High School Chemistry/Quantum Numbers

We've spent a lot of this chapter talking about waves, and electron waves in particular. While most of us know what normal water waves look like, very few people have an understanding of what electron waves look like. In the last lesson, we talked about electron density, and how an electron wave could be thought of as representing the thickness or thinness of the electron density "fog" at any point in space within the atom. We considered the probability pattern for the electron in a hydrogen atom. Now let's consider some more complicated atoms.

Lesson Objectives

 * Explain the meaning of the principal quantum number, n.
 * Explain the meaning of the azimuthal quantum number, ℓ.
 * Explain the meaning of the magnetic quantum number, ml.

Schrödinger's Equation Provides Three Integers Used to Define the Energy States and Orbital for an Electron
You should remember that electrons form standing waves whenever they are trapped within an atom. You should also remember that only certain standing waves are allowed in any confined space because only certain standing waves can fit perfectly inside that space (remember, a perfect fit requires that a wave begin and end where its box begins and ends). In a one-dimensional box, it's easy to picture all of the possible waves that fit perfectly inside. In three dimensions, it's a little harder. Unfortunately when it comes to electrons in an atom, there's an additional complication on top of the fact that electron waves are three-dimensional! It turns out that the electrons in an atom aren't confined to nice square or rectangular boxes. Instead, they're confined to spherical boxes (which should make sense, since atoms are, after all, tiny little spheres). In other words, electron waves inside an atom must begin and end on the surface of a sphere. You may have a really good imagination, and an amazing ability to picture objects in three-dimensions, but for most people, trying to figure out what these spherical three-dimensional waves look like can be quite a challenge.

Luckily, that's how electron wave functions can help. Electron wave functions basically describe the possible shapes that electron waves can take. We won't actually worry about the wave functions. Instead, we'll only worry about specific numbers, called quantum numbers. Quantum numbers are always part of an electron wave function and are extremely important when it comes to determining the shape of a probability pattern.

When electron wave functions were first developed by a man named Erwin Schrödinger (Figure 6.11), his goal was to show how a wave-like description of the electron could be used to understand the behavior of an electron in a hydrogen atom. In order to do this, Schrödinger first defined the size and spherical shape of the hydrogen atom itself. Schrödinger then assumed an electron trapped within a hydrogen atom formed a standing wave that fit perfectly inside without "spilling out" or "doubling over" on itself (it turns out that in circular or spherical boxes, misfit waves don't "spill out" so much as they "double over", as shown in Figure 6.12. Finally, Schrödinger supposed that an electron wave had to be continuous (remember, something that is continuous has no gaps, holes or jumps).



Amazingly, what Schrödinger discovered was that in order to satisfy his basic assumptions about the electron wave both fitting inside the atom and being continuous, certain quantities in the electron wave function had to be "whole number integers". Whenever Schrödinger assigned a whole number to these quantities, he ended up with a wave that fit perfectly inside the hydrogen atom. However, whenever he assigned a decimal number to these quantities, he ended up with a wave function that either "doubled over" on itself, or was discontinuous! The quantities that had to be assigned whole numbers soon became known as quantum numbers. In the wave function for a hydrogen electron there are always three quantum numbers. The first quantum number, n, is called the principal quantum number, the second quantum number, ℓ, is called the azimuthal quantum number, and the third quantum number, ml, is called the magnetic quantum number. Together these three quantum numbers define the energy state and orbital of an electron, but we'll talk more about exactly what that means in the next section.

n = Principal Quantum Number (Energy Level)
The first quantum number, known as the principal quantum number, is given the symbol n. In order to describe a valid standing wave, n has to have integer values, but there's an additional restriction on n as well. The value of n must be a positive integer value (n = 1, 2, 3, . . .). In other words, n can never equal a negative integer. In fact, n can never even equal 0! The principal quantum number gives you two different clues as to what an electron wave looks like. First, it tells you how the electron density spreads out as you move away from the center of the atom. For electron waves with low principal numbers, like n = 1, the electron density is very thick right in close to the center of the atom, but then becomes rapidly thinner as you move out. In contrast, for electron waves with high principal quantum numbers, like n = 6, the electron density isn't as thick near the center of the atom, but is spread quite a bit further out. In general, the higher the principal quantum number, the further away from the nucleus you'll be able to detect a significant amount of electron density (Figure 6.13).

Sometimes, you will hear people say that the principal quantum number determines the "size" of the electron wave function. When people say this, they don't actually mean the absolute or total "size" of the electron wave. What they mean is how big or small the electron wave appears to be based on where most (usually about 90%) of the electron density is concentrated. In an electron wave with a low principal quantum number, the electron density is mostly found close to the center of the atom. Even though there is a tiny bit of electron density at distances far away from the atom's nucleus, there's so little that you can't really tell it's there. As a result, electron waves with low principal quantum numbers appear small. On the other hand, in an electron waves with a high principal quantum number, the electron density is much more spread out, and thus much thicker at distances far from the center of the atom. Therefore, electron waves with high principal quantum numbers appear big.

The principal quantum number also describes the total number of nodes that the electron wave contains. What are nodes? Nodes are places where the electron wave has absolutely no amplitude, or "height". Take a look at the one-dimensional waves in the figure below. Can you spot the nodes (portion a.)? Now take a look at the two-dimensional waves (portion b.). Can you spot the nodes? A three-dimensional wave with nodes is something like an onion. Think of how the onion has layers and how in between the different layers there’s always an empty space or a break. If the onion was a three-dimensional wave, the breaks in between the onion layers would be like the nodes. The higher the principal quantum number, n, the more nodes an electron wave has.



A node is any place where a wave has zero amplitude, or zero height. Both a. and b. illustrate different ways of representing waves. In a. the amplitude corresponds to the height of the yellow line above (or below) the black axis. When the yellow wave crosses the black axis, the wave has zero amplitude and thus a node. In b., the amplitude of the wave corresponds to the thickness of the blue cloud. When there is no blue cloud, the wave has zero amplitude and thus a node.

The principal quantum number is extremely important, not only because it tells you something about the "size" of an electron wave and the number of nodes in an electron wave, but also because it tells you something about the energy of that wave. If you think back, you'll remember that negative electrons like to be close to the positive nucleus because the energy of an electron is lower the closer it is to a positive charge. What does that mean in terms of the principal quantum number? It means that an electron wave with a lower principal quantum number, and electron density centered close to the nucleus of the atom, will have a lower energy, while an electron wave with a higher principal quantum number, and electron density spread further away from the nucleus of the atom, will have a higher energy.

Similarly, as the number of nodes in an electron wave increases, the energy of the wave increases as well. Think about the jump rope experiment. Do you remember how you created standing waves in a jump rope? Did it take more or less energy on your part to get multiple waves to form along the rope? It should have taken you more energy to create more waves. That's because in the process of creating more waves, you also created more nodes and nodes are always associated with an increase in energy. Again, let's take a look at what his means in terms of the principal quantum number. An electron wave with a lower principal quantum number has fewer nodes, and thus will also have a lower energy. On the other hand, an electron wave with a higher principal quantum number has more nodes and thus will have a higher energy.

Notice that as n increases, both the "size" of the electron wave and the number of nodes it has increase as well. As a result, the energy of an electron wave always increases with n.


 * 1) The bigger value of the n, the higher the energy.
 * 2) The smaller the value of n, the lower the energy.

Since the principal quantum number determines the energy of a particular electron wave, n is often thought of as referring to the "energy level" of the electron. The term "energy level" actually comes from Bohr's old solar system model of the atom. Thanks to Schrödinger, and his wave equation, though, we now know that the energy level doesn't correspond to a particular orbit around the nucleus, but rather, to a particular electron wave adopted by the electron when it becomes trapped inside the atom.

