Student indicated by n, and is given by

Student name : Jamie Feerick Student number: G00357938Course: Chemical and Bio-Pharmaceutical Science Maths Journal 1                       Chemical graph theory is used tomathematically model molecules, to gain an understanding of the physical propertiesof these chemical compounds.

The innovators behind this idea where AlexandruBalaban, Ante Graovac, Ivan Gutman, Haruo Hosoya, Milan Randi?, NenadTrinajsti? and Harry Weiner. Some of the physical properties such as polarity,potential energy and boiling point are related to the geometric structure ofthe compound. This is seen to be exceptionally trustworthy on the account ofAlkane. A precedent of an alkane is Ethane. In Ethane every Hydrogen atom as asingle chemical bound and every Carbon has four chemical bonds this means thatthe Hydrogen atoms can be expelled without losing information about theparticle. The subsequent representation of Ethane is also known as a carbontree and can also be shown as a graph by substituting the carbons for dots andthe chemical bonds as straight lines connecting these dots(carbons).

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    Figure 1. Ethane Molecule   Figure 2. Ethane with its Hydrogen atoms removed   Figure 3. Carbon tree of Ethane represented as a graph.          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 London scattering powers because of quick dipole-actuatedattractions. The alkane breaking point must rely upon the relative sub-atomicmass and on how well the particles pack together, which is identified with thegeometry of the atom.

Balaban noticed that for a similar relative sub-atomicmass, the boiling point of the substance decreased as the Carbon tree spreadsout .  Here I have shownsome similar examples of other alkanes, such as Octane and 2,2,3Trimethylpentane to show the difference. Both of these Alkanes are also made upof eight carbon atoms so they also have the same molecular mass as Ethane.                                         I have also included 2,2,4-trimethylpentane carbon tree forreference . 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 Stallingrad whichlasted eight months the Allie forces used 1500 tanks and around one millioncasualties. For the reason the figures of 1400 tanks per month seemed far tohigh. It was back 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 where producingper month. The mathematicians requested that the soldiers record the serialnumber on each German tank that they come across sot that they would be able tocome up with some sort of algorithm to identify the number of tanks that theywere producing. This is one of the types of equations they would have used topredict or estimate the number of tanks that the Germans where producing if thetanks 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. Usingthis information, the Allies knew they had to produce a larger sum of tanksthan the Germans in order to counteract the superior German tanks. Turns outthat 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.

    References M m k kn k


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