11.02: Quantum Numbers for Electrons

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Learning Objectives

Explain what spectra are.
Learn the quantum numbers that are assigned to electrons.
Learn the rules for the allowed quantum numbers for electrons.

There are two fundamental ways of generating light: either heat an object up so hot it glows or pass an electrical current through a sample of matter (usually a gas). Incandescent lights and fluorescent lights generate light via these two methods, respectively.

A hot object gives off a continuum of light. We notice this when the visible portion of the electromagnetic spectrum is passed through a prism: the prism separates light into its constituent colors, and all colors are present in a continuous rainbow (part (a) in Figure $\PageIndex{1}$ - Prisms and Light). This image is known as a continuous spectrum. However, when electricity is passed through a gas and light is emitted and this light is passed though a prism, we see only certain lines of light in the image (part (b) in Figure $\PageIndex{1}$ - Prisms and Light). This image is called a line spectrum. It turns out that every element has its own unique, characteristic line spectrum.

Figure $\PageIndex{1}$: Prisms and Light. (a) A glowing object gives off a full rainbow of colors, which are noticed only when light is passed through a prism to make a continuous spectrum. (b) However, when electricity is passed through a gas, only certain colors of light are emitted. Here are the colors of light in the line spectrum of Hg.

Why does the light emitted from an electrically excited gas have only certain colors, while light given off by hot objects has a continuous spectrum? For a long time, it was not well explained. Particularly simple was the spectrum of hydrogen gas, which could be described easily by an equation; no other element has a spectrum that is so predictable (Figure $\PageIndex{2}$ - Hydrogen Spectrum).

Figure $\PageIndex{2}$: Hydrogen Spectrum. The spectrum of hydrogen was particularly simple and could be predicted by a simple mathematical expression.

In 1913, the Danish scientist Niels Bohr suggested a reason why the hydrogen atom spectrum looked this way. He suggested that the electron in a hydrogen atom could not have any random energy, having only certain fixed values of energy that were indexed by the number n (the quantum number). Quantities that have certain specific values are quantized values. Bohr suggested that the energy of the electron in hydrogen was quantized because it was in a specific orbit. Because the energies of the electron can have only certain values, the changes in energies can have only certain values (somewhat similar to a staircase: not only are the stair steps set at specific heights but the height between steps is fixed). Finally, Bohr suggested that the energy of light emitted from electrified hydrogen gas was equal to the energy difference of the electron’s energy states:

\[E_{light}=h\nu =\frac{hc}{\lambda}=\Delta E_{electron}\]

This means that only certain frequencies - (and thus, certain wavelengths (\lambda)) of light are emitted. Figure $\PageIndex{3}$ - Bohr’s Model of the Hydrogen Atom, shows a model of the hydrogen atom based on Bohr’s ideas.

Bohrs Model of the Hydrogen Atom.png

Figure $\PageIndex{3}$: Bohr’s Model of the Hydrogen Atom. Bohr’s description of the hydrogen atom had specific orbits for the electron, which had quantized energies.

Chemistry Is Everywhere: Neon Lights

A neon light is basically an electrified tube with a small amount of gas in it. Electricity excites electrons in the gas atoms, which then give off light as the electrons go back into a lower energy state. However, many so-called “neon” lights don’t contain neon!

Although we know now that a gas discharge gives off only certain colors of light, without a prism or other component to separate the individual light colors, we see a composite of all the colors emitted. It is not unusual for a certain color to predominate. True neon lights, with neon gas in them, have a reddish-orange light due to the large amount of red-, orange-, and yellow-colored light emitted. However, if you use krypton instead of neon, you get a whitish light, while using argon yields a blue-purple light. A light filled with nitrogen gas glows purple, as does a helium lamp. Other gases—and mixtures of gases—emit other colors of light. Ironically, despite its importance in the development of modern electronic theory, hydrogen lamps emit little visible light and are rarely used for illumination purposes.

Neon lights. The different colors of these “neon” lights are caused by gases other than neon in the discharge tubes. Human/Need/Desire. Neon sculpture by Bruce Nauman (1983), who has been characterized as a conceptual artist. Image used with permission (CC BY-SA 3.0; eschipul).

