Student indicated by n, and is given by

Student name : Jamie Feerick

Student number: G00357938

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Course: Chemical and Bio-Pharmaceutical Science

 

Maths Journal 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chemical graph theory is used to
mathematically model molecules, to gain an understanding of the physical properties
of these chemical compounds. The innovators behind this idea where Alexandru
Balaban, Ante Graovac, Ivan Gutman, Haruo Hosoya, Milan Randi?, Nenad
Trinajsti? and Harry Weiner. Some of the physical properties such as polarity,
potential energy and boiling point are related to the geometric structure of
the compound. This is seen to be exceptionally trustworthy on the account of
Alkane. A precedent of an alkane is Ethane. In Ethane every Hydrogen atom as a
single chemical bound and every Carbon has four chemical bonds this means that
the Hydrogen atoms can be expelled without losing information about the
particle. The subsequent representation of Ethane is also known as a carbon
tree and can also be shown as a graph by substituting the carbons for dots and
the chemical bonds as straight lines connecting these dots(carbons).

 

 

 

Figure 1. Ethane Molecule

 

Figure 2. Ethane with its Hydrogen atoms removed

 

Figure 3. Carbon tree of Ethane represented as a graph.

 

 

 

 

 

 

 

 

The
structure of an alkane decides its physical properties. Physical properties of
alkanes can be displayed utilizing topological indices. Some of these indices
are notable outside of the substance and numerical groups, for example, the
relative atomic mass (Mr) of a compound. For alkanes the relative molecular
mass is a component of the amount of carbon atoms, indicated by n, and is given
by Mr(n) = 12.01115n + 1.00797(2n + 2) atomic mass units (amu). Using these
equations, you can confirm that the relative molecular mass of ethane in Figure
one is 30.0701amu. Boiling points are a measure of the powers of attraction
between like particles. For non-polar compounds , for example alkanes, these
powers are London scattering powers because of quick dipole-actuated
attractions. The alkane breaking point must rely upon the relative sub-atomic
mass and on how well the particles pack together, which is identified with the
geometry of the atom. Balaban noticed that for a similar relative sub-atomic
mass, the boiling point of the substance decreased as the Carbon tree spreads
out .

 

Here I have shown
some similar examples of other alkanes, such as Octane and 2,2,3
Trimethylpentane to show the difference. Both of these Alkanes are also made up
of eight carbon 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 the
boiling point of Isooctane to be lower than that of Ethane and that is the case
as expected. The boiling point for isooctane is 372.4 K or 99.25 degrees
Celsius and the boiling point of octane is 398.7 k or 125.55 degrees Celsius.  From this information you are able to see
that you can graph the boiling of families of alkanes that have similar
geometric structures using their molecular weight as the only index in the
graph.

 

 

How Allie Forces used maths in world war two to give them an
advantage.

Maths was used greatly by the Allie forces in world war two
to help give them the upper hand on the opposing German forces. During Ww2
Allied forces admitted that German tanks where more advanced than the tanks
that the Allied forces had at there disposal. The allied forces needed to
figure out of many tanks the German forces where producing so they would be
able 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 captured
enemy troops.

From this the allies had come to the conclusion that the
German factories were creating around 1400 tanks per month from June 1940 right
through to September 1942, an outstanding figure which just seemed far to high
to be true. To paint a picture of that in the Battle of Stallingrad which
lasted eight 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 was back to drawing board for the Allie forces. This is where the Allie
mathematicians came into play. They believed that there would be some form of
pattern in the serial numbers on the German tanks that would give them the
advantage in being able to indicate the number of tanks that they where producing
per month. The mathematicians requested that the soldiers record the serial
number on each German tank that they come across sot that they would be able to
come up with some sort of algorithm to identify the number of tanks that they
were producing. This is one of the types of equations they would have used to
predict or estimate the number of tanks that the Germans where producing if the
tanks where number from one to  n.

 

 

This equation is usually called a minimum-variance unbiased
estimator. where m is
the largest serial number observed (sample maximum) and k is the number of
tanks observed (sample size). Once a serial number has been observed, it is no
longer in the pool and will not be observed again.

Using an equation like this it is reported that the Allie
forces 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 of tanks
than the Germans in order to counteract the superior German tanks. Turns out
that the mathematician’s serial methodology was fairly exact, after the war
internal German data put the German Factory production at around 256 tanks per
month. This meant that the mathematicians where only out by one tank.

 

 

 

 

References

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