An instance of a mathematical model in chemical kinetics Ivana Milanovic*, Ruzica Vukobratovic ...

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Int. J. Knowledge Engineering and Soft Data Paradigms, Vol. 3, Nos. 3/4, 2012

An instance of a mathematical model in chemical kinetics Ivana Milanovic*, Ruzica Vukobratovic and Vidak Raicevic High School ‘Isidora Sekulic’, Vladike Platona 2, Novi Sad, Serbia E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: The paper presents how we can apply mathematics in teaching of chemistry and chemical kinetics in the classroom. We describe here how chemical reaction order depends on the concentration of the reactants, using differential and integral calculus. Subsequently, we look into solving the problem of determining the order of chemical reaction and the problem is solved by mathematical models, using properties of linear functions. We have used the software package GeoGebra, especially the use of spreadsheets, forming a list of points, their graphical representation, fitting curves for modelling of our examples. Keywords: subject correlation; mathematical model; chemical kinetics; integral; linear function; fitting, GeoGebra. Reference to this paper should be made as follows: Milanovic, I., Vukobratovic, R. and Raicevic, V. (2012) ‘An instance of a mathematical model in chemical kinetics’, Int. J. Knowledge Engineering and Soft Data Paradigms, Vol. 3, Nos. 3/4, pp.294–308. Biographical notes: Ivana Milanovic received her degree in Mathematics from University of Novi Sad, Serbia in 2000. She teaches mathematics at high school ‘Isidora Sekulic’, Novi Sad, since 2000. Her main field of interest is methodics of teaching mathematics. Ruzica Vukobratovic received her degree in Mathematics from University of Novi Sad, Serbia. She received her Magister of Science in Mathematics from University of Novi Sad, Serbia in 2010. She teaches mathematics at high school ‘Isidora Sekulic’, Novi Sad, since 1992, and is currently the Principal of the school since 2007. Her main field of interest is methodics of teaching mathematics. Vidak Raicevic is a fourth grade student at high school ‘Isidora Sekulic’, Novi Sad. His main fields of interest are chemistry and mathematics. This paper is a revised and expanded version of a paper entitled ‘Establishing order of chemical reaction using mathematical modelling’ presented at the Computers in Education Conference, at the 34th International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO 2011), Opatija, Croatia, 23–27 May 2011.

Copyright © 2012 Inderscience Enterprises Ltd.

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Introduction

1.1 Modelling One of the more important functions of mathematics is that of describing and explaining real-world phenomena, researching into the key questions about the observed world and testing corresponding ideas, all of that through the formation of mathematical models. The model can be further used to compose predictions of the behaviour of the specific phenomenon in the given conditions (Tall, 1977). The process of formation of the mathematical model is called mathematical modelling. Modelling presents one of the basic processes of the human mind, expressing our ability to think and imagine, to use symbols and languages, to communicate, to perform generalisations on the basis of experience, to confront the unexpected. It enables us to notice patterns, to estimate and predict, to administer processes and objects, to exhibit the meaning and the purpose. Because of this, modelling is most commonly viewed as the most important conceptual means readily available to man. Modelling is not only science, but skill and art as well (Takači, 2008). Because the real world involves economy, ecology, chemistry, engineering, etc., mathematical modelling can be regarded as the activity which enables a mathematician to become an economist, an ecologist, a chemist. Instead of performing real-world experiments, he or she can perform experiments from the mathematical aspect of the world. Some of the characteristics of mathematics are very important when modelling is concerned, such as: 1

The mathematical language is precise, which helps in the formulation of hypotheses and identifying the basic postulates about phenomena and concepts of the real world which are subjected to mathematical modelling.

2

The mathematical language is concise, with strictly defined regulations for performing specific operations. This helps the construction of the model staying correct and regular.

3

All results previously proven by mathematicians are on disposal. The model can be solved using all available mathematical resources, theorems, definitions, etc.

