[Waffles7]

We learned Quantum Numbers today but I cannot for the life of me figure out how "spin" works. We need to do it for nitrogen. I know that it has something to do with "1/2" but I don't understand how to determine if it is positive or negative. Thanks.

[mikasaur]

Hello and welcome to the forum!

Understanding how "spin" works can actually be very detailed and complicated (though I'm sure Enthalpy [the forum user not the concept] will be happy to chime in and explain it). So here's some simple info for a high school student such as yourself.

Spin is described by the fourth quantum number and for electrons can take on two values, as you stated: -1/2 and +1/2. The positive and negative signs are kind of arbitrary in the way that north and south are arbitrary. Turn a map of the word upside down and now up is down and down is up but it's still the same map.

Each orbital in an atom (e.g. 2s, 2p_x, 3d_xy) can contain at most two electrons. But the Pauli Exclusion Principle states that no two electrons can occupy the same quantum state simultaneously, i.e. they cannot have the same quantum numbers. So if you have two electrons in the same orbital, one must be +1/2 and the other -1/2. If you have just one electron in an orbital then you can just say that it has a +1/2 spin.

[Corribus]

The classical interpretation is of a quantum particle spinning on its axis, which gives it an intrinsic angular momentum. At the same time, quantum particles that are orbiting (not a good word to use for atoms, but again we're in a classical frame of mind) the nucleus have an associated orbital angular momentum. Like most other energy-related quantum properties, both angular momentum values are quantized (can take on only specific values). In the quantum world, we use a 'unit value' that we usually then multiply by a 'quantum number' to indicate the total observable magnitude - the reason relates to the Schrodinger equation, which is probably unimportant to go into right now. Fermions, which include electrons, have half integer spin quantum numbers (1/2, 3/2, 5/2, etc.). The total amount of spin angular momentum is determined via the simple equation SQRT[s(s+1)]ħ. Here ħ is your 'unit value' and the quantum number is based on s. For electrons and other elementary fermions, s can only have a value of 1/2, so the total allowed spin angular momentum for an electron is SQRT[3/4]ħ.

An important thing to understand is that momentum is a vector quantity. For reasons well beyond the scope of general chemistry, we are usually concerned about the portion of the angular momentum that is directed along a single spatial axis, which (for one of the three potential axes) is also quantized. For convenience, we pick the z-axis. A total spin angular momentum value of SQRT[3/4]ħ has z-axis projection of (1/2)ħ. However, while the total spin momentum magnitude is fixed at SQRT[3/4]ħ, we note that there is no reason why it has to be oriented up or down (along the positive z or negative z axes). In the absence of any other particles that might influence it, there is a statistically equal probability of both possibilities. For this reason, we say that the z-axis spin angular momentum is ±(1/2)ħ. The associated quantum number is simply ±(1/2).

(This is, by the way, mostly analogous to the orbital momentum quantum numbers you may have learned about, L and m_L, the former representing the total orbital angular momentum and the latter representing the component of that momentum that is directed along an arbitrary [usually z] axis. The difference being that the orbital angular momentum isn't confined to a single half integer value. Notice that whatever the positive L value, the m_L values have both positive and negative integers, reflecting the fact that the total momentum L is just an absolute magnitude, but m_L represents that magnitude along the z-axis, and there is not preference for it to be directed in one direction or the other. The total spin momentum quantum number, s, is usually not of much importance, but the quantum number that indicates the amount of momentum directed along the same z axis has a lot of importance in chemistry and physics and is one of the four you usually learn about in general chemistry.)

How does this relate to anything important? Well, electrons are charged particles. A moving or rotating charge creates a magnetic field. If a nearby charged particle also generates its own magnetic field, then there becomes an energetic interaction between the two particles. Or, in many cases, between the particle and itself - if an electron generates a magnetic field due to its rotation around a nucleus, and it also generates a magnetic field due to its internal spin, then those fields can interact either positively or negatively, depending on their mutual vector orientation. These kinds of interactions are responsible for the way electrons fill into the orbitals of multi-electron atoms. Also, they impact the energy states that an electron may occupy around an atom. In spectroscopy, we see finely divided spectral bands because of these interactions between the spin and orbital angular momentum, something we call fine structure. Basically, the orbital momentum and spin momentum, even for a single electron, can be aligned in parallel or antiparallel fashion; one of these orientations costs a little more energy than another, which impacts the potential spacings between electron levels that we probe with spectroscopic techniques.

In multielectron atoms, the spin impacts the electron configuration in a number of ways. For one thing, fermions obey the Pauli-Exclusion principle, which means that electrons sharing the same orbital have to line up in antiparallel fashion. The implication of this is that there are only two electrons allowed per orbital. This pretty much determines the physical nature of all matter, including the way substances interact, their chemical and physical properties... everything! Actually, though, spins prefer not to line up in antiparallel fashion if they can help it, which means that when you are filling orbitals that have otherwise the same energy, it is the usual rule to fill them each half empty first. For example, in nitrogen, you have three p orbitals and three electrons. Each electron technically can take on z-directed angular momentum values of ±(1/2). There are a number of ways you can fill the orbitals, but the lowest energy state (all other things equal, which they often aren't) is to put one electron into each of these orbitals, with their spins aligned. Each of these unpaired spins has the potential to combine with unpaired spins from nearby atoms... so now you see why nitrogen often likes to bond three times. (The 4-fold bonding preference of carbon, which has two unpaired spins, is not as simple to explain, but this is a start for you I hope to see why spins have such enormous importance in chemistry.)

[Enthalpy]

[...] I'm sure Enthalpy will be happy to chime in and explain [the spin]

Well, no, because I have no clear representation about the spin, alas. I've a working mental model of the orbital momentum which I misuse for the spin by analogy - not satisfying. But fortunately we have Corribus.

