Mathematics is used widely in chemistry as well as all other sciences. Mathematical calculations are absolutely necessary to explore important concepts in chemistry. Without some basic mathematics skills, these calculations, and therefore chemistry itself, will be extremely difficult. However, with a basic knowledge of some of the mathematics that will be used in your chemistry course, you will be well prepared to deal with the concepts and theories of chemistry. The first two chapters explain basic mathematics. The last chapter explores more complex mathematical methods and concepts that may be of interest to students who are comfortable with the material in the previous sections. It is suggested that even if you are comfortable with a section, that you work through the course material in order.
Role of mathematics in chemistry
Debashis Mukherjee
Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
ALTHOUGH chemistry was practised from the dawn of civilization
as the discipline to create materials, including the
extraction of metals, it was more of a craft of artisans than a
subject of enquiry into the fundamentals. Physics, in contrast,
evolved very rapidly as an ‘exact’ science, in particular
in the hands of Archimedes, Galileo and Newton, where
the fundamental laws of motion got quantified and predictions
were precise enough to distinguish between the
rival conceptual frameworks. The major reason why chemistry
developed into an exact science relatively late is
that the underlying laws of binding and transformations
of chemical substances have their basis in the quantum
behaviour of the constituents of matter. The behaviour of
chemical substances, as isolated species or in bulk – which
dominates our world of senses – are, however, only indirectly
related to their microscopic constitution and this has remained
a problematic ontological issue which deterred an
intellectually satisfying and integrated quantitative conceptual
framework for chemistry. Moreover, chemistry as
a discipline enjoys a degree of autonomy in the sense that
the desirable goals of a chemist (control of emergent chemical
behaviour, designing molecules with specific chemistry,
monitoring chemical transformations into well-defined
channels) are determined by questions and aesthetics of
chemical nature. In this sense, chemistry is more ‘complex’
than physics.
I look upon the less complex science as one which engenders
the fundamental basis of a science which lies one
tier higher in complexity. To borrow the terminology
from biology we may say that the fundamental basis of a
science are genotypes, while the emergent properties arising
out of the genotypical laws are phenotypes. I want to
argue that chemistry is the simplest science of complexity
since the fundamental physical laws are its genotypes and
the emergent chemical expressions are the phenotypes. Chemistry
is thus compatible with physical laws but not reducible
to them. The really interesting problems in chemistry
seem to remain fully unresolved in terms of understanding
from physical principles because scientists have not come
to grips in discerning the pattern, structure and interconversions
displayed by molecules from the fundamentals
of subatomic physics. This is despite the fact that we understand
the quantum and statistical mechanical laws of
physics well enough but it is neither unique nor trivial to
pose questions of chemical nature in terms of physical laws.
The complexity of chemistry has even an underlying extra
degree of freedom in the sense that the superstructure of
chemical functions is to some extent insensitive to the
physical laws underpinning them. Results from a more
quantitative formulation from a more fundamental basis
often lead to qualitatively similar but quantitatively different
conclusions, so that certain empirical generalizations
can well describe chemistry and even lead to an illuminating
understanding, quite independent of the underlying laws
of the substratum. Obviously, by ‘understanding’ we mean
assessing the relative importance of the various processes
reflected in some conceptual constructs which act together
to shape the phenomena of interest. Models emerge when
we tie up understanding and quantitative descriptions of
the conceptual constructs and weave a story out of it. Stories
are complete or convincing to various extents depending
on the mix of understanding and quantification. Another
appealing simile is sculpting. Much is removed but much
remains also for the pattern to emerge. Recognizing what
to remove yet emphasizing the essence in all its splendour is
an orphic endeavour of sorts, involving inspiration, metaphor,
symbolic representation and innovative analogy. The
role of mathematics in chemistry must satisfy this polysemiotic
and polymimetic richness.
We thus distinguish quantum molecular physics as somewhat
distinct from theoretical chemistry when we want to
discuss the role of mathematics in chemistry. An appreciation
of this difference is often not made, leading either to a
perception that brute force computation or even empirical
quantitative simulation would lead to understanding chemical
significance or to the dismissive attitude of the experimental
chemists that theoretical chemistry fails to
provide predictive answers of chemical significance,
when in fact they are probably pointing out the limitations
of computational molecular physics. When I talk about
the role of mathematics in chemistry, I have in mind an evolving,
conceptually integrated and many layered theoretical
framework of molecular science, which subsumes both
molecular/materials physics and chemistry and chemical
biology as subdisciplines. This is a never ending saga but
the stories become more and more complex as we begin to
see more intricate patterns and can relate them more and
more to the physical laws underpinning them.
The genotypes of chemistry are embedded in quantum
mechanics, equilibrium and non-equilibrium statistical
mechanics, and diffusion behaviour in fluids. The phenotypes
are the molecules displaying myriad chemical properties
in isolation and in transformation. The chemical concepts
like bonds, lone pair, aromaticity.
Dimensional analysis between mathematics and chemistry
1. Dimensional Analysis
In engineering the application of fluid mechanics in designs make much of the use of empirical results from a lot of experiments. This data is often difficult to present in a readable form. Even from graphs it may be difficult to interpret. Dimensional analysis provides a strategy for choosing relevant data and how it should be presented.
This is a useful technique in all experimentally based areas of engineering. If it is possible to identify the factors involved in a physical situation, dimensional analysis can form a relationship between them.
The resulting expressions may not at first sight appear rigorous but these qualitative results converted to quantitative forms can be used to obtain any unknown factors from experimental analysis.
2. Dimensions and units
Any physical situation can be described by certain familiar properties e.g. length, velocity, area, volume, acceleration etc. These are all known as dimensions.
Of course dimensions are of no use without a magnitude being attached. We must know more than that something has a length. It must also have a standardised unit - such as a meter, a foot, a yard etc.
Dimensions are properties which can be measured. Units are the standard elements we use to quantify these dimensions.
In dimensional analysis we are only concerned with the nature of the dimension i.e. its quality not its quantity. The following common abbreviation are used:
length = L
mass = M
time = T
force = F
temperature = Q
In this module we are only concerned with L, M, T and F (not Q). We can represent all the physical properties we are interested in with L, T and one of M or F (F can be represented by a combination of LTM). These notes will always use the LTM combination.
Calculus vs chemistry
