High School Chemistry/The Electron Spin Quantum Number

Lesson Objectives

 * Explain what is meant by the spin quantum number, ms.
 * Explain how the spin quantum number affects the number of electrons in an orbital.
 * Explain the difference between diamagnetic atoms and paramagnetic atoms.

Introduction
In the ../Quantum Mechanics Model of the Atom/ chapter, you learned about quantum numbers. Remember, quantum numbers actually come from the wave function. Even though you've never seen a mathematical equation for a wave function (and you'd probably think it was pretty scary if you did!) you can still understand what the different quantum numbers mean, because they all control different aspects of what the electron standing wave looks like. The principal quantum number, n, determines the "size", the azimuthal quantum number, ℓ, determines the "shape" of the electron wave and the number of nodes in the electron wave, and the magnetic quantum number, ml, determines the "orientation" of the electron wave. Now suppose that you were told the exact size, the exact shape and the exact orientation of an electron orbital. Could you draw that object without being told anything else? You should be able to – what could you possibly need to know about an electron wave other than its size, shape and orientation? It seems as if the first three quantum numbers tell us all the information necessary to picture what an electron's probability pattern looks like – but it turns out that there's fourth quantum number! What! How could there be another quantum number? What property could it possibly describe? There's nothing left after size, shape, and orientation. That’s exactly what scientists thought until they discovered spin.

In the early days, when scientists were just beginning to learn about quantum physics, most of what they knew about quantization involved atomic spectra. Remember, in ../The Bohr Model of the Atom/ chapter, you learned that atomic spectra were discontinuous, and that this led Bohr to propose the existence of quantized energy levels. To explain these quantized energy levels, Bohr suggested that electrons in an atom were restricted to specific orbits and that they moved around the nucleus in these orbits just like the planets move around the sun. It turned out that Bohr's model wasn't entirely correct because electrons don't orbit the nucleus. In fact, based on what scientists know of the wave-particle duality and the Heisenberg Uncertainty Principle, scientists now think that electrons can be anywhere inside the atom, although the probability of finding them at any one particular location over another depends on the relative amount of electron density at those locations. Even though Bohr's model had some problems, Bohr was right in his prediction that electrons could only exist at specific "allowed" energy levels, and that all other energies were "forbidden".

In the ../Quantum Mechanics Model of the Atom/ chapter, you learned how the "allowed" and "forbidden" energy levels could be explained in terms of electron standing waves. Only certain electron standing waves will fit perfectly inside the atom without "doubling over" on themselves, or containing discontinuous jumps. Therefore, just like the Bohr Theory of the Atom, the wave function theory of the electron explains the existence of discontinuous atomic spectra. Actually, wave functions themselves are pretty good at predicting exactly what the atomic spectra of different atoms will look like (In other words, at which wavelengths lines of light will appear – if you have forgotten about atomic spectra, go back and read ../The Atomic Theory/ chapter!) Remember, Bohr's model could only predict the atomic spectrum of hydrogen. With a little bit of work, though, scientists could get electron wave functions to predict many different atomic spectra for many different types of atoms. It seemed perfect… but there was a still a slight problem. Every so often when scientists looked carefully at atomic spectra, they would find two separate lines of light at a wavelength where the wave function had only predicted one. The two separate lines were always really close together, so if you didn't look carefully, you'd think it was just one line and you'd think that the wave function was doing a great job. But the fact remained that frequently the wave function was missing some of the finer details of the atomic spectra.

Now if you think back to what we discussed in ../The Bohr Model of the Atom/ chapter, you'll remember that the lines of light in an atomic spectrum result when an electron "falls" from one energy level to another and releases its extra energy in the form of light. Even though we learned about this process in terms of the Bohr model, the same principle applies to the wave function theory as well. Remember, in terms of wave functions, the energy level of an electron is determined by the principal quantum number, n, while the azimuthal quantum number, ℓ, defines the sublevel and also affects electron energy in atoms with more than one electron.

Electrons don't circle the nucleus like planets in a solar system as pictured in the figure below on the left. Nevertheless, when they fall from a higher energy level electron standing wave to a lower energy electron standing wave they release light in much the same way as shown in the figure below on the right.

When an electron begins at one energy (one value of n and ℓ) and then falls to a lower energy (with a different value of n and ℓ), it releases its extra energy in the form of light. Therefore, if there are two very closely spaced lines of light in an atomic spectrum, it must mean that there are two very closely spaced energy states from which (or to which) an electron can "fall". In terms of predicting the finer details of an atomic spectrum, then, it's clear that the problem with the wave function was that in certain situations it would only find one allowed energy state where there should have been two!