The Angular Momentum Quantum Number (Sub-levels, s, p, d, f)(s=sharp, p=principal, d=diffuse and f=fundamental)
The second quantum number, known as the azimuthal quantum number, is given the symbol ℓ. While the principal quantum number told you about the "size" of an electron wave and the number of nodes in an electron wave, the azimuthal quantum number tells you more about the "shape" of an electron wave. In other words, the shape that the electron wave appears to have as a result of its electron density being "thicker" in one place than it is in another. You might be tempted to think that the shape of an electron wave is always spherical because the atom itself is spherical. It turns out, however, that while there are spherical electron waves, there are also waves that look like dumb-bells and waves that look like butterflies, and waves that look so crazy it's almost impossible to describe them! Some of the different possible shapes for electron waves are shown in the figure below.

You might wonder about the various balloon-like shapes in the figure above. One difficulty with using drawings to represent electron waves is that the electron density itself is actually spread over a huge region of space surrounding the center of the atom. As you learned earlier, though, for many electron waves, almost all of the electron density is in close to the nucleus of the atom, with only a tiny bit of electron density further out. Over the years, scientists have developed a standard way of drawing electron waves. Rather than trying to account for all of the electron density in an electron wave, scientists usually just draw "balloons" around the region of space that contains about 90% of the electron wave's total electron density. The figure below shows how scientists convert a cloud of electron density into a balloon. Even though there is a little bit of electron density outside of the cartoon balloons (and thus a small probability of finding the electron outside of its balloon), most electron behavior can be understood by ignoring the tiny bit of electron density that the cartoon balloon doesn't capture.



The exact shape of the cartoon balloon representing an electron wave is determined by the value of ℓ. In other words, the dumb-bell shaped balloon in the image above has one value of ℓ, while the butterfly-shaped balloon has another value of ℓ. Of course, scientists would get pretty tired of saying things like "the dumb-bell shaped wave", or "the butterfly shaped wave", so instead, they name the different waves using letters from the alphabet. The most common shapes for waves are called s, p, d and f. (You would probably find it a lot easier to remember what the different waves looked like if scientists had given them nice descriptive names, like the "dumb-bell wave" or the "butterfly wave", but boring names like s, p, d and f turn out to be much more convenient).

In the next lesson, we'll take a closer look at exactly what some of the common wave shapes look like. First, though, we have to consider the relationship between n and ℓ. Remember that a wave function for an electron always has three quantum numbers. In order to fully describe an electron wave, then, you have to know the values of all three of these numbers (n, ℓ and ml). Now you might think that as long as n, ℓ and ml are all integers, the wave that they describe will be a perfectly good electron wave. But that's not the way it works. It turns out that for a particular value of n, only certain values of ℓ are allowed.

For wave functions that actually describe electrons in an atom, ℓ is always no less than 0, but also no more than n − 1 (ℓ = 0, 1, 2, … n − 1). The following examples should help to clarify the restriction on ℓ.

What are the allowable values of ℓ for an electron wave with n = 3?

Solution:

n = 3

1. Find the minimum value of ℓ.


 * The minimum value of ℓ is always 0.

2. Find the maximum value of ℓ.


 * The maximum value of ℓ is always n – 1
 * maximum ℓ = n − 1
 * maximum ℓ = 3 − 1
 * maximum ℓ = 2

3. List all of the integers (no decimals!) starting from the minimum value of ℓ and ending with the maximum value of ℓ.


 * ℓ = 0, 1, 2

What are the allowable values of ℓ for an electron wave with n = 1?

Solution:

n = 1

1. Find the minimum value of ℓ.


 * The minimum value of ℓ is always 0.

