Partitioning of Molecular Weight - An Elliptic Model for Isomers

Partitioning of any real number has been achieved based on an elliptic model introduced. Then, it has been adopted to isomeric molecules including optically active ones. Certain angles and bounds are defined. A bivariant regression model has been proposed for a set of isomeric molecules and discussed.


Introduction
Decomposition of molecules theoretically is widely used approach to facilitate some calculations in chemistry and physics [1][2][3][4]. Especially, in the past it was employed for the facile solution of Hückel molecular orbital parameters of conjugated systems prior to era of high speed computers. For that purpose group theory, graph theory etc. were the scientific approach to the solution [5][6][7][8][9]. The decomposition or fragmentation means to obtain smaller subgraphs from larger symmetrical molecular graphs.
On the other hand, molecular weight is a kind of mute box which does not tell much except the overall mass number and kind of atoms it possesses. It is a matter of curiosity whether it will keep its silence in the future. Although many properties of matter intensively or extensively depend on the molecular weight, these properties have not been interrelated to parts of it (in the absence of structural information including some gross and fine topology of the molecule). In the present treatment, firstly any real number has been partitioned based on an elliptic model developed. Then it has been employed partitioning of molecular weight of any set of isomeric molecules.

Estimation of ?
Consider Figure 2 and infinitesimally small triangle in which angle is . Area of the triangle is = ( Note that if the ellipse is sufficiently flat ( << ) the area of the triangle, , is very close to area of sector. Apsis @ of point is given by @ = . If is small enough, and can be obtained from eq. (8).
Area of BOA sector can be approximated as Since ( < 1, < 1, then their multiplication has to be much smaller than 1. Consequently, an approximate value for is = (1 − ( ) .
where I is the total area of the ellipse. Evaluating the integral one obtains, Note that area of an ellipse is M , then ( is estimated as

Results and Discussion
The partitioning described above ( ′/ ratio) is a general approach for any number. Many partitioning alternatives mathematically exist for any number. If the model is adapted for molecules then the position of the molecule on the ellipse depends on the way of partitioning of the molecule. Since molecular weight (MW) is considered then for each set of isomeric molecules a specific ellipse has to be considered. Molecular property considered should be implicit function of angle , (by assumption), which is dictated by and ′. Then the molecular property considered should be the function of those radius vectors in different contributions.
In the case of molecules by assumption = 3/2. Since < for the existence of an ellipse < 3/2. Then = 1 could be taken as the lowest upper bound ( 3 of hydrogen is just 2 amu). On the other hand, ( to be real (see eq. (25)) it has to be M /2 < 1. Then, < 2 /M is another upper bound for , namely < 3/ M.
A methodology could be established for partitioning a molecule into two parts. However, if angle is associated with some physical or chemical property, then a restricted case arises. For instance, if the angle of rotation of plane polarized light is considered, then the model predicts some of the related properties (see Figure 3). Since in an ellipse weight ( = 2 ) reside on the ellipse, namely isomeric compounds. Considering the first and second quadrants of the ellipse ( arc), and isomers having the molecular weight meso compounds) gather at point between to ( ( , &)) and their enantiomers along hand, all the meso isomers accumulate at point >, becomes 2 π , then eq. meso compounds (at point vectors are in different quadrants of the ellipse for that point. That is molecule has been partitioned into two equivalent but opposing parts. The associated polarizability values thus should cancel out each other inactive. Diastereomers and enantiomers occupy a position on the ellipse between points and or to ′. To be more precise consider an ellipse of where R is a constant. At point http://www.earthlinepublishers.com meso compounds (at point (0, )), the model yields ′ = , but those radius vectors are in different quadrants of the ellipse for that point. That is molecule has been partitioned into two equivalent but opposing parts. The associated polarizability values thus should cancel out each other's effect. Note that meso compound are optically inactive. Diastereomers and enantiomers occupy a position on the ellipse between points To be more precise consider an ellipse of     [11] based on the elliptical model introduced presently can be developed. Note that those bivariant regression models turn into univariant type models in the case of isomers accumulated at ( , / ).
Brewster had suggested an approach to the calculation of the sign and magnitude of optical rotation based on the consideration of two independent contributions which are mutually responsible for the observed rotation. They are (i) configuration contribution (ii) conformational contribution. For the sign of configuration contribution he proposed an empirical rule as well which is based on polarizability of the substituents around a choral center [12][13][14][15]. Although, he assumed two contributions, his way of approach is completely different from the present one.

Conclusion
The elliptic model presently introduced for the partition of molecular weight opens new horizons in physical sciences because although molecular weight appears in many scientific formulae, its introduction component wise is a brand new approach. Then many bivariant regression equations can be developed for various molecular properties to get better regression statistics compared to univariant regression equations in many cases. It would be a matter of investigation how to relate the way of partitioning of molecular weight to proper dissection of chemical structure.