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Electron configuration: Atomic Orbitals and Quantum Numbers

The Quantum Model of the Atom

Atomic Orbitals and Quantum Numbers

Excerpted from Modern Chemistry (Holt, Rinehart, Winston)

Shorthand. We write electrons as e-

To the scientists of the early twentieth century, Bohr’s model of the hydrogen atom contradicted common sense. Why did hydrogen’s electron exist around the nucleus only in certain allowed orbits with definite energies? Why couldn’t the electron exist in a limitless number of orbits with slightly different energies?

To explain why atomic energy states are quantized, scientists had to change the way they viewed the nature of the electron.

In the Bohr atomic model, e-  of increasing energy occupy orbits farther and farther from the nucleus.

This model only worked well for Hydrogen. For larger atoms, people had to continually modify the Bohr model, by hand, for each individual atom, so they knew that it was only an approximation.

The Schrödinger equation came next – this was a more general model. For Hydrogen, it gave the same answers as the Bohr model. But even better, it worked for every atom!

Like the Bohr model, it shows that e- in orbitals have quantized energies. But this equation doesn’t give a definite orbit for an e-.  Rather – if we interpret the e-  as a solid particle – it only gives us the odds of finding it at a certain place.

Electron probability cloud

The Schrödinger equation revealed other aspects of e-
Each property is called a “quantum number” (Well, we had to call it something, so why not something cool sounding like this?)

Quantum numbers specify the properties of atomic orbitals, and the e- in them.

* main energy level  –  n

* shape  – s, p, d or f

* number of orbitals in this level

* spin quantum number

quantum number and sublevels

The principal quantum number

The principal quantum number, symbolized by n, indicates the main energy level occupied by the electron. Values of n are positive integers only—1, 2, 3, and so on.

energy levels quantum number

As n increases, the electron’s energy and its average distance from the nucleus increase (see Figure 12). For example, an electron for which n = 1 occupies the first, or lowest, main energy level and is located closest to the nucleus.

As you will see, more than one electron can have the same n value. These electrons are sometimes said to be in the same electron shell. The total number of orbitals that exist in a given shell, or main energy level, is equal to n2.

Angular Momentum Quantum Number

Except at the first main energy level, orbitals of different shapes – known as sublevels—exist for a given value of n.The angular momentum quantum number, symbolized by l, indicates the shape of the orbital. For a specific main energy level, the number of orbital shapes possible is equal to n.

The values of l allowed are zero and all positive integers less than or equal to n − 1.

For example, orbitals for which n = 2 can have one of two shapes corresponding to l = 0 and l = 1. Depending on its value of l, an orbital is assigned a letter, as shown in Table 1.

Angular Momentum Quantum Number Orbital levels

As shown below:
s orbitals are spherical
p orbitals have dumbbell shapes
d orbitals are more complex
f orbital shapes are even more complex!

In the first energy level, n = 1, there is only one sublevel possible—an s orbital.

As mentioned, the second energy level, n = 2, has two sublevels—the s and p orbitals.

The third energy level, n = 3, has three sublevels—the s, p, and d orbitals.

The fourth energy level, n = 4, has four sublevels—the s, p, d, and f orbitals.

In an nth main energy level, there are n sublevels.

Each atomic orbital is designated by the principal quantum number followed by the letter of the sublevel.

For example, the 1s sublevel is the s orbital in the first main energy level, while the 2p sublevel is the set of three p orbitals in the second main energy level.

On the other hand, a 4d orbital is part of the d sublevel in the fourth main energy level.

How would you designate the p sublevel in the third main energy level? How many other sublevels are in the third main energy level with this one?

Magnetic Quantum Number

Atomic orbitals can have the same shape but different orientations around the nucleus. The magnetic quantum number, symbolized by m, indicates the orientation of an orbital around the nucleus. Values of m are whole numbers, including zero, from −l to +l. Because an s orbital is spherical and is centered around the nucleus, it has only one possible orientation. This orientation corresponds to a magnetic quantum number of m = 0. There is therefore only one s orbital in each s sublevel.

As shown above, the lobes of a p orbital extend along the x, y, or z axis of a three-dimensional coordinate system. There are therefore three p orbitals in each p sublevel, which are designated as px , py , and pz orbitals.

The three p orbitals occupy different regions of space and those regions are related to values of m = −1, m = 0, and m = +1.

There are five different d orbitals in each d sublevel (see figure above)

The five different orientations, including one with a different shape, correspond to values of m = −2, m = −1, m = 0, m = +1, and m = +2.

There are seven different f orbitals in each f sublevel.

As you can see in Table 2, the total number of orbitals in a main energy level increases with the value of n. In fact, the number of orbitals at each main energy level equals the square of the principal quantum number, n2.

What is the total number of orbitals in the third energy level? Specify each of the sublevels using the orbital designations you’ve learned so far.

Spin Quantum Number

An electron in an orbital behaves in some ways like Earth spinning on an axis. The electron exists in one of two possible spin states, which creates a magnetic field.To account for the magnetic properties of the electron, theoreticians of the early twentieth century created the spin quantum number.

The spin quantum number has only two possible values – (+1/2 , −1/2 ) – which indicate the two fundamental spin states of an electron in an orbital. A single orbital can hold a maximum of two electrons, but the two electrons must have opposite spin states.

Caution analogies

Caution: the idea of electrons having “spin” is really an analogy.
Electrons are not really small billiard balls.

They don’t spin. Rather, they have magnetic moments as if they spin.

See What exactly is the ‘spin’ of subatomic particles? Scientific American

The creating mechanism of all particles. What is the Spin?

Learning Standards

Next Generation Science Standards

HS-PS1-2. Construct and revise an explanation for the outcome of a simple chemical reaction based on the outermost electron states of atoms, trends in the periodic table, and knowledge of the patterns of chemical properties.

Disciplinary Core Ideas
PS1.A: Structure and Properties of Matter
 The periodic table orders elements horizontally by the number of protons in the atom’s nucleus and places those with similar chemical properties in columns. The repeating patterns of this table reflect patterns of outer electron states.
PS1.B: Chemical Reactions
 The fact that atoms are conserved, together with knowledge of the chemical properties of the elements involved, can be used to describe and predict chemical reactions.


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