2. Find the maximum value of ℓ.


 * The maximum value of ℓ is always n − 1
 * maximum ℓ = n − 1
 * maximum ℓ = 0

3. List all of the integers (no decimals!) starting from the minimum value of ℓ and ending with the maximum value of ℓ.


 * ℓ = 0

Often, scientists will refer the different values of ℓ as electron sublevels. In a hydrogen atom with one electron, the value of ℓ has no effect on the energy of the electron. However, in an atom with more than one electron, the value of ℓ does have a small effect on the electron's energy. In other words, the principle quantum number, n, always determines the overall energy level of an electron, but that energy level is actually divided into multiple sublevels based on the value of ℓ. All of these sublevels have equal energy in a hydrogen atom, because the hydrogen atom only has one electron. For atoms with more than one electron, though, the different sublevels split apart, and some of them end up having more energy than others.

ml = Magnetic Quantum Number (Identifies Orbital)
The third, and final quantum number, known as the magnetic quantum number, is given the symbol ml. Remember, the principal quantum number told you about the "size" of an electron wave and the number of nodes in an electron wave, while the azimuthal quantum number told you more about the "shape" of an electron wave. The magnetic quantum number, though, gives you even more information about what the electron wave looks like. The magnetic quantum number tells you how the electron wave is orientated in space.

Orientation in space basically means where the electron wave points. Take a look at the two dumb-bell shaped electron waves shown in Figure 6.14 (in the next lesson, you'll learn that these are actually p orbitals). In the first electron wave, lobes of the wave point "up-and- down" along the z-axis, while in the second electron wave, the lobes of the wave point "in-and-out" along the x-axis. These two electron waves have different orientations in space, and thus different values for the quantum number ml.



Now compare Figure 6.14 with Figure 6.15. In both figures, the electron wave has a different orientation in space, as indicated by the red arrow. But what do you notice about the orientations of the electron wave in Figure 6.15? They look the same, don't they? Obviously, orientation doesn't matter for the spherical electron wave. In other words, because the spherical electron wave looks the same no matter how you rotate it, there's really just one orientation. So how many different ml values should an electron wave with a spherical shape have (remember, ml values are used to describe orientation)? Clearly, an electron wave with a spherical shape should only have one ml value, because it only has one possible orientation. What about an orbital with a dumb-bell shape? Should it have a single ml value, or should it have several different ml values? Well, since different orientations of the dumb-bell shaped wave actually look different, you'd probably expect the dumb-bell shaped wave to have several different ml values, one for each possible orientation. In fact, the dumb-bell shaped wave actually has three possible values for ml, and so it has three possible orientations. (Don't worry about why there are exactly three orientations for the dumb-bell shaped wave. We'll talk a little bit about that in the next lesson, but the full explanation requires a lot of math that you'll learn about if you decide to study quantum physics or quantum chemistry). Hopefully, by comparing Figures 14 and 15, you should be convinced that the shape of the wave (which depends on ℓ) is important when it comes to determining the number of possible orientations (or the number of possible ml values). It shouldn't be surprising, then, that for any given value of ℓ, only certain ml values are allowed if you want to end up with a wave function that makes sense and actually describes electrons in a hydrogen atom.



For a particular value of ℓ, only certain values of ml are allowed.

The rule for ml is that for any value of ℓ, ml can be any integer starting from −ℓ and ending at +ℓ. ml = −ℓ … +ℓ). The following examples should help to clarify the restriction on ml.

What are the allowable values of ml for an electron wave with ℓ = 2?

Solution:

ℓ = 2

1. Find the minimum value of ml.


 * The minimum value of ml is always −ℓ.
 * minimum ml = −2

2. Find the maximum value of ml.


 * The maximum value of ml is always +ℓ
 * maximum ml = +2

3. List all of the integers (no decimals!) starting from the minimum value of ml and ending with the maximum value of ml.


 * ml = −2, −1, 0, 1, 2

What are the allowable values of ml for an electron wave with ℓ = 0?

Solution:

ℓ = 0

1. Find the minimum value of ml.


 * The minimum value of ml is always −ℓ.
 * minimum ml = 0

2. Find the maximum value of ml.


 * The maximum value of ml is always +ℓ
 * maximum ml = 0

3. List all of the integers (no decimals!) starting from the minimum value of ml and ending with the maximum value of ml.