[Corribus]

Just to elaborate slightly - there is a tendency to apply classical frames of reference to quantum phenomena. This isn't just a learning tool for today's students. It was also a useful part of the historical development of quantum mechanics.

The idea of electrons (or other charged elementary particles) having an angular momentum conjures up pictures of electrons having absolute rotating trajectories around nuclei. The closest analogy people use is often of planets orbiting the sun in more or less discrete orbits. This planetary picture may have served as inspiration to Neils Bohr's "stationary orbit" model of the atom. Of course Bohr's model is completely wrong, and we now know very well that electrons do not behave this way. They do not have discrete positions or deterministic trajectories. We speak in terms of probability densities.* Nevertheless, these electrons DO have discrete orbital angular momentum and energy values, which gives rise to the orbitals (also idealized constructs) that we learn about in chemistry. Quantum and classical physics are clearly related, but the way that quantum reality condenses into the classical world when masses are increased beyond a certain threshold is still a matter that physicists wrestle with.

When it became apparent through famous experiments like those by Stern and Gerlach that electrons also have an intrinsic angular momentum (i.e., one not related to orbital motion), it was natural for those early scientists to try to interpret the experimental findings through their classical understandings of the world. The obvious solution to electrons having their own intrinsic angular momentum was to interpret it as spinning motion. Moving back to the planetary model, it would be akin to planets spinning on their intrinsic axes while at the same time revolving around the sun. It's a pretty picture but no physicist today takes it too seriously. Electrons we now know with great confidence are not "point particles". They have tremendous wavelike character. What does it mean for a wave to spin on its axis? Does a wave even have an axis around which to physically spin?

So, I think it's important to differentiate the facts from helpful classical interpretations of the facts. The facts are (1) that electrons have an extrinsic angular momentum associated with their mostly electrostatic interactions with nearby nuclei and (2) they also have intrinsic angular momentum that is completely independent from any interactions with nearby particles. The former we interpret as being the quantum version (whatever that is) of particle-like electrons revolving around the particle-like nucleus, and the latter we interpret as being the quantum version (whatever that is) of particle-like electrons spinning around their own axes. The angular momenta of electrons have great ramifications to chemistry and physics, but these simple interpretations of the facts largely ignore other facts about quantum physics, notably that small-mass particles are dominated by wavelike character that renders "particle-based" Newtonian notions of trajectory, rotation, motion, and velocity highly inaccurate or even irrelevant. In short, words like "revolve, spin, move, collide" all derive from classical understanding of the way things like baseballs work. Their meaning as related to very small particles is largely undefined.

Students should also realize that these are not issues that are completely solved scientifically or philosophically by the physics community, so their confusion is completely understandable - even inevitable. When it comes to untangling quantum physics, at some point everyone has to draw a line somewhere and just accept things as axiomatic. This line is typically drawn lower for physicists than chemists, so if you want deeper insight, you might consult some physicists. But we warned, those people speak in tongues.

You might find this link somewhat useful for interesting ruminations on the topic: http://www.scientificamerican.com/article/what-exactly-is-the-spin/

*(Those involved in the foundation of quantum physics wrestled with the implications as this gradually became apparent, and not only the scientific implications; a non-deterministic universe has a lot of philosophical ramifications that early quantum physicists found unsettling, leading some to question whether the theory could possibly be right or complete. Einstein famously remarked "But an inner voice tells me that this [quantum theory] is not yet the right track. The theory yields much, but it hardly brings us closer to the Old One's secrets. I, in any case, am convinced that He does not play dice." To which Neils Bohr supposedly replied by asking Einstein to stop telling God what to do. So there you go - two of the 20th centuries smartest physicists couldn't agree on it, so what hope have we? )

[Vidya]

We learned Quantum Numbers today but I cannot for the life of me figure out how "spin" works. We need to do it for nitrogen. I know that it has something to do with "1/2" but I don't understand how to determine if it is positive or negative. Thanks.

If some ask you to spin ...then you know only two directions are feasible ...clockwise and anticlockwise .. all charged particles like electron revolving around the nucleus are also spinning on their axis( just like earth is also revolving around the sun and spinning on the axis to form days and night).An electron can spin in two directions ...clockwise and anticlockwise and they are represented by +1/2 or -1/2...you can take any sign for any direction.

[Enthalpy]

If some ask you to spin [...] all charged particles like electron revolving around the nucleus are also spinning on their axis( just like earth is also revolving around the sun and spinning on the axis to form days and night). [...]

If you imagine the spin like a rotation then you miss the essential feature that the electron has intrinsic angular and magnetic momenta around all axes.

"Electron revolving around the nucleus" is very misleading. For all spherical orbitals, there is no revolution at all. The other orbitals are stationary wave functions, that is, their amplitude is independent of the time. This doesn't resemble a classical revolution.

[Vidya]

I agree with you Enthalpy and salute your knowledge also..however for high school students we bring down the concept to simplest manner....

[Enthalpy]

When I was a high school student too, I preferred a correct explanation to a simple wrong one.

And for those who go on studying, debunking wrong knowledge costs a huge effort and often leaves harmful remnants. A bigger effort than learning the correct theory right from the beginning.

That's especially obvious in quantum mechanics, where so many newspapers, books and courses contribute to false representations that people have to fight against and are mislead.

So: better a difficult explanation, or none at all, than a wrong one.

[pm133]

I agree with you Enthalpy and salute your knowledge also..however for high school students we bring down the concept to simplest manner....

I appreciate it's difficult teaching these sorts of concepts to kids but you really need to emphasise to them that the electron it isn't physically doing this.
As others have said above, you can do serious damage to the confidence of students in the future if they are led to believe that electrons are classically spinning around an axis.