Fire Assay

Fire Assay is a technique which analyzes the amount of precious metal (gold, silver, palladium and platinum) in a sample of ore or scrap. It is generally regarded as the most accurate (7-10 ppb or 0.1g/t), economical, and consistent method for gold analysis, though it is time consuming. It works with all forms of gold, and there is no such thing "un-assayable" gold, provided the method is done correctly

It is mentioned in the Doomsday Book and was invented as far back as 2000 BC, when very pure silver was produced from lead ore in Asia Minor. The earliest detailed written record known is De Rey Metallica by Georgius Agricola in 1556, which served as the standard book on the subject for about two centuries.

The process can be split into three stages: fusion, cupellation, and analysis.

Fusion: A sample of 10-30 g is blended into a fine powder. The most common sample is 30 g, which is roughly equivalent to a traditional "assay tonne". Up to 150 g can be used for particularly low concentrations.

This is added into a crucible along with lithard (PbO), borax (sodium borate), silica (sodium dioxide) and soda ash (sodium carbonate) to form a melt, and heated to 1000-1200° C. The exact temperature and composition of the ingredients depends on what trace elements are present in the sample. For instance, a siliceous ore requires a basic flux, and a basic ore requires an acid flux; soda ash is good at removing sulfur; etc. The details are adjusted to suit the ore, using an approach based on long empirical experience. The whole process from start to finish has a high degree of skill and benefits from having an assayer with knowledge and intuition.

The main purpose of these extra ingredients are to create a "flux" which mixes with the sample. This flux lowers the melting point of the mixture (mainly required for the metal oxide impurities) and imparts a homogeneous fluidity.

A small amount of carbon is added causing some of the lithard to be oxidized to lead and sink to the bottom of the melt along with any silver. Gold platinum and palladium in the melt will then drift into the lead-silver alloy at the bottom, while the impurities are collected into the liquid slag. The reactions here are complex and much of it has not been theoretically mapped out, but the basic principle is that the hot PbO is very oxidizing, and the slag is very soluble to metal oxides, so anything not a precious metal will tend to be oxidized and drift into the slag. In addition, most precious metals exhibit a good affinity for lead by themselves, and even the ones which do not will be dense enough to sink into the pool at the bottom, forming a mechanical mixture.

The melt is added to a mold and allowed to cool, then the glassy slag separates cleanly by tapping with a small hammer, leaving a lead button.

Cupellation:  The lead button is placed inside a cupel, which is a crucible made of pourous bone ash or magnesia. When heated to about 800-1000° C in air, the molten lead is oxidized back to PbO which then melts and is drawn into the cupel by capilliary action, leaving a small bead (called a dore bead) of precious metal. Cupels are rated by the grams of lead oxide they will absorb.

Analysis: The dore bead can be analyzed directly, or the silver can first be separated using nitric acid, dissolving the silver as silver nitrate. Separating the metals allows gravimetric analysis. The most common analytical method used today is AA (Atomic absorption spectroscropy). Other methods used are ICP (Inductively coupled plasma atomic emission spectroscopy), ICP/MS (Inductively coupled plasma mass spectroscopy) or for very low concentrations, INAA (Instrumental Neutron Activation Analysis).

Stable Cyclic Carbenes

By "stable" we mean carbenes which can exist at room temperature without being bonded to a metal ion. Stable cyclic carbenes are classified into five families: NHCs (N-heterocyclic carbenes), Thiazolylidenes, PHCs (P-heterocyclic carbenes), Cyclopropenylidenes and CAACs (Cyclic alkyl amino carbenes).

The neutral carbene exists in either a singlet or triplet state:

This depends on what is larger, the pairing energy or the difference between the p and sp orbitals. From 1990-2010, stable singlet carbenes have been isolated at room temperature, while stable triplets have been observed cold.

The stable singlet carbenes are more important, as they considered extremely useful as catalysts when complexed to a transition metal.

NHCs are by far the largest group with hundreds of publications, so review articles tend to focus on these.

We can see the singlet-triplet gaps of the five families (kcal/mol):

This suggests that NHCs are mostly likely to produce singlets, which might explain why they are so popular with researchers.

σ-electron withdrawing groups favor the singlet state, σ donating groups favor triplet state. But the most important factor in stablizing singlets is the presence of a π-donor (an element or a pi bond) next to the carbene carbon. Bulky substituents can also offer kinetic stability and force the molecule into a shape which encourages π-donation onto the carbene carbon. The more full the empty carbene p-orbital is, the higher the singlet-triplet gap, the more stable the carbene.

The carbene also gains stability by linking its side atoms into a 3-5 membered ring, like in the examples above. First, because the shape of the ring can force s-character into the carbene lone-pair orbital. Second because it forces π-orbitals of π-donors to align with the empty carbene p orbital, increasing overlap/donation.

The first stable NHCs was made in 1991 by Arduengo. Who grew crystals of this thing, now known as Arduengo's Carbene:

Those side groups are called adamantane groups. They tend to be shorthanded as "Ad". This molecule has all the stabilization attributes mentioned above.

The most well known NHC is the 2nd generation Grubbs catalyst used for olefin metathesis, which won him the nobel prize in 2005.

Another feature of NHCs is that some are used as organic catalysts in their own right, without being complexed to a metal.

Going through the other four classes, stable thiazolylidenes came in 1997. The inferior singlet-triplet gap is from the inferior pi-donating capacity of sulfur compared to nitrogen. They tend to require very bulky side groups. They have some utility in metal complexes, but no highly effective thiazolylidene-based catalysts have been reported.