4

Computers can be used for the display of graphical forms and numerical calculation. The usage of the computer technology enables the model’s visualisation and simulation; the computer is engaged in the operative role, while the user is left more time for critical and logical judgment in solving the problem. Further on, computers take part in the presentation, analysis and verification of the model (Anon., n.d.).

The advantages of professionally using models are numerous. A model is a simplified and idealised picture of reality. It enables us to face the real world (system) in a simplified way, avoiding its complexity and irreversibility, alike all dangers that can arise from experiments conducted on the real system. By this process of simplification, it can probably predict with great accuracy certain aspects on which it concentrates. Scientific phenomena can be studied through modelling, forms and connections can be determined; globally speaking, modelling can be integrated in various disciplines of contemporary scientific study.

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Certain limitations also exist while in the process of mathematical modelling. A model presents the abstraction of reality, in the sense of being able to capture only some of its aspects. The fact that the model by its very nature concentrates on specific aspects of the situation, disregarding others or being partially imprecise, can present a problem. The existence of a model is not enough on its own; it requires focusing on the most important parts of the observed situation and understanding of its limitations and conditions in which it should be treated (Tall, 1977; Anon., n.d.; Takači, 2008). It is necessary to carefully select the level of abstraction so the resulting model can illustrate the system as precisely as possible, while the complexity of the same model must not present a limiting factor (Takači, 2008). Exceedingly complex or perfect models are commonly very expensive and/or inadequate for experimentation. On the other hand, very banalised models do not illustrate the specific system accordingly, while the results obtained via them can be scarce and incorrect. It can be thus concluded that the challenge in such modelling is “not to obtain the most extensive describing model, but to produce the simplest model possible which includes the main features of the observed phenomenon” (Howard Emmons) (De Vries, 2001). Formal, scientifically accepted laws should be followed when describing a model. An informal description of a model provides the basic concepts of the model, and although there is a tendency concerning its completeness and accuracy, these states are never fully reached. During the composition of the informal description, in order to eliminate the mentioned imperfections, a categorisation to objects, descriptive variables and object interaction rules is conducted. Objects are parts on which the model is assembled; descriptive variables describe states in which the objects exist in certain periods of time (where parameters describing the constant characteristics of models are also included); rules define the manner in which objects of the model can affect one another, in order to change the state they are in. Anomalies that occur in the informal description of the model are mostly concerned with the incompleteness, inconsistence or obscurity of the description. Unless the model encompasses all the situations that can come about, the description is deemed incomplete. If the description includes two or more rules for a single situation, the usage of which gives rise to contradictory actions, the description is inconsistent; and if it is needed to carry out two or more actions in one situation, while the order of their execution is not defined, then the description is unclear (Takači, 2008). Taking this into consideration, need arises for using formal descriptions of models. A formal model description uses the methodology of modelling; it determines the type of the given objects in a clear and unambiguous way. It provides greater precision and completeness of the description of a model. Abstraction is used in a formal description; attention is given to important abilities of the model; an engineering and scientific approach is present, the construction of the model is based on formalisation, and the usage of the model on the analysis of the obtained results.

1.2 Mathematical modelling in teaching Contemporary conceptions of education in the gymnasium prescribe two new aspects of work of both teachers and students.

