Factors affecting orbital overlap, boron halides

There are three main factors which affect orbital overlap, the first two have been described earlier.

1. Symmetry

2. Proximity of energy levels

3. Size of the orbitals

This third factor helps explain why boron trifluoride is a weaker lewis acid than boron trichloride:

From simple electronegativity arguments we would expect BF₃ to be a stronger acid, since electrons would be pulled away from the boron atom hence encouraging it to accept an electron pair. 

The reason BF₃ is weaker is that the halogen atoms contain electrons in p orbitals, which in boron trihalides have a small overlap with the empty p orbital of boron. This effect is weaker in heavier halogens because of the size difference:

The p-orbital overlap can also represented as resonance structures:

This overlap/resonance/extra stability cannot happen in an adduct, since in an adduct the orbital is already occupied by a lone pair.

Acids

Acids have two main definitions. The Bronsted-Lowly definition is the one used an A-level.

Bronsted-Lowly acid: A proton donor.

Lewis acid: A substance which accept a pair of electrons.

H+ is a Lewis acid, it has an empty 1s orbital to take in a lone pair:

Boron trichloride is also a Lewis acid. The boron atom in this is sp² hybridized, leaving an unoccupied p orbital. This p orbital can accept an electron pair.

A Lewis base is a lone-pair donor. When a Lewis base donates a lone pair to a Lewis atom it creates a complex called an adduct. An example of an adduct you would have seen is using AlCl₃ as a friedel-crafts catalyst in the electrophillic substitution of benzene:

Lewis acids in water are similar to Bronsted acids, because the adduct they form with water allows easy disassociation of H+:

Doping and corundum

Intrinsic defects occur in a pure crystal, without having to add other elements.

Extrinsic defects are from the presence of impurities, from atoms not contained in the normal stoichiometry of the pure crystal. These can be substitutional or interstitial. Even trace amounts of these impurities can radically alter macroscopic properties. When these impurities are purposely added, they are often called "dopants", and the process is called "doping". Non-crystalline compounds can also be doped.

Steel is an example of a doped crystal structure mentioned already. Another good example is aluminium oxide, also called alumina. This is a white power known for its strength and commonly used as an abrasive, such as in grinding tools, sandpaper, and toothpaste.

The most stable crystalline form of aluminium oxide is corundum. Pure corundum is transparent:

If corundum contains a fraction of Al3+ ions substituted for Cr3+, it will produce ruby:

Other colors of corundum are called sapphires. A blue sapphire is created by substituting a mixture of Fe²⁺ and Ti⁴⁺ ions into the place of Al³⁺:

Anti-site defect, coloured gold

An anti-site defect can also be called "atom interchange". It consists of an interchanged pair of atoms:

It is a common defect in metal alloys, which consist of neutral atoms, such as in CuAu above. It is very unfavourable in binary ionic compounds because it would cause similarly-charged ions to be neighbouring.

Gold, silver and copper blend together smoothly in a range of alloys varying in colour.

It is interesting that the color many people associate with gold is not the color of pure gold, since marketplace gold usually has copper and silver added, making it yellow or yellowish. 

Many ancient cultures had high copper content in their gold, so they associated it more with red than yellow.

Gold can also be made white by mixing it with about 10% nickel, palladium or manganese. This alloy is what "white gold" means in jewellers, it is unlikely to refer to a gold/silver alloy, though they might mix a small amount of silver in.

Generally a small touch of zinc is added to harden the alloy.

Pure gold is 24 carat. Mixing 14 parts of gold to 10 parts something else will create 14 carat gold, 18 to 6 creates 18K gold, and so on.

Frenkel defects

Frenkel defects are when an atom or ion in a crystal is displaced into a interstitial site.

Frenkel defects are more commonly found in compounds with low coordination numbers.

Vacancy and Schottky defects

A vacancy defect in a crystal is simply when one of the atoms is missing from a lattice site:

In ionic solids these are called Schottky defects:

Schottky defects retain the stoichiometry of the compound, since any any ions removed have to be balanced by counter-ions. Schottky defects are mostly found in compounds which have:

1. High ionic character

2. High coordination numbers

3. Similarly sized ions.

Effect of defects on Gibbs free energy

Below is the relationship of Gibbs free energy of a solid to its number of defects:

Adding a defect is normally endothermic so it makes the enthalpy less negative, but it also increases disorder and so increases in the entropy. Since initial defects increase entropy at a faster rate, then initial defects will increase TS more than they raise H. Therefore any crystal above T=0 will have its minimum Gibbs energy at a non-zero number of defects.