Want to explore neon lights and electron transitions further? See how different electron transitions result in different wavelengths and therefore different colors in this excellent interactive simulation: https://interactives.ck12.org/simulations/chemistry/bohr-model-of-electron/app/index.html?lang=en&referrer=ck12Launcher&backUrl=https://interactives.ck12.org/simulations/chemistry.html

Food and Drink App: Artificial Colors

The color of objects comes from a different mechanism than the colors of neon and other discharge lights. Although colored lights produce their colors, objects are colored because they preferentially reflect a certain color from the white light that shines on them. A red tomato, for example, is bright red because it reflects red light while absorbing all the other colors of the rainbow.

Many foods, such as tomatoes, are highly colored; in fact, the common statement "you eat with your eyes first" is an implicit recognition that the visual appeal of food is just as important as its taste. But what about processed foods?

Many processed foods have food colorings added to them. There are two types of food colorings: natural and artificial. Natural food colorings include caramelized sugar for brown; annatto, turmeric, and saffron for various shades of orange or yellow; betanin from beets for purple; and even carmine, a deep red dye that is extracted from the cochineal, a small insect that is a parasite on cacti in Central and South America. (That's right: you may be eating bug juice!)

Some colorings are artificial. In the United States, the Food and Drug Administration currently approves only seven compounds as artificial colorings in food, beverages, and cosmetics:

FD&C Blue #1: Brilliant Blue FCF
FD&C Blue #2: Indigotine
FD&C Green #3: Fast Green FCF
RD&C Red #3: Erythrosine
FD&C Red #40: Allura Red AC
FD&C Yellow #5: Tartrazine
FD&C Yellow #6: Sunset Yellow FCF

Lower-numbered colors are no longer on the market or have been removed for various reasons. Typically, these artificial colorings are large molecules that absorb certain colors of light very strongly, making them useful even at very low concentrations in foods and cosmetics. Even at such low amounts, some critics claim that a small portion of the population (especially children) is sensitive to artificial colorings and urge that their use be curtailed or halted. However, formal studies of artificial colorings and their effects on behavior have been inconclusive or contradictory. Despite this, most people continue to enjoy processed foods with artificial coloring (like those shown in the accompanying figure).

Artificial food colorings.png

Artificial food colorings. Artificial food colorings are found in a variety of food products, such as processed foods, candies, and egg dyes. Even pet foods have artificial food coloring in them, although it's likely that the animal doesn't care!. Source: Photo courtesy of Matthew Bland, http://www.flickr.com/photos/matthewbland/3111904731.

Quantum Numbers

Bohr’s ideas were useful but were applied only to the hydrogen atom. However, later researchers generalized Bohr’s ideas into a new theory called quantum mechanics, which explains the behavior of electrons as if they were acting as a wave, not as particles. Quantum mechanics predicts two major things: quantized energies for electrons of all atoms (not just hydrogen) and an organization of electrons within atoms. Electrons are no longer thought of as being randomly distributed around a nucleus or restricted to certain orbits (in that regard, Bohr was wrong). Instead, electrons are collected into groups and subgroups that explain much about the chemical behavior of the atom.

In the quantum-mechanical model of an atom, the state of an electron is described by four quantum numbers, not just the one predicted by Bohr. The first quantum number is called the principal quantum number(n). The principal quantum number largely determines the energy of an electron. Electrons in the same atom that have the same principal quantum number are said to occupy an electron shell of the atom. The principal quantum number can be any nonzero positive integer: 1, 2, 3, 4,….

Within a shell, there may be multiple possible values of the next quantum number, the angular momentum quantum number(ℓ). The ℓ quantum number has a minor effect on the energy of the electron but also affects the spatial distribution of the electron in three-dimensional space—that is, the shape of an electron’s distribution in space. The value of the ℓ quantum number can be any integer between 0 and n − 1: ℓ = 0, 1, 2,…, n − 1

Thus, for a given value of n, there are different possible values of ℓ:

If n equals	ℓ can be
1	0
2	0 or 1
3	0, 1, or 2
4	0, 1, 2, or 3

and so forth. Electrons within a shell that have the same value of ℓ are said to occupy a subshell in the atom. Commonly, instead of referring to the numerical value of ℓ, a letter represents the value of ℓ (to help distinguish it from the principal quantum number):

