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Student name: Jamie Feerick Student number: G00357938Course: Chemical and Bio-Pharmaceutical Science Maths Journal 1                     Chemical graph theory is often usedto mathematically graph and depict molecules, to enhance our understanding ofthe physical properties of these chemical compounds.The innovators behind this idea where Alexandru Balaban, Ante Graovac, IvanGutman, Haruo Hosoya, Milan Randi?, Nenad Trinajsti? and Harry Weiner. Some ofthe physical properties such as polarity, potential energy and boiling pointare related to the geometric structure of the compound. This is seen to be exceptionallytrustworthy on the account of Alkane. A precedent of an alkane is Ethane. InEthane every Hydrogen atom as a single chemical bound and every Carbon has fourchemical bonds this means that the Hydrogen atoms can be expelled withoutlosing information about the particle.

The subsequent representation of Ethaneis also known as a carbon tree and can also be shown as a graph by substitutingthe carbons for dots and the chemical bonds as straight lines connecting thesedots(carbons).    Figure 1. Ethane Molecule   Figure 2. Ethane with its Hydrogen atoms removed   Figure 3. Carbon tree of Ethane represented as a graph.

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         Thestructure of an alkane decides its physical properties. Physical properties ofalkanes can be displayed utilizing topological indices. Some of these indicesare notable outside of the substance and numerical groups, for example, therelative atomic mass (Mr) of a compound. For alkanes the relative molecularmass is a component of the amount of carbon atoms, indicated by n, and is givenby Mr(n) = 12.01115n + 1.00797(2n + 2) atomic mass units (amu).

Using theseequations, you can confirm that the relative molecular mass of ethane in Figureone is 30.0701amu. Boiling points are a measure of the powers of attractionbetween like particles. For non-polar compounds, for example alkanes, thesepowers are dispersion forces due to quick dipole-actuated attractions. Thealkane breaking point must rely upon the relative sub-atomic mass and on howwell the particles pack together, which is identified with the geometry of theatom. Balaban noticed that for a similar relative sub-atomic mass, the boilingpoint of the substance decreased as the Carbon tree spreads out .  Here I have shownsome similar examples of other alkanes, such as Octane and 2,2,3Trimethylpentane to show the difference. Both Alkanes are also made up of eightcarbon atoms, so they also have the same molecular mass as Ethane.

                                         I have also included 2,2,4-trimethylpentane carbon tree for reference. Figure 6.  2,2,4-trimethylpentane carbon tree              From the points I have already stated you would expect theboiling point of Isooctane to be lower than that of Ethane and that is the caseas expected.

The boiling point for isooctane is 372.4 K or 99.25 degreesCelsius and the boiling point of octane is 398.

7 k or 125.55 degrees Celsius.  From this information you are able to seethat you can graph the boiling of families of alkanes that have similargeometric structures using their molecular weight as the only index in thegraph.  How Allie Forces used maths in world war two to give them anadvantage.

Maths was used greatly by the Allie forces in world war twoto help give them the upper hand on the opposing German forces. During Ww2Allied forces admitted that German tanks where more advanced than the tanksthat the Allied forces had at there disposal. The allied forces needed tofigure out of many tanks the German forces where producing so they would beable to produces more in order to be able to defeat the superior German tanks.To tackle this issue the Allie forces first used the usual methods of spying,intercepting and translating transmissions and of course interrogating capturedenemy troops.From this the allies had come to the conclusion that theGerman factories were creating around 1400 tanks per month from June 1940 rightthrough to September 1942, an outstanding figure which just seemed far to highto be true.

To paint a picture of that in the Battle of Stalingrad which lastedeight months the Allie forces used 1500 tanks and around one million casualties.For the reason the figures of 1400 tanks per month seemed far to high. It wasback to drawing board for the Allie forces. This is where the Alliemathematicians came into play. They believed that there would be some form ofpattern in the serial numbers on the German tanks that would give them theadvantage in being able to indicate the number of tanks that they whereproducing per month. The mathematicians requested that the soldiers record theserial number on each German tank that they come across sot that they would beable to come up with some sort of algorithm to identify the number of tanks thatthey were producing.

This is one of the types of equations they would have usedto predict or estimate the number of tanks that the Germans where producing ifthe tanks where number from one to  n.   This equation is usually called a minimum-variance unbiasedestimator. where m isthe largest serial number observed (sample maximum) and k is the number oftanks observed (sample size). Once a serial number has been observed, it is nolonger in the pool and will not be observed again.

Using an equation like this it is reported that the Allieforces predicted the Germans where producing a number of 255 tanks per month.Using this information, the Allies knew they had to produce a larger sum oftanks than the Germans in order to counteract the superior German tanks. Turnsout that the mathematician’s serial methodology was fairly exact, after the warinternal German data put the German Factory production at around 256 tanks permonth. This meant that the mathematicians where only out by one tank.


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