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The first once concerns on developing interdisciplinarity in teaching. Concrete and evident correlation between subjects that would provide adoption and establishment of students’ knowledge regarding two or more scientific fields simultaneously, and those segments of knowledge would permeate one another, so that the student can gain a comprehensive and complete image regarding specific concepts and phenomena. A student should solve problems from one subject by using knowledge acquired from another subject. When it comes to this sort of connections, mathematics holds a key position. The second one concerns the need to use mathematical models in teaching. It is a result of scientific, technological and social progress which renders the world more complex, with new and more difficult problems that need solutions. These problems require very specific solutions that entail specific information; the identification of requirements of the solutions also becomes important, and not only solutions. In such an environment, mathematical modelling has a very important role. That is why it is important for students to familiarise themselves with the process of modelling, to learn what can be modelled and how, as well as what is the final goal. It is necessary for students to master the basic principles of this process, because it is very likely that they will encounter modelling in further education and professional practice. Traditional teaching most commonly includes the method of oral presentation by the teacher, with a frontal form of student work in class. The lectures are strictly defined, while the goal is reduced simply to implementing knowledge structures. The teaching with modelling, as one representative of contemporary teaching encompasses a different approach. The heuristic method, the problem method and the experimental method are present here, and the form of student work is directed at working in groups or pairs. The prominent goals are: orientation on application, modelling, authenticity and problem solving; aspects of presentation and interpretation in mathematics, concentration on adequate concept construction is emphasised, as well as discussion about the possibilities and limits of mathematical treatment, orientation on fundamental mathematical procedures, interdisciplinarity, inclusion of historical and socio-psychological aspects and various social goals of mathematics teaching. The possibility of experimenting, connecting different types of displays, dynamical displays, executing algorithmic and numerical operations with the aid of computers broadens old content and introduces new which was not visually available in traditional teaching. A student that is thought modelling in class adopts knowledge of a certain level and enhances his or her creativity while he adopts new lecture material. The student forms specific skills and habits, develops abilities, and generally is prompted to form attitudes and beliefs. Taking that into consideration, modelling is justifiably one of the key mathematical competences of students. Mathematical modelling in class involves a specific methodology and foundation on didactical and methodical principles. This process consists of the following stages: 1

The real problem: By studying some phenomena from the real world, the real problem is noticed; it can be connected with any science, and it can even be an everyday situation. This is possible for the student to realise in accordance with his or her interests and affinities, and with the help of a teacher. The following cognitive activities are concerned: problem understanding, possible simplification of the

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The mathematical problem: Mathematical modelling is approached, and the construction of the mathematical model begins. Here, a mathematisation is taking place: the real problem is adjusted to mathematical contents, using the required assumptions, formulations and other means. The mathematical problem becomes the representation of the real problem. This is the most important, though the most difficult step in modelling.

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The solution of the mathematical problem: The corresponding mathematical concepts and various methods are applied on the model, with the goal of obtaining a mathematical solution. Interpretation of the solution of the mathematical problem is then approached. An analysis of the obtained results is made, while the model is evaluated and verified.

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The solution of the real problem: If the model is accepted, demathematisation follows; the mathematical solution is restored to the environment in which the problem has been formulated, thus obtaining the solution of the real problem. If the model is not accepted, we return to the real problem, and the procedure is repeated.

The cognitive activities while proceeding to the next step of modelling are performed together by the student and teacher, though student independence and creativity is stressed. The teacher helps the student overcome problems that occurred during the process, directs him or her in the right direction in case any conflicting situations occur, which is not impossible, because it is commonly needed to adapt very abstract contents. It is necessary for the students to adopt, practice and apply specific knowledge, abilities and habits through the process of modelling. Initially, it is the ability of perception and creativity in the course of selecting a certain problem. Later, it is the ability of collecting and organising data and its selection, as well as arranging and distributing data that was found to be of importance. The basis of mathematical modelling lies in mathematical knowledge, which gives rise to correct reasoning, critical and logical thinking, as well as the student’s self-confidence, sureness when using numbers, understanding relations, forms and spatial constructions, etc. This way, the student develops persistence and systematicity, the ability of communication and cooperation with other participants in the process; keeping of an orderly technical documentation is practiced; a certain general culture of labour is adopted. Here, it is also possible to point out the possibilities of using various mathematical methods in problem solving. Lastly, students learn how to interpret the obtained results. The most important point throughout the process is the development of the ability of generalisation of the observed rules and connections and an abstract way of thinking. After the realisation of the process of modelling and solving the problem, the student will be able to relate the given problem and its result more easily with another similar problem from the same or close field of study. The process of mathematical modelling can be presented with the following scheme (Kraljević and Čižmešija, 2009).

