# Tag Archives: algebra

## Initial conditions

Recall from last week that our Periodic Table Smoothie contains the following species:

 Substance Amount present (moles) He(g) 1.00000 Be(s) 0.51435 LiH(s) 0.27670 Li2C2(s) 0.27165 B2H6(g) 0.23300 Be2C(s) 0.17470 H2(g) 0.14267 BeC2(s) 0.13625 CH4(g) 0.00949

Pressure: 718 kPa
Temperature: 350 °C

## Reactions of nitrogen in our 10-litre vessel

Our freshly-added 1.00 mol of nitrogen gas, N2(g), reacts with hydrogen gas to make ammonia in the following reversible (equilibrium) reaction. We will assume that the interior metal surface of the vessel is a suitable catalyst for this reaction (e.g. iron).

There are three other reactions below that might have occurred at higher temperature, but I’ve chosen not to raise the temperature of the vessel at this point. Rather, we’ll keep it at 350 °C to keep things manageable.*

*I was tempted at this point to elevate the temperature of our vessel to 500 °C so that the second reaction could take place as well. This would produce copious amounts of smelly ammonia gas, which would allow for larger quantities of interesting organic compounds to be produced later on. To keep our simulation safe and (relatively) simple, I’ve decided to keep the vessel at 350 °C. Interesting compounds organic will still form – only in smaller amounts.

## Equilibria

The ammonia reaction above (the first equation) is actually an equilibrium reaction. That means that the reactants are never completely used up, and the yield is not 100%.

Recall from Le Châtelier’s principle that removing product from an equilibrium reaction causes the position of equilibrium to shift to the right, forming more product. This is because:

“If an equilibrium system is subjected to a change, the system will adjust itself to partially oppose the effect of the change.” – Le Châtelier’s principle

There are three reactions that will remove ammonia from our vessel while it’s being produced, and I’ve put all three of these into the simulation. One of these is the reverse of the reaction above (producing hydrogen and nitrogen gases) and the other two are described below. Let’s take a look at those other two reactions.

## With what will the ammonia react in our vessel?

Ammonia can undergo the following reactions with the other things in our vessel**

**The ammonia does react with methane and beryllium as well, but only at temperatures of 1200 °C and 600 °C, respectively.

Two compounds will be formed: lithium amide and borazine.[1] Lithium amide reacts with nothing else in the vessel, so the reaction chain stops there. Borazine, on the other hand, is much more interesting.

Borazine is a colourless liquid at room at temperature. It boils at 53 °C and has a structure that resembles that of benzene.

Because of the electronegativity difference of about 1.0 between the B and N atoms in the ring, borazine has a mesomer structure:

Like benzene, there is partial delocalisation of the lone pair of electrons on the nitrogen atoms.

## Borazine polymerises into polyborazine!

Fascinatingly, borazine polymerises into polyborazine at temperatures above 70 °C, releasing an equal number of moles of hydrogen gas.[2] Polyborazine isn’t particularly well-understood or well-documented, but one recent paper suggested it might play a role in the creation of potential ceramics such as boron carbonitrides. Borazine can also be used as a precursor to grow boron nitride thin films on surfaces, such as the nanomesh structure which is formed on rhodium.[3]

Like several of the other compounds we’ve created in our Periodic Table Smoothie, polyborazine has also been proposed as a hydrogen storage medium for hydrogen cars, whereby polyborazine utilises a “single pot” process for digestion and reduction to recreate ammonia borane.

The hydrogen released during the polymerisation process will then react further with a little bit of the remaining nitrogen to produce a little more NH3(g) – but not much. Recall from earlier that the ammonia reaction is an equilibrium one, and the yield of NH3(g) at pressures under 30 atmospheres is very low. Pressure in our vessel is still only around 7 atmospheres.

## Once polymerised, this would form about 12 grams of polyborazine:

As far as I’m aware, no further reactions will take place in the vessel this week.

## Conclusion after adding 1.00 mole of nitrogen gas

 Substance Amount in mol He(g) 1.000 Be(s) 0.514 LiH(s) 0.000 Li2C2(s) 0.272 B2H6(g) 0.000 Be2C(s) 0.175 H2(g) 0.007 BeC2(s) 0.136 CH4(g) 0.009 N2(g) 0.552 NH3(g) 0.154 LiNH2(s) 0.277 polyborazine 12.194 grams

Pressure: 891 kPa (higher than before due to the addition of nitrogen gas)
Temperature: 350 °C (vessel is still being maintained at constant temperature)

Next week, we’ll add a mole of oxygen gas to the vessel. Warning: it might explode.

## References

1. Stock, Alfred and Erich Pohland. “Borwasserstoffe, VIII. Zur Kenntnis Des B 2 H 6 Und Des B 5 H 11”. Berichte der deutschen chemischen Gesellschaft (A and B Series) 59.9 (1926): 2210-2215. Web.
2. Mohammad, Faiz. Specialty Polymers. Tunbridge Wells: Anshan, 2007. Print.
3. Toury, Berangere and Philippe Miele. “A New Polyborazine-Based Route To Boron Nitride Fibres”. Journal of Materials Chemistry 14.17 (2004): 2609. Web. 4 May 2016.

# Matchsticks and n-dimensional hypercubes

Here’s an addictive maths challenge for you to try.

It’s separated into four stages:

• blue (easy);
• green (medium);
• yellow (hard); and
• orange (very hard).

k = 3 in each example.