Chemical graph theory is often used
to mathematically graph and depict molecules, to enhance our 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
Figure 1. Ethane Molecule
Figure 2. Ethane with its Hydrogen atoms removed
Figure 3. Carbon tree of Ethane represented as a graph.
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 dispersion forces due to 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 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.
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
How Allie Forces used maths in world war two to give them an
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
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 Stalingrad 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.