An instance of a mathematical model in chemical kinetics Figure 1

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Modelling diagram (see online version for colours)

The process of mathematical modelling while determining the order of a chemical reaction

This paper presents an instance of mathematical modelling of chemical phenomena, more precisely the determination of the chemical reaction order. During this project, the following points have been realised: •

In chemical kinetics, one of the fields of chemistry, chemical reactions are studied as phenomena that take place in the real world. The rate of the chemical reaction is related to the order of the reaction.

•

The mutual dependence and conditioning of these concepts are presented as real problems that need to be studied. Calculus is a branch of mathematics that focuses on limits, functions, derivatives and integrals. The knowledge of these concepts can become meaningful and more durable if it is related with the corresponding concepts of another scientific discipline. Practice and better understanding of the concepts of further mathematics is also a result. The example of such a correspondence with chemical concepts is shown during the modelling.

•

Later on, the formation of the mathematical model was approached; it consisted of two parts: the first one being a theoretical solution, and the second one a practical solution of the real problem.

•

Then, the respectful mathematical contents were applied to the model. The modelling of the rate and order of a chemical reaction was done using differential equations and integrals; and while studying the reaction order, a relation between the definite integral and the logarithmic function was observed. The linear function takes part in describing the reaction order. In the second part, the modelling was done by the means of a computer, where experimental data was treated, which gave the solution of the problem. The model simulated the changes that happened in specific time

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•

Lastly, an analysis of the produced graphs was made; and on the basis of their properties, a certain interpretation of solutions of the mathematical problem was composed, as well as an interpretation of analytical and numerical solutions of the real problem. The model was accepted.

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Rate of a chemical reaction

3.1 Definition Chemical kinetics is the discipline of chemistry that deals with the kinetics of chemical and physicochemical processes. In other words, it examines the rate at which these processes occur (House, 2007). Since the chemical reaction is the chemical transformation of reactants into products, reactants disappear in this process, while products arise. The rate of a chemical reaction is the value that describes how quickly reactants disappear and how quickly products arise. For this purpose, the rate of a chemical reaction is defined as the change in concentration of reactants or products per unit of time: v=

c −c Δc = t +Δt t Δt Δt

(1)

where ν is the reaction speed, ∆c is concentration changes in the time interval ∆t and ct+∆t and ct are concentrations in the moments t + ∆t and t. The change of concentration ∆c is negative if the reactant is measured and positive if the product is measured. For practical reasons, it is more common to measure the concentration of reactants. If the time interval ∆t is so small that it can be considered to approach zero, and if we present the rate of a chemical reaction as a function of time v = f(t), we can derive a formula for calculating the instantaneous rate of chemical reactions vt:

vt = lim

Δt → 0

ct +Δt − ct dc =± Δt dt

(2)

where the obtained expression is positive if c is the concentration of a product or negative if c is the concentration of a reactant. Molar concentrations are used, and their standard unit is mol/L which can also be abbreviated as M. The rate of a chemical reaction always has the unit mol/Ls (House, 2007).

3.2 Rates of chemical reactions according to collision theory It was noted that the rate of a chemical reaction is directly dependent on the concentration of reactants that participate in it, which is explained by the collision theory: if the concentration of a certain substance is higher, a collision between molecules that starts a chemical reaction is more likely to occur. In line with this, the rate of a chemical reaction involving reactants A and B can also be defined as:

An instance of a mathematical model in chemical kinetics v = k[ A]m [ B ]n

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where k is the chemical reaction rate constant, and [A] and [B] are concentrations of substances A and B respectively, and m and n are the corresponding exponents. The reaction rate constant is a value which is specific for each reaction and conditions in which it occurs. The values of the exponents of concentrations of substances depend on the mechanism of a reaction, and they often correspond to the coefficients of these reactants in a balanced chemical equation. The sum of all exponents of concentrations of the reactants of a chemical reaction is also called the order of a reaction. Reactions are usually zero-, first- or second-order. If we indicate the order of the reaction as r = m + n, the unit of the reaction rate constant will be: mol1− r L3−3r s

(4)

because the unit of reaction rate of a reaction of any order is always mol/Ls. The chemical reaction rate expressed as (3) also presents an instantaneous chemical reaction rate if the chosen values for concentrations of the substances are those which were present in the reaction system at a certain point in time. Suppose that the chemical reaction rate depends on the concentration of only one reactant, A. This may mean that the reaction involves only one reactant or that the exponents of concentrations of other reactants in the formula of chemical reaction rate are equal to zero. Equalling the instantaneous rates of chemical reactions as in (2) and (3) gives: −

d [ A] = k[ A]m dt

(5)

where m is the exponent of the concentration of the reactant A and also the value that represents the reaction order. We will solve the obtained differential equation for the most common cases, i.e., when the reaction is first-, second- or zero-order (House, 2007).

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Order of a reaction

4.1 First-order reaction If we take that m = 1 in the equation (5), i.e., it is a first-order reaction, by transforming we get: −

d [ A] = kdt [ A]

(6)

We can integrate equation (6) within the time points 0 and t and the concentration at the zero point, [A]0, and the concentration at the moment t, [A]: [ A]

−

∫

[ A]0

t

d [ A] = k dt [ A]

∫ 0

(7)

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that yields: ln

[ A]0 = kt [ A]

(8)

4.2 Second-order reaction If we take that m = 2 in the equation (5), i.e., it is a second-order reaction, by transforming we get: −

d [ A] = kdt [ A]2

(9)

which by the integration within the limits of time at the moment 0 and t and the concentration at the zero point, [A]0, and the concentration at the moment t, [A]: [ A]

∫

(10)

1 1 − = kt [ A] [ A]0

(11)

−

∫

t

d [ A] = k dt [ A]2 0

[ A]0

further gives:

4.3 Zero-order reaction If we take that m = 0 in the equation (5), i.e., it is a zero-order reaction whose rate is independent of the concentration of any reactant, by transforming we obtain: −d [ A] = kdt

(12)

which by the integration within the limits mentioned in the previous cases: [ A]

∫

t

∫

(13)

[ A] = [ A]0 − kt

(14)

−

[ A]0

d [ A] = k dt 0

yields:

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Processing of experimental data

5.1 Graphical representation It is possible to acquire data of the values of concentration of a reactant in the periods of time using different experimental methods. Then, a graph of reactant concentration over

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time is drawn. Most often, the best case for data processing occurs when the functions of the concentration change over time are linear. However, given that the reactions of the first-, second- and zero-order do not show the variation of concentration change over time [which can be seen from the equations (8), (11) and (14)] the ordinate of the graph on which the function is linear is not the same for the reactions of first-, second- and zero-order. In order to be linear, the function must have the form y = bx + c. By transformation of equation (8) we get: ln[ A] = − kt + ln[ A]0

(15)

and since [A]0 is the initial concentration of a reactant and therefore a constant, the graph abscissa of the first-order reaction flow in which the concentration change over time is linear is t, while the ordinate is ln[A]. Analogously, by the transformation of equation (11) we obtain: 1 1 = kt + [ A] [ A]0

(16)

and the abscissa of linear concentration change over time is t and the ordinate is 1 / [A], when a second-order reaction is in question. Finally, from equation (14) it follows that the abscissa of linear dependence of concentration of reactants in the reaction of the zero-order is t, while the ordinate is [A]. On this basis, we can easily determine the order of the reaction by plotting the change of ln[A], 1 / [A] and [A] over time. The graph in which the function is linear, depending on its ordinate, shows the reaction order. Based on the graph, k can be determined as the function slope (House, 2007).

