Wednesday, January 24, 2018

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Chemistry - Electronic Structure

Somewhere in your education, you were probably shown a model of the atom showing a dense nucleus (comprised of protons and neutrons) surrounded by electrons whirling around like satellites around a planet. While there are some useful features of this model, it is largely in error. A more realistic view of atomic structure begins with the realization that electrons do not exist as discrete particles, but rather as wave packets. In a sense, they orbit the nucleus within certain areas of probability, as described by the principles of quantum mechanics. One way to envision this is to think of an electron’s placement around the nucleus in the same way you might picture a city shrouded by a layer of fog. The electron does not have a discrete location (even though there is a discrete number of them found in every atom), but rather may be found anywhere within a certain region to varying degrees of probability.

Things get even stranger as we encounter atoms having multiple electrons. No two electrons may share the same quantum states in the same atom – a principle called the Pauli Exclusion Principle. This means the electrons surrounding a nucleus must exist in distinct patterns. Just a few of these patterns are shown here as orbitals (regions of high probability where up to two electrons may be found surrounding a nucleus):1


Electrons situate themselves around the nucleus of any atom according to one basic rule: the minimization of potential energy. That is, the electrons “try” to get as close to the nucleus as they can. Given the electrostatic attraction between negative electrons and the positive nucleus of an atom, there is potential energy stored in the “elevation” between an orbiting electron and the nucleus, just as there is gravitational potential energy in any object orbiting a planet. Electrons lose energy as they situate themselves closer to the nucleus, and it requires an external input of energy to move an electron further away from its parent nucleus.

In a sense, most of chemistry may be explained by this principle of minimized potential energy. Electrons “want” to “fall” as close as they can to the positively-charged nucleus. However, there is limited “seating” around the nucleus. As described by Pauli’s Exclusion Principle, electrons cannot simply pile on top of each other in their quest for minimum energy, but rather must occupy certain regions of space where their quantum states will be unique.

An analogy for visualizing this is to think of it in terms of an amphitheater, having concentric rows of seats where spectators may view the event on stage. Everyone wants to be as close to the action as possible, but each person is constrained to sitting in one seat. As a result, all the inner seats are filled, with the only empty seats being further away from the stage. The concept of energy fits neatly into this analogy as well: just as electrons give up energy to “fall into” lower-energy regions around the nucleus, people must give up money to sit in the seats closest to the action on stage.

The energy levels available for orbiting electrons are divided into categories of shells and subshells. A “shell” (or, principal quantum number, n) describes the main energy level of an electron. In our amphitheater analogy, this is the equivalent of seating sections. A “subshell” (or, subsidiary quantum number, l) further divides the energy levels within each shell, and assigns different shapes to the electrons’ probability “clouds.” In the amphitheater analogy, this would be pairs of seats within each section having varying degrees of comfort, each identical seat pair being one orbital. Just as people want to sit as close to the action as possible (electrons occupying the lowest-value shell), they also desire to sit in the most comfortable seats they can find in each section (electrons occupying the lowest-energy subshell within each shell).

Chemists identify electron shells both by number (the value of the quantum number n) and/or by capital letters: the first shell by the letter K, the second by L, the third by M, and the fourth by N. Higher-order shells exist for atoms requiring a lot of electrons (high atomic number), and the lettering pattern is alphabetic (fifth shell is O, sixth is P, etc.). Each successive shell has a different number of subshells available, like amphitheater seating sections having different numbers of seats (the sections closest to the stage having the fewest seats, and the furthest sections having the most seats per section).

A numbering and lettering system is also used by chemists to identify subshells within each shell (the l quantum number value starting with zero, and lower-case letters beginning with “s”): the first subshell (l = 0) in any shell represented by the letter s, the second (l = 1) by p, the third (l = 2) by d, the fourth (l = 3) by f, and all others by successive lower-case letters of the alphabet. Each subshell of each shell has an even-numbered capacity for electrons, since the electrons in each subshell are organized in “orbital” regions, each orbital handling a maximum of two electrons. The number of orbitals per shell is equal to twice the l value plus one. An “s” subshell has one orbital holding up to two electrons. A “p” subshell has three orbitals holding up to six electrons total. A “d” subshell has five orbitals holding up to ten electrons total. An “f” subshell has seven orbitals holding up to 14 electrons total.