The story got even stranger when scientists put magnets around different atoms and looked at what happened to their atomic spectra as a result. All of those pairs of closely spaced spectral lines that the wave function couldn't predict actually split apart even further! In other words, magnetic fields affected the two different energy levels differently. It made the low energy level drop even lower, and it made the high energy level rise even higher. After the magnet experiments, scientists came to the realization that their wave function description was not as complete as they had hoped. There was some other property that had to be included – but what?

In 1925, two scientists by the names of Samuel Goudsmit and George Uhlenbeck (Figure 7.1) suggested extending the wave function equation so that it included a fourth quantum number, called the "spin" quantum number. When any charged object (like, for example, a negatively charged electron) spins in a magnetic field, its energy is determined by the direction in which it rotates. In other words, if the object rotates clockwise, it will have a different amount of energy than it would have if it had rotated counterclockwise. Goudsmit and Uhlenbeck argued that if the electron was spinning, then it was easy to explain why those two closely spaced spectral lines split further apart in the presence of a magnetic field.



One line corresponded to the energy level for the electron spinning clockwise, and the other corresponded to the energy level for the electron spinning counterclockwise. (If you're really clever, you might wonder why these two energy levels were different and could be seen in atomic spectra even when there weren't any external magnets around. It turns out that there are always tiny internal magnetic fields created by matter itself. Usually, though, these internal magnetic fields are very small, so their effect is very small as well.)

Pictured below are two negatively charged particles spinning in a magnetic field. The left particle will have a different energy than the right particle, because they are spinning in different directions.



By incorporating electron spin into the electron wave function, scientists found that the fourth quantum number, also known as the spin quantum number, ms, could take on two different values – they were ms = +1/2 and ms = −1/2. Notice that unlike n, ℓ and ml, which can only be integers, ms can be half-integers. The value of ms will never be some crazy decimal like ms = 0.943895, but it can be the decimal ms = 0.5, because that corresponds to one-half, or a half-integer.

Now it's tempting to think of ms = +1/2 as the electron spinning in one direction, and ms = −1/2 as the electron spinning in the other direction. When you do that, though, how are you picturing the electron? You're picturing it as a particle, aren't you? And how do you explain the wave-like nature of an electron in terms of rotation clockwise or counterclockwise? Well, you can't. So despite the fact that the idea for including electron spin in the wave function came from picturing electrons as tiny little spinning objects, scientists try not to make any direct comparisons between the spin quantum number and a particle rotating clockwise or counterclockwise. Instead, scientists usually make vague statements like "spin can only truly be understood as a quantum property. There is no directly comparable macroscopic (large scale) property". Rather than saying anything about a clockwise rotation, or a counterclockwise rotation, scientists call electrons with ms = +1/2 "spin-up" and electrons with ms = −1/2 "spin down".

When Two Electrons Occupy the Same Orbital, They must Have Opposite Spins
Do you remember what an orbital is? An orbital is a wave function for an electron with a specific set of quantum numbers, n, ℓ, and ml. For example, the numbers n = 1, ℓ = 0, and ml = 0 define one particular orbital, while the numbers n = 2, ℓ = 1, and ml = −1 define another orbital, and the numbers n = 2, ℓ = 1 and ml = 0 define a different orbital again. Orbitals are very important because any time you know the values of the first three quantum numbers n, ℓ and ml, you know the region in space where there is a high probability of finding the electron. In other words, if you know which orbital an electron is in, you know exactly what its "box", or "territory" looks. The orbital is really just a description of where the electron spends most of its time.

For example, if you're told that the electron is in an orbital with n = 1, ℓ = 0, ml = 0, you automatically know that the electron is probably going to be found in a spherical shaped "territory" that's pretty close to the nucleus of the atom. On the other hand, if you're told that the electron is in an orbital with n = 5, ℓ = 1 and ml = 0 you know that the electron is probably going to be found in a dumb-bell shaped "territory" that extends quite far from the nucleus of the atom, along the y-axis (technically speaking ml = 0 could be the orbital along the x-axis, the orbital along the y-axis or the orbital along the z-axis, but whichever axis you choose for the ml = 0 orbital, the ml = 1 and ml = −1 orbitals will automatically point along the other two axes).

The spin quantum number doesn't tell you anything about the region in space where you're likely to find the electron. Therefore, if you want to know which orbital an electron is in, all you need to know is the values of the first three quantum numbers – you don't need to know the value of the fourth. So why is the value of the fourth quantum number important? Well, the spin quantum number determines whether or not an electron is allowed to share an orbital with another electron that's already there. It turns out that each region of the atom (or each orbital) can actually be shared by two electrons.