5.2 Establishing the existence of linear dependence Very often, when some of the mentioned graphs are plotted, it is clear at first sight whether they are linear or not. However, it should be noted that the data obtained through an experimental method always exhibits an experimental error. In some cases, when using the visual method it is difficult to determine whether it is linear or not. In this case, it is best to use a software package with the possibility of determining the ‘success’ of fitting the entered points to a linear function. GeoGebra is a free mathematical software package that encompasses geometry, algebra and calculus. GeoGebra is a computer tool that offers numerous possibilities, relating the symbolic and iconic representations. With this tool, the principle of activity, the principle of operation, the principle of experimenting and the principle of learning by exploration are made visible (Herceg and Herceg, 2007). In a recent version of GeoGebra, GeoGebra 4.0 Beta, it is possible to use the command RSquare. It is a form of a statistical test that checks how much a given point corresponds to the function graph by calculating the coefficient of determination (R2), the values of which range from 0 to 1. The more the points correspond to the type of dependency of the chosen function, the closer the value is to 1 (Steel et al., 1997).

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The problem of determining the order of a specific chemical reaction

In mathematics, a function presents the dependence of one variable (y) on another variable (x); and that (functional) dependence is what abstracts this concept from others in mathematics. The concept of a function as a representative of the contemporary mathematical thought indicates variability, interconnection and conditionality of mathematical values. This kind of relationship and dependence is just that of real-world phenomena. Real-world phenomena that encompass causality between two or more variable values, when modelled, produce a function as a mathematical model for the studied phenomenon (Takači et al., 2010). The following part of this study will explain how chemical reactions can be treated as real phenomena that surround us, while modelling will explain concepts of chemical reactions. Students were given experimental data (House, 2007) of the reaction A → products, as shown in Table 1. Three problems were put in place: a

the determination of the reaction order

b

the determination of the chemical reaction rate constant

c

the determination of the amount of time needed for the concentration of the reactant A to reach 0.038 M.

A graphical model that simulates concentration change of the reactant A over time was constructed afterwards. Visualisation of the dependence of [A] in respect to time t is done in order to obtain a linear function. Creating students’ ability to solve the presented chemical kinetics problem using the knowledge about linear functions and the analysis of the produced model is the goal of such mathematical modelling. Table 1

Experimental data

Measurement count

Time, t(h)

Concentration, [A] (mol/L)

1

0

1.24

2

1

0.96

3

2

0.78

4

3

0.66

5

4

0.56

6

5

0.5

7

6

0.44

8

7

0.4

9

8

0.37

10

9

0.34

11

10

0.31

Task: a

Determine the reaction order (done in GeoGebra).

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Solution: a

Test 1 •

The Spreadsheet view is used, the table values are entered for the time t and concentration [A] in order to check whether the reaction is zero-order.

•

Then, on the basis of this data, a list of points was made, using the command create list of points.

•

A list of points is then assigned to the graphical display, where the value of time is on the abscissa, and values of concentration are on the ordinate.

•

Then the fitting was started, using the command FitLinear, in order to determine whether a graph of a linear function was obtained.

•

By the visual method, it was determined that the obtained graph is not the graph of a linear function, and the advanced version of GeoGebra also determined the coefficient of determination of 0.86.

Figure 2

b

The interface of GeoGebra in the first test (see online version for colours)

Test 2 •

The Spreadsheet view is used, the table values are entered for the time t and for the natural logarithm of concentration, ln[A], in order to check whether the reaction is first-order.

•

Then, on the basis of this data, a list of points was made, using the command create list of points.

•

A list of points is then assigned to the graphical display, where the value of time is on the abscissa and the values of ln[A] are on the ordinate.

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Then the fitting was started, using the command FitLinear, in order to determine whether a graph of a linear function was obtained.

•

The visual method was slightly less obvious in determining that it is not a linear graph of the function. The coefficient of determination of 0.96 was calculated.

Figure 3

c

The interface of GeoGebra in the second test (see online version for colours)

Test 3 •

The Spreadsheet view is used, the table values are entered for the time t and for the reciprocal of the concentration, 1 / [A] in order to check whether the reaction is second-order.

•

Then, on the basis of this data a list of points was made, using the command create list of points.

•

A list of points is then assigned to the graphical display, where the value of time is on the abscissa and the value of 1 / [A] is on the vertical axis.

•

Then the fitting was started, using the command FitLinear, in order to determine whether a graph of a linear function was obtained.

