitals that each describe a specific pattern of electron density, meaning each orbital has a specific shape and energy. There are three quantum numbers that are used to describe an orbital:
  1. principal quantum number, n, can be any positive integral (1,2,3, etc.) As n increases, so does the size of the orbital and its energy, meaning that the electrons are less tightly bound to the nucleus.
  2. azimuthal quantum number, l, can have any value from 0 to (n-1). This defines the shape of the orbital. Zero defines the s orbital, 1 defines the p orbital, 2 defines the d orbital, and 3 defines the f orbital.
  3. magnetic quantum number, ml, can have any value from -l to +l, including zero. This number describes the orientation of the orbital in space. In general, in conjunction with n and l it is used to identify a single orbital, which holds two electrons. The total number of ml for each l tells you how many orbitals are found in general subshell. For example, an l value of 1 designates the p orbital, and gives ml values of -1, 0 and 1. This tells us that there are three total orbitals in one p shell, which can hold six electrons.
Note: The collection of orbitals with the same n value is referred to as a shell. The collection of orbitals with the same n and l values is referred to as a subshell.

The inclusion of a fourth quantum number allows us to designate specific electrons. The spin quantum number, ms,  can have a value of -1/2 or +1/2, indicating the two opposite directions that an electron can spin. This brings up an important concept known as the Pauli Exclusion Principle: no two electrons in an atom can have the same set of four quantum numbers.

Energy Levels

As we've said, each orbital has a specific energy. To be more specific, in a multi-electron atom, energy increases as the value (n+l) increases. For orbitals with the same (n+l) values, the lower energy orbitals have the lower n values. This means that orbitals with the same n increase in energy as l increases - so the 2s orbital (l=0) is lower in energy than the 2p (l=1). All orbitals of a given subshell have the same energy; in other words, these orbitals are degenerate.

Real-World Application of Energy Levels

The concept of energy levels helps explain something that you encounter everyday but probably don't even think about - light! When an atom is excited, meaning there is an input of energy, electrons may be "bumped up" to higher energy levels. When the electron(s) then return to their lower energy state, they must release that energy difference between the higher and lower levels. This energy is given off in the form of photons. Depending on the amount of energy, the photons may have wavelengths that are in the visible light spectrum, meaning that we can see them. This phenomenon proves to us that electrons do reside in distinct energy levels and orbitals.

Electron Configurations

An atom's electron configuration is the way in which its electrons are distributed among its various orbitals. An atom's ground state configuration, is where the electrons are in the lowest possible energy states. The Pauli Exclusion Principle tells us that not all the electrons can be found in that 1s orbital (even though it's lowest in energy), so instead the orbitals are filled in order of increasing energy (see energy levels above) with no more than two electrons per orbital.This is known as Aufbau's Principle.

Another important concept is Hund's Rule, which states that for degenerate orbitals (such as the three of the 2p subshell, for example) the lowest energy configuration is achieved when the number of electrons with the same spin (or parallel spins) is maximized. This guarantees that electrons will occupy different orbitals (until the number in the subshell requires pairing), and minimizes electron-electron repulsions. Let's take a look at the chart below of electron diagrams of some of the lighter elements:

Notice the electron configurations are reported in two different ways. The format on the right is known as an orbital diagram, where each orbital is represented by a box and each arrow represents an electron. Take a look at carbon's configuration. Notice that this adheres to Hund's Rule - instead of having its two 2p electrons paired in a single orbital or in two different orbitals with opposite spins, they have parallel spins occupying different orbitals. This provides the lowest energy configuration.

The format on the left is the written electron configuration. This is written according to the filling order, which goes from lowest energy orbital to highest. Refer back to the Energy Level section above to review this trend.

We can also write electron configurations in a condensed form. The noble gases each have completely filled outer shells, with the stable octet of electrons. The next element following a noble gas marks the beginning of a new period on the table with an electron in a new shell, and so we can abbreviate the core electrons (or the electrons of the noble gas configuration) to write a condensed electron configuration.

Expanded for Sodium: 1s2 2s2 2p6 3s1
Condensed: [Ne] 3s1

So how does the periodic table help?

The periodic table is designed so that elements with the same valence electron configurations are in the same columns, or groups. Notice that all Group 2 elements have 2 valence electrons, giving a full s orbital, for example. So, the periodic table is the best resource for the order in which orbitals are filled. After all, it was designed with just that in mind. To come up with an element's electron configuration, simply start from hydrogen and go from left to right writing the corresponding subshells and filling them with electrons appropriately until you reach that element.

Let's go through an example.

Write the electron configuration for bromine.
First, identify where bromine is on the periodic table.
Bromine is in Group 17, Period 4.

Then start from hydrogen and write down the orbitals as you move across the periodic table in order of increasing atomic number.
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p5

Notice that there are only 5 electrons in that last subshell. Bromine is the fifth element in that 4p block, meaning it has only 5 electrons occupying that subshell.

We could also write the condensed electron configuration.
[Ar] 4s2 3d10 4p5
4s was the beginning of a new period, so the rest of the electrons could be represented by the noble gas configuration of argon.

Ion Electron Configurations

When forming anions, as is done by many p-block elements, simply add the number of "extra" electrons - or the value of the negative charge - to your existing electron configuration for the neutral element. Usually, this then creates a stable octet, or a noble gas configuration, which can be condensed as such.

When forming cations, the situation gets trickier, especially when you've got d and f orbitals involved. The general rule is this: subtract the number of electrons - the value of the positive charge - from the orbital(s) with the highest principal quantum number. A good example of this principle is gallium. It can form a 3+ ion. Using the periodic table, we can determine that its condensed electron configuration for the neutral element is [Ar] 4s2 3d10 4p1. When we form the ion we take from the highest n value orbitals. That means that all electrons from the n=4 shell are removed before any from the n=3 shell are touched. This means that Ga3+ has an electron configuration of [Ar] 3d10, NOT [Ar] 4s2 3d8.