TS will also rise with temperature. You can tell from the above graph that this will shift the Gibbs free energy minimum to the right, thus increasing the number of defects, since any substance will spontaneously change towards the structure with the lowest Gibbs free energy.

Diminishing returns

Imagine adding defects to a perfectly ordered crystal, the first defects added will increase entropy much more than the others:

Likewise, increasing temperature near absolute 0 will increase entropy much more than increasing the temperature of a hot object:

One analogy I've seen used is that shouting in a silent room of people produces more disorder than shouting into a room full of people talking. Actual explanations are given by statistical arguments.

For a really simple example, picture a 4-atom crystal and assume a defect means a missing atom. One defect means 4 microstates, two defects means 6 microstates, the first defect is therefore more significant.

Gibbs free energy

Exothermic reactions are not always spontaneous, and spontaneous reactions are not always exothermic. We can't say that a reaction will happen just because it releases energy.

But if we know the change of both enthalpy and entropy in a reaction, we can work out whether it would happen using the formula for Gibbs free energy:

G = H + TS

Where
G = Gibbs free energy
H = Enthalpy of formation of system
T = Temperature
S = Entropy of system

Reactions precede if the change in Gibbs free energy is negative, so in chemistry the following equation is more common:

ΔG = ΔH + TΔS

Where
ΔG = Change in Gibbs free energy
ΔH = Reaction enthalpy change
ΔS = Reaction entropy change

Gibbs free energy equation assumes a constant temperature and pressure, so we are assuming that any temperature change from the heat of ΔH is negligible.

We can try applying this equation to the evaporation of water, which has a positive enthalpy change (since bonds are being broken) and a negative entropy change (gaseous molecules can occupy more microstates).

To be spontaneous, ΔG has to be negative, which requires ΔH - TΔS to be below 0


Using data from wikipedia:

Heat of vaporisation of 1 mole of water = 40680 J / mol
Entropy of vaporisation of water = 108.9 J/(K mol)

So temperature has to be 373.56 Kelvin for ΔH = TΔS, and higher than this for the reaction to proceed, corresponding perfectly with the known boiling point of water.

The laws of thermodynamics

Remember that scientific laws are not proven from principles, they are just statements shown to be true via repeated experiments.


The zeroth law:

If two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

The first law:

The energy of an isolated system remains constant.

The second law:

The entropy of an isolated system increases in the course of a spontaneous change.

The third law:

The entropy of a perfect crystal approaches 0 as the temperature approaches 0.


Definitions:

Isolated system: System which matter or energy is not being transferred into. The universe is seen as an isolated system. Smaller things can be modeled as an isolated system, such as a insulated box.

Spontaneous: Happens without interference, such as the transfer of heat from a hot object to a cold object, or some phases changes like the evaporation of water.

Entropy: Often vaguely called a "measure of disorder", or more precisely, the "the number of possible states at the same energy level". Increasing the volume of a gas will increase entropy, since higher volume mean more possible locational arrangements of gas molecules. Increasing temperature will too, since it allows kinetic, vibrational and rotational energy levels to be filled in more possible states. In thermodynamics these possible states are called microstates. Microstates are ways of arranging a system without changing macrostates such as temperature and volume.

Consider a group of atoms in a small separate universe. Being a gas will increase the volume of the atoms, increasing entropy. But exothermic bond formation from being becoming a solid would increase temperature, hence also increasing entropy. How do we know what increase in entropy is higher, and hence what will spontaneously happen?

The answer to this kind of question is given by equations, coming in a later post.

Open system: Energy and matter can transfer between the systems and surroundings

Closed system: Energy but not matter can be transferred between the system and surroundings.

Extra note:

The second law can also be written as "changes which increase entropy happen spontaneously" or "entropy always increases". This has interesting consequences for speculation about the future of the universe in the idea of a "heat death", which is the universe reaching the state of maximum entropy - a dark endless sea of photons.

This the theme of "The Last Question", widely regarded as Isaac Asimov's best short story. It is available here.

Nondirectional bonding and co-ordination numbers

This graph depicts the dependence of ionic crystal structure (empirically determined) on principle quantum number and electronegativity difference between ions.

Co-ordination number increases as n increases. Low co-ordination is preferred by crystals with covalent character. High co-ordination is preferred by crystals with metallic (non-directional) character. Non-directional bonding increases with n according to this quote:

The principal quantum number of an atom is a measure of the directional character of the bonds formed by this atom. As n increases, the atomic orbitals involved in the bond formation and hence the bonds themselves gradually lose their directional properties.