Two electrons can share their territory, or their orbital, provided each electron does something different. This means that sharing an orbital is only possible for electrons which have different spin quantum numbers. An electron that has a spin quantum number of ms = +1/2 can share an orbital with an electron that has a spin quantum number of ms = −1/2. However, an electron that has a spin quantum number of ms = +1/2 cannot share an orbital with another electron that also has a spin quantum number of ms = +1/2. Similarly, an electron that has a spin quantum number of ms = −1/2 cannot share an orbital with another electron that has a spin quantum number of ms = −1/2.

When Electrons are Paired, They are Diamagnetic (i.e. No Magnetic Attraction)
In the last section, you learned that any time two electrons share the same orbital, their spin quantum numbers have to be different. In other words, one of the electrons has to be "spin-up", with ms = +1/2, while the other electron is "spin-down", with ms = −1/2. This is important when it comes to determining the total spin in an electron orbital. Technically speaking, an electron's spin isn't exactly the same as its spin quantum number, but the difference isn't important when it comes to figuring out whether or not two electron spins cancel out. In order to decide whether or not electron spins cancel all you need to do is add their spin quantum numbers together. If the total is 0, then the spins cancel each other out. If the total is greater than 0, or less than 0, then the spins do not cancel each other out. Let's take a look at two examples:

Electron A has ms = +1/2 while electron B has ms = −1/2. Do the spins of these two electrons cancel each other out?

Solution:


 * ms for A = +1/2
 * ms for B = −1/2


 * Total ms = ms for A + ms for B
 * Total ms = (+1/2) + (−1/2)
 * Total ms = 1/2 − 1/2
 * Total ms = 0

In this case, the spins of electron A and electron B do cancel each other out.

Electron A has ms = −1/2 while electron B has ms = −1/2. Do the spins of these two electrons cancel each other out?

Solution:


 * ms for A = −1/2
 * ms for B = −1/2


 * Total ms = ms for A + ms for B
 * Total ms = (−1/2) + (−1/2)
 * Total ms = −1/2 − 1/2
 * Total ms = −1

In this case, the spins of electron A and electron B do not cancel each other out.



The idea of "canceling out" should make sense to you. If one spin is "up", and the other is "down", then the "up" spin cancels the "down" spin and there is no leftover spin at all. The same logic applies if we think of spins as "clockwise" and "counterclockwise". If one spin is "clockwise" and the other is "counterclockwise", then the two spin directions balance each other out and there is no leftover rotation at all. Notice what all of this means in terms of electrons sharing an orbital. Since electrons in the same orbital always have opposite values for their spin quantum numbers, ms, they will always end up canceling each other out! In other words, there is no leftover (or "net") spin in an orbital that contains two electrons. Whenever two electrons are paired together in an orbital, we call them diamagnetic electrons. On the other hand, an orbital containing only one electron will have a total spin equal to the spin of the electron that it contains.

Even though electron spin can only be truly understood using quantum physics, it does produces effects that we can actually see in our everyday lives. Electron spin is very important in determining the magnetic properties of an atom. If all of the electrons in an atom are paired up and share their orbital with another electron, then the total spin in each orbital is zero and, by extension, the total spin in the entire atom is zero as well! When this happens, we say that the atom is diamagnetic because it contains only diamagnetic electrons. Diamagnetic atoms are not attracted to a magnetic field. In fact, diamagnetic atoms are slightly repelled by magnetic fields.

Since diamagnetic atoms are slightly repelled by a magnetic field, it is actually possible to make certain diamagnetic materials float! To the right you can see a thin black sheet of pyrolytic graphite floating above the gold colored magnets.

When an Electron is Unpaired, Paramagnetism will be Observed
If atoms that contain only paired electrons are slightly repelled by a magnetic field, what do you think is true of atoms that contain unpaired electrons? If you guessed that atoms with unpaired electrons are slightly attracted to a magnetic field, then you guessed right! Whenever electrons are alone in an orbital, we call them paramagnetic electrons. Remember, if an electron is all by itself in an orbital, the orbital has a "net" spin, because the spin of the lone electron doesn't get canceled out. If even one orbital has a "net" spin, the entire atom will have a "net" spin as well. Therefore, we say an atom is paramagnetic when it contains at least one paramagnetic electron. Notice that the definition of diamagnetism and the definition of paramagnetism are subtly different. This can be confusing if you aren't careful. Be sure to note: In order for an atom to be diamagnetic, all of its electrons must be paired up in orbitals. In order for an atom to be paramagnetic, at least one of its electrons must be unpaired.