If ℓ equals	The letter is
0	s
1	p
2	d
3	f

The next quantum number is called the magnetic quantum number (m_ℓ). For any value of ℓ, there are 2ℓ + 1 possible values of m_ℓ, ranging from −ℓ to ℓ:

\[−ℓ \lt m_ℓ \lt ℓ\]

or $| m_{ℓ} | \leq ℓ$

$| m_{ℓ} | \leq ℓ$ \[|m_ℓ|\lt ℓ\]

The following explicitly lists the possible values of m_ℓ for the possible values of ℓ:

If ℓ equals	The m_ℓ values can be
0	0
1	−1, 0, or 1
2	−2, −1, 0, 1, or 2
3	−3, −2, −1, 0, 1, 2, or 3

The particular value of m_ℓ dictates the orientation of an electron’s distribution in space. When ℓ is zero, m_ℓ can be only zero, so there is only one possible orientation. When ℓ is 1, there are three possible orientations for an electron’s distribution. When ℓ is 2, there are five possible orientations of electron distribution. This goes on and on for other values of ℓ, but we need not consider any higher values of ℓ here. Each value of m_ℓ designates a certain orbital. Thus, there is only one orbital when ℓ is zero, three orbitals when ℓ is 1, five orbitals when ℓ is 2, and so forth. The m_ℓ quantum number has no effect on the energy of an electron unless the electrons are subjected to a magnetic field—hence its name.

The ℓ quantum number dictates the general shape of electron distribution in space (Figure $\PageIndex{4}$ - Electron Orbitals). Any s orbital is spherically symmetric (Figure $\PageIndex{4a}$ - Electron Orbitals, and there is only one orbital in any s subshell. Any p orbital has a two-lobed, dumbbell-like shape (Figure $\PageIndex{4b}$ - Electron Orbitals); because there are three of them, we normally represent them as pointing along the x-, y-, and z-axes of Cartesian space. The d orbitals are four-lobed rosettes (Figure $\PageIndex{4c}$ - Electron Orbitals they are oriented differently in space (the one labeled d_z² $_{^{}}$ has two lobes and a torus instead of four lobes, but it is equivalent to the other orbitals). When there is more than one possible value of m_ℓ, each orbital is labeled with one of the possible values. It should be noted that the diagrams in Figure $\PageIndex{4}$ are estimates of the electron distribution in space, not surfaces electrons are fixed on.

Figure $\PageIndex{4}$: Electron Orbitals. (a) The lone s orbital is spherical in distribution. (b) The three p orbitals are shaped like dumbbells, and each one points in a different direction. (c) The five d orbitals are rosette in shape, except for the $d_{z^{2}}$ orbital, which is a “dumbbell + torus” combination. They are all oriented in different directions.

The final quantum number is the spin quantum number (m_s). Electrons and other subatomic particles behave as if they are spinning (we cannot tell if they really are, but they behave as if they are). Electrons themselves have two possible spin states, and because of mathematics, they are assigned the quantum numbers +1/2 and −1/2. These are the only two possible choices for the spin quantum number of an electron.

Example $\PageIndex{1}$:

Of the set of quantum numbers {n, ℓ, m_ℓ, m_s}, which are possible and which are not allowed?

{3, 2, 1, +1/2}
{2, 2, 0, −1/2}
{3, −1, 0, +1/2}

Solution

The principal quantum number n must be an integer, which it is here. The quantum number ℓ must be less than n, which it is. The m_ℓ quantum number must be between −ℓ and ℓ, which it is. The spin quantum number is +1/2, which is allowed. Because this set of quantum numbers follows all restrictions, it is possible.
The quantum number n is an integer, but the quantum number ℓ must be less than n, which it is not. Thus, this is not an allowed set of quantum numbers.
The principal quantum number n is an integer, but ℓ is not allowed to be negative. Therefore this is not an allowed set of quantum numbers.

Exercise $\PageIndex{1}$

Of the set of quantum numbers {n, ℓ, m_ℓ, m_s}, which are possible and which are not allowed?

{4, 2, −2, 1}
{3, 1, 0, −1/2}

Answers

Spin must be either +1/2 or −1/2, so this set of quantum number is not allowed.
allowed

Rules for Quantum Numbers

Now that you know that electrons have quantum numbers, how are they arranged in atoms? The key to understanding electronic arrangement is summarized in the Pauli exclusion principle: no two electrons in an atom can have the same set of four quantum numbers. This dramatically limits the number of electrons that can exist in a shell or a subshell.

