Lewis diagrams, octet rule

In A-level chemistry you are taught that second-row atoms tend to complete their outer shell of electrons when bonding. The second-row shell can fit 8 electrons, and this method is often called "the octet rule". You would also have learnt how to draw Lewis diagrams:

A single line can also represent a shared pair of electrons:

The octet rule is great way to feel like a knowledgeable chemist, until someone asks you to draw the structure of carbon monoxide. You could draw this:

But the carbon is still two electrons away from an octet. If that structure is allowed, then why aren't these?

The structures are allowed, but they would react vigorously when in contact with other atoms to produce states which satisfy the octet rule - lower energy states.

Carbon monoxide can be burnt to produce CO₂ - a lower energy state which satisfies the octet rule for both atoms - but it doesn't happen immediately. And there are no obvious ways to tell what less-stable states can still exist at RTP (room temperature and pressure) or not.

In conclusion, the octet is useful but not infallible. The sure method of knowing what bonds an atom will form and what their shape are requires using computers to calculate wave equations. The results of these are drawn using molecular orbital diagrams, described later.

Mendeleev, polarizability, Fajan's rules

Dmitri Mendeleev devised the periodic table in 1869.

Dmitri Mendeleev (1834 - 1907)

The polarizability of an atom is the ability of its electron cloud to be distorted by an electric field, such as by neighboring ions. It usually denoted as α. Bonds which have a heavily polarized electron cloud between them are considered covalent.

As a chemist you may come across two ions and be expected to have some intuition about whether their bonding would be ionic or covalent, factors which affect this are summarized in Fajan's rules:

1. Small, highly charged cations have polarizing ability.
2. Large, highly charged anions are easily polarized.
3. Cations that do not have a noble-gas electron configuration are easily polarized.

One analogy for the first rule is trying to suck up water from a lake with a vacuum cleaner. You'd create more of a distortion by using a narrow tip to focus on a small area.

For the second rule we should remember that large anions usually have high principle quantum numbers. The gap between energy levels decreases as we increase n:

The textbook writes that polarizability is high when separation between frontier orbitals is low. I suppose electrons have an easier time moving through multiple small energy gaps then one large gap. Intuition for this may be found in the equations for quantum tunneling, where the chance of an electron passing through a barrier falls very rapidly as the length of the barrier is increased.

For the third rule, it makes sense that electrons in a more stable configuration are harder to drag away.

Hund's rule

Electrons will fill the lowest energy level, but there is another effect that influences which orbitals they fill, called Hund's rule:

When more than one orbital has the same energy, electrons occupy separate orbitals and do so with parallel spins (↑↑). 

Up and down arrows are a useful shorthand for the +1/2 or -1/2 spin of electrons. If we apply Hund's rule to a carbon atom, we know the electrons would be arranged like this:

And not like this:

The 3 orbitals inside the 2p shell are degenerate (same energy level) so it doesn't matter what direction they are pointing in, but it's common to use the order p_x, p_y and p_z. 

Using subscript for directions (i.e. the magnetic quantum number), and superscript for number of electrons, a free carbon atom can be denoted as 1s² 2s² 2p_x¹ 2p_y¹ 2p_z¹. This is usually simplified to 1s² 2s² 2p³.

Orbital penetration

In the previous post I mentioned energy level splitting of subshells:

In multiple-electron systems these are split into different energy levels, in that case the lower values are usually lower in energy level (i.e. S becomes lower than P)

The energy levels of multi-electron atoms are mostly found by spectroscopy. Understanding why the levels are arranged that way can be solved by computers using wave equations, but "a computer confirmed it" isn't a fun answer. A less reliable answer, but one more relatable to humans, can be found by considering orbital penetration.

Consider the radial distribution functions shown below, these show for each orbital how the probability of finding an electron varies with distances from the nucleus.

1S subshell

2S subshell

3S subshell

2P subshell

3P subshell

You can view more at the orbitron, but you should be able to see a pattern in these. Going up in principle quantum numbers, the first S has one hump, the second S has two humps, and so on. The other subshells follow the same pattern.

For an atom containing 2S and 2P electrons, such as carbon, we know from A-level chemistry that the S orbitals are filled first. The reason is electron shielding, electrons close to the nucleus reduce the attraction of electrons outside the nucleus, since negative charge repels. 2S electrons experience relatively less shielding then 2P electrons because they have an extra hump close to the nucleus, and you can see it by superimposing their radial distribution functions:

Superimposition of the 2S and 2P subshells

The penetration of the 2S subshell allows electrons in it to experience more nuclear charge, which is enough to dip the orbital lower in energy level. I'm aware that it isn't entirely obvious from the above graph, but this answer is enough for undergrad exams.

Note: The S subshell has the unique property of having a non-zero chance of being found on the nucleus itself. This means you should draw the functions touching the Y-axis at just above 0, with other orbitals hitting it right on 0. It is common for some exam marks to be based on this.

Quantum mechanics 2

Consider a single proton and add an electron orbiting around it, the electron will be termed to have negative energy. This is numerically the same amount of energy you need to put in to take the electron out of the orbital. An orbital with "lower energy" requires more energy to take an election out.

When describing atomic orbitals scientists prefer to consider an atom with one electron, either hydrogen, or a "hydrogenic atom" which can have multiple protons so long as it has just one electron. This is because a system with multiple electrons has electron-electron repulsion changing the energy levels of atomic orbitals, and working out this effect can be really hard.