Non-directional character also increases with Δχ:

In compounds the directional character of the bonds does not depend only on n but also on the electronegativity difference, because the bonds become increasingly ionic and hence more nondirectional as Δχ increases.

Heavy group 14 compounds do not fit the pattern well in the above graph:

They are sometimes 6-coordinate when 4-coordinate is expected. This can be attributed to the inert pair effect. Lower group-14 elements are less likely to involve their two S electrons in bonding. This is the reason carbon and silicon tend to form a +4 oxidation state while lead and tin tend to form +2. Or the reason carbon dioxide is CO₂ while lead oxide is PbO.

Lack of involvement from the s orbital in bonding means the valance orbitals have p³ symmetry (two electrons resonating through three p orbitals), rather than sp³ symmetry.

p³

sp³

The p³ symmetry fits well into the octahedral coordination of the rock-salt structure.

Pauling electronegativity

Electronegativity (denoted χ) is not calculated from principles, the scales for electronegativity are empirical. The Pauling scale is the most commonly used one, in which he relates the electronegativity difference of two elements to the bond strength between two elements:

Where
B(A-B) = Disassociation energy of of A-B bond
B(A-A) = Disassociation energy of A-A bond
B(B-B) = Disassociation energy of B-B bond
χ_A = Electronegativity of A
χ_B = Electronegativity of B
C = Constant. Takes the value of 96.5 if disassociation energies are in kJ mol^-1

The strength of A-B bonds are higher than the average of A-A and B-B bonds, he attributed that extra strength to an ionic resonance structure:

The true structure therefore lying somewhere in between

The stability of this resonance structure increases with the electronegativity difference between the two atoms, therefore increasing the difference in energy. This ionic contribution is the C(X_A+X_B)² component.

Perovskite structure

Perovskite is the mineral form of calcium titanate (CaTiO₃).

Black cubes with a metallic luster.

Uses the general formula ABX₃

Stresses the octahedral environment of Ti⁴⁺

Some textbooks and lectures use a unit cell with titanium in the middle. I won't put it here, since if you learn one unit cell then you can reconstruct the other.

Cadmium chloride

Cadmium chloride (CdCl₂) has a ccp structure of chloride ions with cadmium inside every other octahedral hole, forming sheets:

Here is the unit cell:

And here an easy-to-learn picture of a sheet:

The sheet repeats horizontal and inwards. Layers are below and above.

Cadmium Iodide structure

Cadmium iodide (CdI₂) has a hcp arrangement of iodide ions, with cadmium inside the octahedral holes of every other layer (so half the octahedral holes). It is often shown as sheets:

Here is the unit cell:

While you can just repeat the unit cell if you need the structure, the picture below will allow you to memorise it more directly:

Since the structure is composed of sheets, the patten repeats both parallel to and in towards the screen.

Hexagonal confusion

Most hexagonal unit cells, despite being called "hexagonal" will be depicted as this shape:

But you can reproduce the hexagon just by repeating it four times:

Below is a different type of unit cell representing hexagonal close packing:

It is important to remember that these two representations don't need to paired together, they both separately describe the same structure. This is visible by repeating the second unit cell further:

The circled atoms are higher up

Rutile structure

Titanium dioxide is used in skin products as a sunscreen, pigment, or a thickener. Its most obvious use is as a pigment in white paint. It has multiple mineral forms, the most common form is rutile. The name is from the latin rutilus (red):

Rutile in quartz

The structure is produced by a ccp arrangement of oxygen, with titanium inside half the octahedral holes.

The structure is also adopted by Cassiterite. The sole mineral form of tin oxide and the main ore of tin:

The name is from the greek word for tin - kassiteros.

Fluorite structure

Fluorite is a name for the mineral form of CaF₂

This mineral has been known since ancient times as a flux for smelting ion. The name fluorite is derived from the latin root fluo, meaning "to flow", which eventually ended up in the name of the element fluorine. It also game its name to fluorescence, which the crystal emits when it contains certain impurities. The current main use of fluorite is dissolving it in sulphuric acid to produce HF.

The structure of fluorite is a ccp array of calcium ions with (smaller) fluorine ions inside every tetrahedral hole. Reversing the locations of the cations and ions creates an antifluorite structure.

The calcium ions are the ones in dark green. This is also a good way of learning the tetrahedral hole locations in a ccp unit cell, since here they are all filled by fluoride ions. The plan view below will help make the structure clear: