Importance of Mathematical Tools in Chemical and Life
Sciences
Anjna Kumari1, Ruchi Sangal1.2, Dr. Shashi
Kumar Sharma3
1Assistant Professor, Dept. of Chemistry, Govt. College Hamirpur (H.P).
1,2Assistant Professor, Dept. of Zoology, Govt. College Nadaun (H.P).
3Associate Professor, Dept. of Chemistry, Govt. College Hamirpur (H.P).
*Corresponding
Author E-mail:
ABSTRACT:
Knowledge is a
bonus in today's extremely changing educational environment. The primary driver
of development in the present and the future is the capacity to generate new
information, which may be accomplished through an understanding of mathematics.
Every sphere of life and every field of study can benefit from the use of
mathematics.
Chemistry and biology are influenced by mathematics. For these
fields, various tools developed in mathematics are useful. Calculations in
mathematics are unquestionably used to examine key ideas in biology and
chemistry. Chemistry and chemical calculations will be very challenging if one
don't know certain basic mathematical concepts. It is a precise science that is
based on numerical models that can be expressed and used using mathematical
terminology. Basic calculator skills, knowledge of numbers and their
properties, ratios and properties, units, diversions and conversions, percents,
simple statistics, indices and logarithms, as well as the ability to interpret
graphs, are just a few of the abilities required. Other skills include
knowledge of the mean, the arithmetical average of numbers, median and mode,
range, dispersion, standard deviation, coefficient of variation, etc. This
paper attempts to review the importance of mathematical tools in chemistry and
biology.
KEY WORDS: Biology,
dispersion, terminology, median.
INTRODUCTION:
The mathematics'
usefulness in modeling phenomena and its widespread usage in science and
engineering, education scholars from the fields of science and math have been
examining the ways how learners understand and use mathematics.
Symbols and symbolic notations are difficult from a purely mathematical
standpoint because of the amount of information they may encode and the variety
of ways that they can be utilized and understood (Bain et al., 2019). According
to Olafare
(2016), the study looked at educators' perceived knowledge of mathematics, its
correlation to chemistry and biology,
and the apparent influence of mathematics knowledge on educators'
performance in both the areas. Chemistry and mathematics are inseparable
companions in nurturing science and technology education. Due to the implementation of mathematical
ideas to societal growth, both humanity and mathematics are improved. Chemistry
and biology are influenced by mathematics. Several tools developed in
mathematics are beneficial for both the fields (Olafare, 2016). Chemists
examine several areas of their discipline using group theory which is the one
of the mathematical tools (Malkevitch ,2017). A
resource that will not only give an overview of crucial algebraic ideas and
concepts as well as give examples of the different sorts of chemistry
challenges that necessitate mathematics is needed, according to research
conducted by Grove and Pugh (2015), to explore the unique characteristics of
the mathematical problems concerning chemistry educators. It was stated that
there should be ongoing chances for students to utilize and enhance their
mathematical skills across all elements of their course and that important
mathematical ideas, concepts, and conclusions should not be kept a secret from
them but rather should be provided in ways that they can comprehend and enjoy.
This framework was
chosen because it is frequently necessary to understand occurrences using
knowledge of chemistry, mathematics, and biology. The understanding that arises
from the combination of these two mental spaces: chemistry - mathematics
and biology -mathematics are larger than the sum of its components; it
necessitates the careful mixing of information from each place.
The role of
mathematics to chemistry:
Chemistry is a rich context to
evaluate researchers understanding of mathematics, encouraging instructors to
use chemistry contexts in their mathematics courses and suggesting researchers
expand their interests to include the application of mathematics that occurs
outside of mathematics courses (Bain et al., 2019). According
to Olafare(2016), calculations in mathematics are clearly used to explore key
chemistry ideas. Chemistry and chemical calculations are quite difficult to
perform without a basic understanding of mathematics. Basic knowledge of
calculators, numbers and their characteristics, ratios and properties, units,
diversions and conversions, precents, elementary statistics, indices and
logarithms, as well as graph interpretation, are some of the skills required .Chemistry
involves the synthesis of molecules from naturally occurring atoms, and
measurement is another branch of mathematics that is extremely helpful in this
process. There are 92 naturally occurring chemical elements with complicated
properties, some of which are gases, liquids, or solids at room temperature,
according to Malkervitch (2017). Mathematics has been utilised in chemistry
since its inception to develop quantitative and qualitative models that help
scientists understand the world of chemistry by recognising the components of
molecules. Protons, neutrons, and electrons are the fundamental building blocks
of an atom. Protons, neutrons, and elections all have measurable masses and
electrical charges, which raises mathematical concerns in chemistry (Olafare,
2016).
According to Olafare, (2016) the 92
naturally existing elements have been divided into families with related
chemical properties using numbers to explain their structure, leading to the
creation of the periodic table. This aids in grouping the elements into groups
that share similar features and provides the knowledge needed to comprehend
these elements' characteristics. Calculations in mathematics are clearly used
to explore and understand chemistry ideas. Restrepo and Pachón (2006), examined
the periodic nature of the characteristics of chemical elements when they are
all taken into account collectively. In other words, they demonstrated the
existence of a mathematical structure in the properties of chemical elements. He
discovered that an oscillating plot results from plotting a physico-chemical
property against the atomic weight, or in contemporary language, the atomic
number Z. These oscillations referred as "periods" because they
resembled periodic trigonometric function like sine and cosine. Also,
scientists refer to such plots as periodic since each oscillation, assuming the
elements are arranged according to their atomic number, includes a number of
them. As a result, we discover the following cardinalities for the first seven
oscillations (or periods): 2, 8, 8, 18, 18, 32, and 32. Although it is clear
that the elements are not equal and the periods are not absolutely periodic
(Babaev and Hefferlin, 1996), there is symmetry in their distribution that
leads us to believe that the chemical element attributes have an underlying
mathematical structure Moreover, Scerri
et al.,(1998) created a mathematical analysis of the Periodic Law based on the
Madelung rule and the concept of total order. The work of Lewis was regarded as
mathematical. As they result from the solution of the eigenvalue equation
HΨ=EΨ, Madelung's work and all subsequent quantum chemistry research can also
be regarded as mathematics. Each chemical element is described mathematically
as an ordered 7-tuple if we define each one using seven of its qualities,
creating a seven-dimensional space (Restrepo and Pachón, 2006). The progressive
electron shells of the noble gases have the corresponding numbers of electrons:
2, 8, 8, 18, 18, and 32, according to Pauling. These numbers correspond to the
number of elements in the periodic system's subsequent periods (Pauling, 1970).
The set of numbers that appear in the series of cardinals is given by 2n2,
but not the sequence itself. As a result, if we defined C=x | x=2n2
where n is a positive integer, we would obtain C= (2,8,18,32,..). but not the
ordered set (2,8,8,18,18,18,32,32) (Restrepo and Pachón,2006). Weise, (2003)
uses the formula Zn=((-1)n (3n+6)+2n3 +12n2
+25n-6)/12 to indicate the total number of electrons in the atoms of noble
gases, where n is an integer (n=1,2,...). He also created a technique using
modifications to Pascal's triangle to determine the total number of electrons
in noble gases. Weise did not create a mathematical equation for the sequence
of the cardinalities of the periods in the Periodic Law, despite his success in
replicating the sequence of total electrons in noble gases. He also modified
Pascal's triangle and applied the triangular numbers to replicate the entire
electron sequence of the noble gases.
In other words, a mathematical
explanation must be used to describe the behaviour and characteristics of atoms
and molecules. Hence, chemistry is an application of mathematics. Malkevitch
(2017), continued by stating that the development of an experimental method to
chemistry, which involves conducting procedures in controlled conditions and
determining whether one obtains repeated findings, was revolutionary.
Controlled experiments give researchers access to statistical and experimental
design mathematic tools. Another component of mathematics which is extremely
helpful in chemistry is measurement. In chemistry, many units of measure are
employed, including centimetres (cm), inches (in), feet (ft), metres (m), and
kilometres (km) for length, and grams (g) and kilograms (kg) for mass. Time
units include second(s), minute(s), hour(hr), area(unit) sq, density(kg/m3) and
acceleration (m/g2). The observation and quantitative measurement of elements
like length, volume, temperature, density, normality. molality, equivalent
weight and chemical solution concentration are all part of the natural
sciences. The measurement is a tool used in mathematics, and chemistry and
other scientific disciplines can use it. The majority of the above-mentioned
quantities have units that must be kept in mind when being used in
calculations. Bain et al.,( 2019) mentioned that a
purely mathematical standpoint, symbols and symbolic notations are complex due
to the amount of information they may encode and the range of possible
interpretations and applications. Similar to this, it might be difficult to
comprehend the information conveyed in a graph using mathematical reasoning.
The usage and conceptualization of mathematics by mathematicians and physical
scientists differ significantly, nevertheless. When considering the additional
layer of conceptual reasoning introduced in physical science areas, where
learners are expected to describe complex processes using abstract mathematical
formalisms, the capacity to reason using equations and graphs is further
difficult (Bain et al., 2019; Kozma et al.,1997; Bain et al.,2016)).
Also, we were
interested in mathematical materials in addition to conceptual resources, such
as concepts related to chemistry. The symbolic structures outlined by Sherin (2001),
which describe intuitive mathematical concepts about equations, are one
instance of a mathematical resource. At the beginning, the symbolic forms
analytical framework was created to describe mathematical concepts and the
majority of the symbolic processes that Sherin (2001), found in these forms are
algebraic (e.g., considering proportional relationships, the influence of a
coefficient, dependence of terms on other values, etc.). He admitted that his
initial list was not exhaustive, and academics from many fields have used the
framework to illustrate a variety of mathematical concepts, including more
complex subjects like integrals, differential equations, and vectors (Bain et
al., 2019; Jones et al., 2015; Dreyfus et al., 2017). Also, Chemists
view mathematics as a valuable tool, thus all chemistry learners should focus
on learning it in order to access and benefit from their science. Many types of
mathematics are utilised in chemistry, from proportional reasoning to complex
differential equations and Fourier analysis. The study of any of the
underpinning mathematics reduces mathematical activity to a set of orderly,
methodical routines and procedures. To be useful as a requirement for general
chemistry, one should have the knowledge and abilities mentioned in the domains
of fundamental mathematics, calculus, and three-dimensional geometry (Olafare, 2016).
Table-1: Correlation
of Mathematics and Chemistry (Ali and Bhaskar, 2016)
|
Mathematics
|
Chemistry
Context
|
|
Ratios
|
Mixing solutions
with certain molarities, making dilutions
|
|
Proportional
reasoning
|
Analysis of
molecular structure; moles
|
|
Algebra and
graphs
|
Analysis of
experimental plots of reaction rates; gas laws
|
|
Calculus
|
Predicting and
measuring rates of reaction in measurable experiments
|
|
Units of
measurements
|
Making sense of
real, complicated measurements
|
|
Logarithms
|
Understanding pH
|
Hewson (2011), claimed that some basic
mathematics topics were also stressed: unit conversions, significant figures,
proportions, and concentrations; expressions involving exponents and
logarithms; basic trigonometry and algebra, including graphing; summation
notation; probability and statistics; and applications of all of these to word
problems (Table:1). It was also stressed that it is unlikely that one cannot excel
in chemistry without a strong understanding of and facility with these topics. According
to Zambrini (2006), chemistry is a precise discipline that is dependent on
mathematical modeling that may be expressed and used by employing the language
of mathematics. Complex mathematical tools are needed to solve problems in the
theory of chemical bonding and molecular structure, rates and equilibria of
chemical processes, molecular thermodynamics, relationships involving energy,
structure, and reactivity, and modelling of systems. Mathematics is essential
to applied chemistry and chemical engineering; examples include atmospheric
composition, biotechnology, and computational methods (Olafare, 2016). Korcz et
al., (2008) elaborated that the dependability of the results is significantly
impacted by the quality of the studied data. The use of statistical techniques
enables the reduction of some stages of a chemist's work, such as the
classification of the large number of data sets. For an initial assessment of
the data quality, statistical approaches are used. In this situation, it's
important to make sure that the raw dataset is free of significant errors or
outliers that can skew the results of the experiment. Chemometric strategies
for data analysis concentrate on identifying the traits that are most connected.
Chemometry is used to create a mathematical model of the relationship between
an investigated attribute and several different sets of stated variables
(parameters that affect measure). Calculations are necessary for modelling
in order to identify the model, verify its applicability, assess the
appropriateness, and develop the model's predictive capacity. The established
relationship model may be applied to technological process system optimization,
forecasting of subsidiary values based on described known values, and control
of the analytical system. For more effective information flow management,
statistical approaches are used in chemical investigations for data gathering
and analysis of chemical compounds. They enable the physical and biological characteristics
of chemical substances to be predicted. For quality control, statistical
approaches are also used in the chemical examination of pollutants, such as
pesticide residues in food (Korcz et al., 2008).
According to Winters et al., (2010)
Science's field of statistics deals with gathering, organising, analysing, and
extrapolating data from samples to the entire population. The mean, the mathematical average of numbers, median, mode, range,
dispersion, standard deviation, interquartile range, coefficient of variation,
etc. are the most well-known statistical tools. Software programmes like SAS
and SPSS are also available, and they are helpful in understanding the findings
for big sample sizes (Begum and Ahmed ,2015). The summation of
all the numbers divided by the total number of scores is the mean (also known
as the arithmetic average). Extreme variables may have a profound impact on
mean. The mean can be used to quickly get a snapshot of
your data or to identify the general trend of a data set. The mean also has the
benefit of being quite simple and quick to evaluate. For instance, a
single patient who stays in the ICU for almost 5 months due to septicaemia may
have an impact on the average length of stay for organophosphorus poisoning
patients. Outliers are values that are at the extremes. The mean is calculated
as follows:
Mean (`x) =
where n is the number of
observations and x represents each observation (Begum and
Ahmed, 2015; Ali
and Bhaskar,2016).
The median (Manikandan, 2011) of a
distribution is the point where half of the variables in the data fall above
and below the median value. In contrast, the mode is the variable that occurs
the most frequently in a distribution. A sample's spread or variability is defined
by its range (Myles and Gin, 2000). The variables' minimum and maximum values
serve as a description of it. The distribution’s variability (Myles and Gin
,2000) is a measurement of how skewed it is. It reveals how closely a single
observation clusters around the mean value. The following formula describes how
a population's variance is calculated:
where N is the number of elements
in the population, x is the population mean, xi is the i-th element
from the population, and 2 is the population variance.
Also, the standard
deviation is a measurement of the range of data from the mean, and it is
sometimes represented by the Greek letter sigma. A low standard deviation
indicates that more data are in line with the mean, even though a high standard
deviation indicates that data are spread more thinly from the mean. The
standard deviation is helpful for immediately determining the dispersion of
data points in a variety of data analysis techniques. (Begum and Ahmed ,2015). The
square root of variance is employed to simplify data interpretation and
maintain the fundamental unit of observation. The standard deviation is equal
to the variance squared (SD) (Binu et al.,2014). The following formula
determines the SD of a population:
where, `x = population
mean, xi= ith element from the population and N is the number of
elements in the population.
Begum and Ahmed, (2015)
mentioned that the associations between dependent and explanatory variables,
which are often plotted on a scatter diagram, are modelled via regression. The
regression line also indicates how strong, or weak such associations are.
Regression is frequently covered in high school or college statistics courses,
with applications in identifying patterns over time in science or business.
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