The number of subshells in any shell is the same as that shell’s n value. Thus, the first (K) shell has but one subshell, “s”. The second (L) shell has two subshells, an “s” and a “p”. The third (M) shell has three subshells available, an “s”, a “p”, and a “d”; and so on.

Here is a list of the first few shells, their subshells, and electron capacity for each:


   Subshell electron capacity   
 n = 1 ; K  l = 0 ; s  2
 n = 2 ; L  l = 0 ; s  2
   l = 1 ; p  6
 n = 3 ; M  l = 0 ; s  2
   l = 1 ; p  6
   l = 2 ; d  10
 n = 4 ; N   l = 0 ; s  2
   l = 1 ; p  6
   l = 2 ; d  10
   l = 3 ; f  14

The complete electron configuration for an atom may be expressed using spectroscopic notation, showing the shell numbers, subshell letters, and number of electrons residing within each subshell as a superscript. For example, the element Helium (with an atomic number of 2) would be expressed as 1s2, with just two electrons in the “s” subshell of the first shell. The following table shows the electron structures of the first nineteen elements in the periodic table, from the element hydrogen (atomic number = 1) to potassium (atomic number = 19):

  Atomic number 
  Electron configuration  
 Hydrogen 1  1s1
 Helium 2  1s2
 Lithium 3  1s22s1
 Beryllium 4  1s22s2
 Boron 5  1s22s22p1
 Carbon 6  1s22s22p2
 Nitrogen 7  1s22s22p3
 Oxygen 8  1s22s22p4
 Fluorine 9  1s22s22p5
Neon 10  1s22s22p6
 Sodium 11  1s22s22p63s1
 Magnesium 12  1s22s22p63s2
 Aluminum 13  1s22s22p63s23p1
Silicon 14  1s22s22p63s23p2
 Phosphorus 15  1s22s22p63s23p3
 Sulfur 16  1s22s22p63s23p4
 Chlorine 17  1s22s22p63s23p5
 Argon 18  1s22s22p63s23p6
 Potassium 19  1s22s22p63s23p64s1

In order to avoid having to write unwieldy spectroscopic descriptions of each element’s electron structure, it is customary to write the notation only for subshells that are unfilled. For example, instead of writing the electron structure of the element Aluminum as 1s22s22p63s23p1, we might just as well write a condensed version showing only the last subshell (3p1), since all the previous subshells are completely full.

Another way to abbreviate the spectroscopic notation for elements is to condense all the shells below the newest (unfilled) shell as the corresponding noble element, in brackets. To use the example of Aluminum again, we could write its spectroscopic notation as [Ne]3s23p1 since its shell 1 and shell 2 configurations are completely described by the electron configuration of Neon.

Re-writing our electron shell table for the first nineteen elements using this condensed notation:

  Atomic number 
  Electron configuration  
 Hydrogen 1 1s1
 Helium 2 1s2
 Lithium 3 [He]2s1
 Beryllium 4 [He]2s2
 Boron 5  [He]2s22p1
 Carbon 6 [He]2s22p2
 Nitrogen 7 [He]2s22p3
 Oxygen 8 [He]2s22p4
 Fluorine 9 [He]2s22p5
Neon 10 [He]2s22p6
 Sodium 11 [Ne]3s1
 Magnesium 12 [Ne]3s2
 Aluminum 13 [Ne]3s23p1
Silicon 14 [Ne]3s23p2
 Phosphorus 15 [Ne]3s23p3
 Sulfur 16 [Ne]3s23p4
 Chlorine 17 [Ne]3s23p5
 Argon 18 [Ne]3s23p6
 Potassium 19 [Ar]4s1

If we progress from element to element in increasing atomic number, we see that no new shell begins to form until after we reach the noble element for that period2 at the far right-hand column. With the beginning of each new period at the far-left end of the Table, we see the beginning of the next higher-order electron shell. The shell(s) below are represented by whichever noble element shares that same configuration, indicating a “noble core” of electrons residing in extremely stable (low-energy) regions around the nucleus.

The beginning of the next higher-order shell is what accounts for the periodic cycle of ionization energies we see in elements of progressing atomic number. The first electron to take residence in a new shell is very easy to remove, unlike the electrons residing in the “noble” configuration shell(s) below:

Not only is the “noble core” notation a convenience for tersely describing the electron structure of an element, but it also represents an important concept in chemistry: the idea of valence. Electrons residing in lower-order shells are, by definition, at lower energy states than electrons residing in higher-order shells. Therefore, the electrons in unfilled shells are more readily pulled away from the atom than those lying in completely full shells below. These “outer” electrons are called valence electrons, and their number determines how readily an atom will chemically interact with another atom.