•

It is obvious that this is a graph of linear function, which was confirmed by the coefficient of determination, which amounted to 1 (higher than in the previous two cases, which certainly removes any doubt).

•

Based on the results obtained in all tests, it was found that the reaction whose experimental data was treated was second-order.

An instance of a mathematical model in chemical kinetics Figure 4

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The interface of GeoGebra in the third test (see online version for colours)

Task: b

Determine the reaction rate constant.

Solution: •

Since it was established that the reaction is second-order, in the further consideration we observe the graph obtained in the third test.

•

The reaction rate constant corresponds to the slope of the function obtained in the Test 3, and using the command Angle a value α = 13.5° is obtained (the angle that the line forms with the positive part of the x-axis).

•

Using the command Tan(α) the value k = tg(α) = 0.24 is obtained.

•

Based on the equation (4), given that it is a second-order reaction and that the time is measured in hours, the chemical reaction rate constant is 0.24 L/molh.

Task: c

Determine the time in which the concentration of the reactant A is [A] = 0.38 M.

Solution: •

With regard to the information given, it follows that the ordinate of the required point is y = 1 / 0.38 = 2.63. As this point is an element of the function line, its coordinates satisfy the line equation which GeoGebra showed, so it follows that the abscissa of the required point is x =7.59 h.

•

On the graph, the required point is marked with M (Milanović and Raičević, 2011).

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Conclusions

In contrast to the traditional teaching of mathematics and other sciences, a different approach was presented in this paper. The basis is studying theory and solving problems in the area of one subject, chemistry, with the aid of mathematical modelling. The presented model is adapted to the conditions and needs of gymnasium teaching, while all the requirements of mathematical modelling are fulfilled. The project was realised during additional classes of mathematics. Twelve students of the third grade took part, aged 17, who were familiar with the usage of GeoGebra and its general possibilities. These students exhibited a particular affinity and interest for science and computing. Not all of the chemical concepts discussed here are studied on the gymnasium level. Therefore, the goal for this and other similar projects is creating a scientific basis for students who will have contact with chemistry in their further education. The goal was also to present the possibility of the application of mathematical concepts in other fields of study. At the same time, knowledge from chemistry was adopted and mathematical contents were practiced. A practical and evident link was formed between the concepts from the real world and mathematics, together with the link of theory and praxis. Knowledge acquired in this manner becomes more permanent, functional and applicable.

References Anon. (n.d.) ‘An introduction to mathematical modeling’, available at http://staffweb.cms.gre.ac. uk/~st40/Books/MathematicalModelling (accessed on 9 April 2011). De Vries, G. (2001) What is Mathematical Modeling, Department of Mathematical Sciences, University of Alberta, Alberta. Herceg, D. and Herceg, Đ. (2007) GeoGebra-dinamička geometrija i algebra, Prirodnomatematički fakultet u Novom Sadu, Novi Sad. House, J.E. (2007) Principles of Chemical Kinetics, 2nd ed., Academic Press, London. Kraljević, H. and Čižmešija, A. (2009) Matematika u kurikulumu-ishodi učenja, PMF-Matematički odjel, Sveučilište u Zagrebu, Zagreb. Milanović, I. and Raičević, V. (2011) Utvrđivanje reda hemijske reakcije pomoću matematičkog modela, godina LVII, broj 1–2, Pedagoška stvarnost, Novi Sad. Steel, R.G. et al. (1997) Principles and Procedures of Statistics, McGraw-Hill, New York. Takači, A. (2008) Development of Computer-aided Methods in Teaching Mathematics and Science, Prirodno-matematički fakultet u Novom Sadu, Novi Sad. Takači, Đ. et al. (2010) Matematičko modeliranje zastupljenosti pušenja, Vol. 56, broj 7–8, Pedagoška stvarnost, Novi Sad. Tall, D. (1977) Cognitive Conflict and the Learning of Mathematics, Mathematics Institute, University of Warwick, Coventry.