The first stable P-heterocyclic carbenes was isolated in 2005:

Mes* = 2,4,6-tri-(t-butyl)phenyl.

Only a few other PHCs are known. They all require enormous groups to be stable.

Also in 2005 were the first cyclic (alkyl) (amino) carbenes. These also have low orbital splitting, partly from having only one heteroatom next to the carbene carbon, and partly from the sigma donating character of the sp³ carbon. The orbital splitting of CAACs and PHCs are similar, suggesting that one animo group has about the same effect as two phospino groups with bulky substituents.

The adjacent carbon has to be quaternary to prevent a 1,3 hydride shift.

CAACs are being heavily researched and arguably have the most potential uses, since the sp³ carbon can produce sigma-donor effects on the carbene carbon more powerful than all the other cyclic carbenes. The sp³ carbon also has more potential for steric bulk protecting the carbene atom.

The relatively empty p orbital in CAACs also give this class some stranger reactivity and unique side-reactions. They form stable aminoketenes when exposed to carbon monoxide, and split H₂ and NH₃ under normal laboratory conditions.

We have the final class produced in 2006, Cyclopropenylidenes. People were surprised to see a stable carbene without even one pi-donating heteroatom adjacent to the carbene carbon. Only one has ever been isolated:

The strained flat shape of the ring puts the empty p orbital of the carbene planar with the double bond, so both adjacent carbons act as pi donors. This is enhanced by the isopropylamino groups, for they can donate their lone pairs into the pi* antibonding orbital, which has the right symmetry to overlap with the empty carbene p orbital. Finally the carbene lone pair is forced into a sp² shape, which enhances the sigma-pi split even further.

The close relation of work and heat

Work and heat are both measured in joules, and are considered different ways of transferring energy. They are found experimentally to be easily intercovertable. Consider the pistons and cylinders below:

We can perform work on the left piston to compress it. Or heat the gas on the right to increase the pressure, pushing the piston, hence performing work. You could even use some of the work gained from the expanding gas to push current through a resistor, producing heat, and that heat can be used to expand a gas... and so on.

Work is the transfer of energy which makes use of ordered motion. The ordered direction of a falling piston introduces random kinetic motion into the gas molecules. So the piston is doing work which is converted to heating the gas.

Heat is the transfer of energy which makes use of random motion. The expansion of a gas in the cylinder will convert the random motion of gas particles into the ordered upwards motion of the piston. So heat from the gas is transformed into work.

The ordered and random motion does not have to be kinetic. An electric current is considered to move in an ordered motion, so using energy to drive a current would be considered doing work.

van der Waal equation of state example

You are given the temperature, pressure, and constants a and b of a gas. Estimate the molar volume:

We simply multiple both sides by (V_m - b)V_m² to get:

Then collect the powers of V_m to obtain:

Then plug the numbers into software to get a solution to this cubic equation.

van der Waals equation of state derivation

This attempts to account for some of the assumptions of the perfect gas equation of state. It is a good example of scientific thinking about a mathematical model, "model building" in other words. a and b are empirical parameters, but they can also be estimated.

First we take the ideal gas equation and try to account for volume taken up by the molecules themselves, by subtracting nb from the volume, where b is a constant.

This approximates molecules as hard spheres. So each molecule has a sphere of exclusion around it of 2r, anything less and the spheres would penetrated eachother.

This gives a total excluded volume of (4/3)π(2r)³ per molecule, which is 8V_molecule.

This number is then halved "to prevent overcounting" to give 4V_molecule. I have never been able to understand why this step is done.

But the end result is that b is roughly 4VN_A. Since many molecules are quite soft, this number is typically the upper limit of empirical measurements.

The term on the right is because attractive forces will reduce both the rate of collisions of the molecules with the side of the container, and reduce the speed of these collisions. These forces are found to act with a strength proportional to the square of the molar concentration (n / V) of the molecules.

Remember that this is overwhelming an empirical law. And the justification given for it is only vague. There are other more satisfactory ones which you may come across later.

Real gases

A real gas approaches a perfect gas at 0 pressure. Attractive forces dominate at moderate pressures and repulsive forces at high pressure - since the repulsive forces have a shorter range.

Hence a real gas is expected to be more compressible at moderate pressures and less compressible at high pressures.

Compression factor

The compression factor is the ratio of the molar volume a gas compared to the molar volume of a perfect gas, at the same pressure and temperature:

From the equation you can see that Z of a perfect gas is 1. Real gases with a larger than perfect volume have   Z > 1, and real gases with a lower than perfect volume have Z < 1.

From the argument in the first section we can expect Z to approach 1 and 0 pressure, be below 1 at moderate pressures, and above 1 at high pressure:

We can see this is true for most gases - H₂ being a commonly-shown exception.

Partial pressures

In a mixture of gases, perfect or real, the partial pressure of an individual gas defined as:

p_i = Patial pressure
p = Total pressure
x_i = mole fraction of the gas

n_i = moles of individual gas

n = total moles of all gases in the mixture

Note the following:

- Each mole fraction is between 1 and 0
- All the mole fractions in a mixture add up to one
- Total pressure is the sum of each partial pressure

These can also been understood intuitively by plugging in numbers.

If the gas is perfect, then Dalton's law applies:

The pressure exerted by a mixture of gases is the sum of the pressures that each one
would exist if it occupied the container alone.