If we examine the electron structures of atoms with successively greater atomic numbers (more protons in the nucleus, therefore more electrons in orbit to balance the electrical charge), we notice that the shells and subshells fill up in an interesting pattern. One might think that all the lower order shells get completely filled before any electrons go into a higher-order shell – just as we might expect people to fill every seat in all the lower seating sections of an amphitheater before filling any of the higher seats – but this is not the case for every elements. Instead, the energy levels of subshells within shells is split, such that certain subshells within a higher shell will have a lower energy value than certain subshells within a lower shell. Referring back to our amphitheater analogy, where seating sections represented shells and seats of varying comfort represented subshells, it is as though people choose to fill the more comfortable seats in the next higher seating section before taking the less-comfortable seats in lowest available section, the desire for comfort trumping the desire for proximity to the stage.

A rule commonly taught in introductory chemistry courses called the Madelung rule (also referred to as Aufbau order, after the German verb aufbauen meaning “to build up”) is that subshells fill with increasing atomic number in such an order that the subshell with the lowest n+l value, in the lowest shell, gets filled before any others.

The following graphic illustrates this ordering:


Madelung filling order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f → 5d 6p 7s 5f 6d 7p 8s (etc.)

It should be noted that exceptions exist for this rule. We see one of those exceptions with the element chromium (24Cr). Strictly following the Madelung rule in progressing from vanadium (atomic number = 23, valence electron structure 3d34s2) to chromium (atomic number = 24), we would expect the next electron to take residence in the “3d” subshell making chromium’s valence structure be 3d44s2, but instead we find two more electrons residing in chromium’s 3d subshell with one less in the 4s subshell (3d54s1). The sequence resumes its expected progression with the next element, manganese (atomic number = 25, valence electron structure 3d54s2). The general principle of energy minimization still holds true . . . it’s just that the relative energies of succeeding subshells do not follow a simple rule structure. In other words, the Aufbau order is an over-simplified view of reality. To use the amphitheater analogy again, it’s as if someone gave up one of the nice chairs in seating section 4 to be closer to a friend who just occupied one of the less comfortable chairs in seating section 3.

The practical importance of electron configurations in chemistry is the potential energy possessed by electrons as they reside in different shells and subshells. This is extremely important in the formation and breaking of chemical bonds, which occur due to the interaction of electrons between two or more atoms. A chemical bond occurs between atoms when the outer-most (valence) electrons of those atoms mutually arrange themselves in energy states that are collectively lower than they would be individually. The ability for different atoms to join in chemical bonds completely depends upon the default energy states of electrons in each atom, as well as the next available energy states in the other atoms. Atoms will form stable bonds only if the union allows electrons to assume lower energy levels. This is why different elements are very selective regarding which elements they will chemically bond with to form compounds: not all combinations of atoms result in decreased potential energy.

The amount of energy required to break a chemical bond (i.e. separate the atoms from each other) is the same amount of energy required to restore the atoms’ electrons to their previous (default) states before they joined. This is the same amount of energy released by the atoms as they come together to form the bond. Thus, we see the foundation of the general principle in chemistry that forming chemical bonds releases energy, while breaking chemical bonds requires an input of energy from an external source. We also see in this fact an expression of the Conservation of Energy: all the energy “invested” in breaking bonds between different atoms is accounted for in the energy release occurring when those atoms re-join.

In summary, the whole of chemistry is a consequence of electrons not being able to assume arbitrary positions around the nucleus of an atom. Instead, they seek the lowest possible energy levels within a framework allowing them to retain unique quantum states. Atoms with mutually agreeable electron structures readily bond together to form molecules, and they release energy in the process of joining. Molecules may be broken up into their constituent atoms, if sufficient energy is applied to overcome the bond. Atoms with incompatible electron structures do not form stable bonds with each other.


1These orbitals just happen to be the 1s, 2p, 3d, and 4f orbitals, as viewed from left to right. In each case, the nucleus lies at the geometric center of each shape. In a real atom, all orbitals share the same center, which means any atom having more than two electrons (that’s all elements except for hydrogen and helium!) will have multiple orbitals around one nucleus. This four-set of orbital visualizations shows what some orbitals would look like if viewed in isolation.

2Recall the definition of a “period” in the Periodic Table being a horizontal row, with each vertical column being called a “group”.

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