 * ml = 0

Now that we’ve talked about all three of the different quantum numbers, you should have a good understanding of how different electron waves can be described. You can describe the "size" of an electron wave, and the number of nodes in an electron wave using the principal quantum number, n. You can describe the shape of an electron wave using the azimuthal quantum number, ℓ. Finally, you can describe the orientation of the electron wave using the magnetic quantum number, ml. Since you know how to describe a general electron wave, it's time to look at several examples of specific electron waves. That’s what we'll talk about in the next lesson.

Lesson Summary

 * Schrödinger discovered that in order for a wave function to describe a standing wave that was continuous, and that didn't "doubled back" on itself, certain quantities in his wave function had to have integer values.
 * The quantities in the wave function which must have integer values are known as quantum numbers.
 * In the wave function for hydrogen, there are three quantum numbers. They are called the principal quantum number (n), the azimuthal quantum number (ℓ), and the magnetic quantum number (ml).
 * The principal quantum number can only have positive integer values, (n = 1, 2, 3 …).
 * The principal quantum number determines how far the bulk of the electron density extends from the center of the atom. The higher the value of n, the further away from the nucleus you will be able to detect a significant amount of electron density.
 * The principal quantum number also determines the number of nodes in an electron standing wave. The higher the value of n, the more nodes there are in the electron wave.
 * The higher the principal quantum number, the greater the energy of the electron. Therefore, the principal quantum number is determines the energy level of the electron.
 * The azimuthal quantum number, ℓ, determines the shape of the electron wave. The values of ℓ are also called the electron sublevels. They are labeled with the letters s, p, d, f, g, h, etc.
 * For a wave function that actually describes an electron in an atom, ℓ is always no less than zero, but also no more than n − 1 (ℓ = 0, 1, 2 … n − 1).
 * In atoms with more than one electron, ℓ has a small effect on the electron's energy.
 * The magnetic quantum number, ml, determines how the electron wave is orientated in space. For any given value of ℓ, ml can be any integer from −ℓ to + ℓ (ml = −ℓ … +ℓ).
 * 1) Match each quantum number with the property that they describe.
 * 2) A point in an electron wave where there is zero electron density is called a _________.
 * 3) Choose the correct word in each of the following statements.
 * (a) The (higher/lower) the value of n, the more nodes there are in the electron standing wave.
 * (b) The (higher/lower) the value of n, the less energy the electron has.
 * (c) The (more/less) energy the electron has, the more nodes there are in its electron standing wave.
 * 1) Fill in the blank. For lower values of n, the electron density is typically found ________ the nucleus of the atom, while for higher values of n, the electron density is typically found __________the nucleus of the atom.
 * 2) Circle all of the statements that make sense: Schrödinger discovered that certain quantities in the electron wave equation had to be integers, because when they weren't, the wave equation described waves which…
 * (a) were discontinuous
 * (b) were too small
 * (c) were too long and narrow
 * (d) were too short and fat
 * (e) "doubled back" on themselves
 * 1) What are the allowed values of ℓ for an electron standing wave with n = 4?
 * 2) How many values of ℓ are possible for an electron standing wave with n = 9?
 * 3) What are the allowed values of ml for an electron standing wave with ℓ = 3?
 * 4) How many different orientations are possible for an electron standing wave with ℓ = 4?
 * 5) What are the allowed values of ml for n = 2?

Vocabulary

 * azimuthal quantum number (ℓ)
 * Defines the electron sublevel, and determines the shape of the electron wave.


 * magnetic quantum number (ml)
 * Determines the orientation of the electron standing wave in space.


 * node
 * A place where the electron wave has zero height. In other words, it is a place where there is no electron density.


 * principal quantum number (n)
 * Defines the energy level of the wave function for an electron, the size of the electron's standing wave, and the number of nodes in that wave.


 * quantum numbers
 * Integer numbers assigned to certain quantities in the electron wave function. Because electron standing waves must be continuous and must not "double over" on themselves, quantum numbers are restricted to integer values.