Orbitals are described by physicists using wave functions (usually denoted as Ψ) which contain their properties, such as their shapes and energy levels.

A consequence of using wave functions is that the orbitals of a hydrogenic atom can be described by three quantum numbers:

1. The principle quantum number, denoted as n. This is an integer which can range from 1 to infinity. The energy level of the orbital increases as n increases.

This number also indicates what shell the orbital is in. In A-level chemistry you are taught that atoms are made more stable by completing a shell, either 2 electrons (n = 1), 8 electrons (n = 2) or 18 electrions (n =3).

Shells with a higher principle quantum number are also larger and more diffuse.

2. The angular momentum quantum number (also called the azimuthal quantum number), denoted as ℓ. "ℓ" is just a lower case "L". This exists at the values 0 ≤ ℓ ≤ n − 1

This describes the shape of the orbital. It also denotes the subshell of n which the orbital is in. In A-level chemistry you learn these as S, P and D subshells, or visually as sphere, dumbbell, and double-dumbbell. The S subshell corresponds to ℓ=0, P is ℓ=1, and D is ℓ=2.  In multiple-electron systems these are split into different energy levels, in that case the lower values are usually lower in energy level (i.e. S becomes lower than P).

So far we have:

N = 1,    ℓ = 0
N = 2,    ℓ = 0, 1
N = 3,    ℓ = 0, 1, 2

This should make sense compared to A-level, where the first shell only had a S subshell, the second had S and P, and the third had S, P and D.

3. The magnetic quantum number, denoted as m_ℓ. This can take the values −ℓ ≤ m_ℓ ≤ ℓ. This describes the direction of the orbital. Now we have:

N = 1,    ℓ = 0,     m_ℓ = 0
N = 2,    ℓ = 0      m_ℓ = 0
             ℓ = 1      m_ℓ = 1, 0, -1
N = 3     ℓ = 0     m_ℓ = 0
              ℓ = 1    m_ℓ = 1, 0, -1
              ℓ = 2    m_ℓ = 2, 1, 0, -1, -2

So the ℓ = 0, or S subshell, can only point one direction. The P subshell can point three directions, and the D subshell can point in five directions, as we already knew.

The Pauli exclusion principle tell us:

No two fermions (which includes electrons) can occupy the same space with the same spin. 

Electrons can have a spin of + 1/2 or -1/2, so a maximum of two electrons can fit into an orbital. If you apply that to the above table and work out the number of possible electrons you could fit into each shell, you get 2 electrons at n = 1, 8 electrons at n = 2, and 18 electrons at n = 3. So this system of notation fits perfectly with filling up the shells in A-level chemistry.

Electrons should also be assumed to fall down to orbitals with lower energy levels, provided they are not prevented by the Pauli exclusion principle. This is summed up in the Aufbau principle.

The orbitals of lower energy are filled in first with the electrons and only then the orbitals of high energy are filled.

The German word "Aufbau" means "building up, construction".

Quantum mechanics

The maths and ideas used in deriving quantum mechanics are very complicated, but many conclusions reached from it are very understandable and practically useful.

This means that students are first taught quantum-mechanical conclusions long before the ability to work them out for ourselves, if we ever do. For training a practical chemist, that's a good idea. For some students that can be unsatisfying.

Vocabulary can sometimes mask our ignorance. "orbital angular momentum quantum number" sounds a lot more clever then "mysterious number which partly describes an atom, which follows slightly different rules to the other numbers".

But the second quote more accurately describes what the number means to people learning the conclusions first. Of all the people who use quantum mechanical vocabulary, few do it with confidence.

Technetium, sodium lamps, indium oxide

The first synthetic element was technetium. It is from the greek word for artificial, "τεχνητός". (teknitos). The longest known half-life of a technetium isotope is 4.2 million years, so any techenetium when the earth was formed would have long-since decayed, and there isn't a natural process on earth that replenishes it.

Shiny, radioactive, tarnishes slowly in moist air

I would hold this in my hand for a moment, even with it's emitted alpha particles destroying my skin. There is something mysterious about a metal which does not naturally exist on earth.

Another isotope of technetium is used is medicine for its two useful properties: emission of gamma rays (for medical imaging), and a half-life of only 6 hours.

Consider low-pressure sodium lamps, as in orange streetlights:

Sodium-lit snow, my second-favorite weather after thunderstorms

Low-pressure sodium lights contain a vacuum with gaseous sodium atoms inside, and a high-voltage electrical current is run through it. This trick can be used to get light out of any element. Along with being a good way of producing specific colors, it is also very efficient at producing light relative to ordinary light-bulbs.

The sodium light before it is turned on contains some neon, argon, and solid sodium. Initially the argon and neon will be excited to produce a dark pink glow, it will then turn orange as it is warmed and the sodium evaporates.

Something which helps this process is the coating of the outside glass with indium tin oxide, typically In₂O₃ combined with 10% by mass SnO₂. This reflects infrared (heat) while allowing visible light to pass through.

Indium tin oxide also has the very valuable property of being able to conduct electricity while transparent, so you are likely to come across it in various displays, such as computer monitors, televisions and phones. Indium is a very rare metal, and its use in conductive conductive coating has caused it's price to rise rapidly in the past decade.