In other words, an atom could have 10 paired (diamagnetic) electrons, but as long as it also has 1 unpaired (paramagnetic) electron, it's still considered a "paramagnetic atom". In order to be a "diamagnetic atom", the atom would have to have 10 paired (diamagnetic) electrons and no unpaired (paramagnetic) electrons. Just as diamagnetic atoms are slightly repelled from a magnetic field, paramagnetic atoms are slightly attracted to a magnetic field.

Lesson Summary

 * If you only consider the first three quantum numbers, the wave function model for the electron will sometimes predict one spectral line where there are actually two closely spaced spectral lines.
 * This led to the proposal of a fourth quantum number, the spin quantum number ms.
 * ms can have two possible values for an electron. It can be "spin-up" with ms = +1/2 or "spin-down" with ms = −1/2
 * When two electrons occupy the same orbital, they must have different spin quantum numbers.
 * An orbital containing two electrons will have no net spin. When this is the case, the two electrons are called diamagnetic electrons.
 * An orbital containing only one electron will have a total spin equal to the spin of the electron that it contains. When this is the case, the electron is called a paramagnetic electron.
 * Electron spin helps to determine the magnetic properties of an atom.
 * If all electrons in an atom are diamagnetic, the entire atom has no net spin, and is termed a "diamagnetic atom". Diamagnetic atoms are slightly repelled from a magnetic field.
 * If an atom contains even one paramagnetic electron, the entire atom has a net spin and is termed a paramagnetic atom. Paramagnetic atoms are slightly attracted to a magnetic field.

Review Questions

 * 1) Choose the correct statement.
 * (a) The spin quantum number for an electron can only have the values ms = +1 and ms = −1
 * (b) The spin quantum number for an electron can only have the value ms = 0
 * (c) The spin quantum number for an electron can have any integer value between −ℓ and +ℓ
 * (d) The spin quantum number for an electron can only have the values ms = +1/2 and ms = −1/2
 * (e) The spin quantum number does not apply to electrons
 * 1) Choose the correct statement.
 * (a) When two electrons share an orbital, they always have the same spin quantum numbers
 * (b) When two electrons share an orbital, they always have opposite spin quantum numbers
 * (c) Two electrons cannot share the same orbital
 * (d) When two electrons share an orbital there is no way to predict whether or not they will have the same spin quantum numbers
 * 1) Fill in the blanks in the following statement using numbers.
 * When scientists used the Schrödinger equation with only __ quantum numbers, they found that the Schrödinger equation was pretty good at predicting atomic spectra, except that there were occasionally __ closely spaced lines of light where the Schrödinger equation predicted only __. This led scientists to suggest that a complete description of an electron, which required ___ quantum numbers.
 * 1) In many atomic spectra, there are two very closely spaced lines of light which can only be predicted by including the spin quantum number into the Schrödinger equation. Decide whether the following statements about these two lines are true or false.
 * (a) the two lines spread further apart when the atom is placed in a magnetic field
 * (b) the two lines move closer together when the atom is placed in a magnetic field
 * (c) the two lines are the result of an experimental error. If scientists are careful, they find that there is really just one line.
 * (d) the two lines actually result from the fact that there are two very closely spaced energy states
 * 1) Goudsmit and Uhlenbeck proposed the existence of
 * (a) the principal quantum number
 * (b) the azimuthal quantum number
 * (c) the spin quantum number
 * (d) the magnetic quantum number
 * 1) Circle all of the quantum numbers that tell you about the region in space where you're most likely to find the electron.
 * (a) the spin quantum number
 * (b) the magnetic quantum number
 * (c) the principal quantum number
 * (d) the azimuthal quantum number
 * 1) Select the correct statement from the list below. An electron with a spin quantum number of ms = −1/2
 * (a) cannot share an orbital with an electron that has a spin quantum number of ms = +1/2
 * (b) prefers to share an orbital with an electron that has a spin quantum number of ms = −1/2
 * (c) cannot share an orbital with an electron that has a spin quantum number of ms = −1/2
 * (d) cannot share an orbital with another electron
 * 1) What is the total spin in an electron orbital if
 * (a) the orbital contains one "spin-up" electron
 * (b) the orbital contains one "spin-down" electron
 * (c) the orbital contains two "spin-up" electrons
 * (d) the orbital contains one "spin-up" electron and one "spin-down" electron