Electrons are typically organized around an atom by starting at the lowest possible quantum numbers first, which are the shells-subshells with lower energies. Consider H, an atom with a single electron only. Under normal conditions, the single electron would go into the n = 1 shell, which has only a single s subshell with one orbital (because m_ℓ can equal only 0). The convention is to label the shell-subshell combination with the number of the shell and the letter that represents the subshell. Thus, the electron goes in the 1s shell-subshell combination. It is usually not necessary to specify the m_ℓ or m_s quantum numbers, but for the H atom, the electron has m_ℓ = 0 (the only possible value) and an m_s of either +1/2 or −1/2.

H (hydrogen): 1 electron

{n = 1, l = 0, m_ℓ = 0, m_s= +1/2} or {n = 1, l = 0, m_ℓ = 0, m_s= -1/2}

The He atom has two electrons. The second electron can also go into the 1s shell-subshell combination but only if its spin quantum number is different from the first electron's spin quantum number. Thus, the sets of quantum numbers for the two electrons are {1, 0, 0, +1/2} and {1, 0, 0, −1/2}. Notice that the overall set is different for the two electrons, as required by the Pauli exclusion principle.

He (helium): 2 electrons

electron 1: {n = 1, l = 0, m_ℓ = 0, m_s= +1/2}

electron 2: {n = 1, l = 0, m_ℓ = 0, m_s= -1/2}

The next atom is Li, with three electrons. However, now the Pauli exclusion principle implies that we cannot put that electron in the 1s shell-subshell because no matter how we try, this third electron would have the same set of four quantum numbers as one of the first two electrons. So this third electron must be assigned to a different shell-subshell combination. However, the n = 1 shell doesn't have another subshell; it is restricted to having just ℓ = 0, or an s subshell. Therefore, this third electron has to be assigned to the n = 2 shell, which has an s (ℓ = 0) subshell and a p (ℓ = 1) subshell. Again, we usually start with the lowest quantum number, so this third electron is assigned to the 2s shell-subshell combination of quantum numbers.

Li (lithium): 3 electrons

electron 1: {n = 1, l = 0, m_ℓ = 0, m_s= +1/2}

electron 2: {n = 1, l = 0, m_ℓ = 0, m_s= -1/2}

electron 3: {n = 2, l = 0, m_ℓ = 0, m_s= -1/2} or {n = 2, l = 0, m_ℓ = 0, m_s= +1/2}

The Pauli exclusion principle has the net effect of limiting the number of electrons that can be assigned a shell-subshell combination of quantum numbers. For example, in any s subshell, no matter what the shell number, there can be a maximum of only two electrons. Once the s subshell is filled up, any additional electrons must go to another subshell in the shell (if it exists) or to higher-numbered shell. A similar analysis shows that a p subshell can hold a maximum of six electrons. A d subshell can hold a maximum of 10 electrons, while an f subshell can have a maximum of 14 electrons. By limiting subshells to these maxima, we can distribute the available electrons to their shells and subshells.

Example $\PageIndex{1}$: Carbon Atoms

How would the six electrons for C be assigned to the n and ℓ quantum numbers?

Solution

The first two electrons go into the 1s shell-subshell combination. Two additional electrons can go into the 2s shell-subshell, but now this subshell is filled with the maximum number of electrons. The n = 2 shell also has a p subshell, so the remaining two electrons can go into the 2p subshell. The 2p subshell is not completely filled because it can hold a maximum of six electrons.

Exercise $\PageIndex{1}$: Sodium Atoms

How would the 11 electrons for Na be assigned to the n and ℓ quantum numbers?

Answer

two 1s electrons, two 2s electrons, six 2p electrons, and one 3s electron

Key Takeaways

Electrons in atoms have quantized energies.
When electrons are excited to higher energy levels (higher n quantum number) they absorb energy
When electrons drop from higher energy levels to lower energy levels (lower n quantum number) they release energy in the form of light.
Electrons will arrange to fill the lowest possible energy levels first.
The state of electrons in atoms is described by four quantum numbers.
No two electrons can have the same set of quantum numbers.