Modeling Chemical Kinetics Graphically

Modeling Chemical Kinetics Graphically
Mozahedul Islam
Abstract In literature on chemistry education it has
often been suggested that students, at high school level
and beyond, can bene¯t in their studies of chemical ki-
netics from computer supported activities. Use of sys-
tem dynamics modeling software is one of the suggested
quantitative approaches that could help to develop stu-
dents' knowledge about chemical kinetics and chemi-
cal equilibrium and to remediate alternative concep-
tions. The methodology, strengths, and weaknesses of
the implementation of graphical system dynamics soft-
ware for modeling chemical kinetics are presented in
this paper. An extension of classical graphical modeling
is proposed that is closer to how chemists think about
chemical reactions and that could make it easier for
students to investigate chemical kinetics, especially in
cases of non-trivial reaction mechanisms. The proposed
approach has also useful applications in other subject
areas. Illustrative examples are given throughout the
paper.
Keywords Chemical kinetics ¢ Graphical, system
dynamics based modeling ¢ computer based learning
1 Introduction
Chemical equilibrium and chemical kinetics are impor-
tant concepts in general chemistry, both in secondary
education (e.g., UK and the Netherlands) as well as
in higher education. The study of chemical equilibrium
aims at a better understanding of incomplete, reversible
chemical reactions that lead to a stable mixture of re-
actants and products and of the factors that infuence


the stability of this dynamic equilibrium. The study
of chemical kinetics seeks to provide insight into the
way chemical reactions proceed, both regarding the ob-
served relationships between reaction rate and the vari-
ables that exert in°uence on them as well as the reac-
tion mechanisms that could explain an experimentally
determined rate law. These topics are related to each
other and a kinetic approach to chemical equilibrium is
quite popular in education.
Chemical equilibrium and chemical kinetics are on
the other hand considered di±cult topics to teach and
to learn, no matter whether a qualitative or (semi-)quan-
titative approach has been adopted. A short review of
common students' alternative conceptions and instruc-
tional strategies is included in Section 2. This is done
for the purpose of underpinning the potential of graphi-
cal computer modeling of chemical kinetics with regard
to addressing some of these alternative conceptions.
It has often been suggested that students can ben-
e¯t in their studies of chemical kinetics and chemical
equilibrium from computer supported activities. Pro-
posed activities range from computer-assisted instruc-
tion (Hameed et al 1993; Reid et al 2000), data log-
ging (Chairam et al 2009; Choi and Wong 2004; Cort¶es-
Figueroa and Moore 1999), use of dedicated packages
(Allendoerfer 2003; Halpern 2006; Lee and Briggs 1978),
modeling with computer algebra systems and other sci-
enti¯c computing environments (Alberty 2004; Ferreira
et al 1999; Harvey and Sweeney 1999; Maher et al 2003;
Mira et al 2004; Mulquiney and Kuchel 2003; Ogilvie
and Monagan 2007; Zielinski 1995), use of a graphi-
cal calculator (Cort¶es-Figueroa and Moore 1999, 2002)
and a spreadsheet program (Blickensderfer 1990; Bruist
1998; de Levie 2002) to computer simulations (Fermann
et al 2000; Halkides and Herman 2007; Huddle and
White 2000; Seybold et al 1997; Solomonidou and Stavri-
dou 2001; Stie® and Wilensky 2003) and system dynam-
2
ics based computer modeling (Chonacki 2004; Kosin-
sky 2001; Ricci and van Doren 1997; Ste®en and Holt
1993). All of these approaches attempt to make the
chemical concepts accessible or comprehensive for stu-
dents, for example by giving students ¯rst-hand experi-
ence with reactions through laboratory work or by sim-
ulating and visualizing the reaction dynamics and/or
the dynamic nature of chemical equilibrium. Although
the possibility of using system dynamics software like
STELLA (www.iseesystems.com) has been advocated
in the past for a quantitative approach to chemical ki-
netics, it seems that it has never expanded enormously
and that the use of spreadsheets in kinetics courses is
dominant (at least at American college chemistry fac-
ulties around the year 2000, as Miles Jr. and Francis
(2002) reported).
This paper puts forward the point that one of the
reasons for the unpopularity of system dynamics model-
ing software may be that teachers and students quickly
¯nd out that the underlying model of graphical model-
ing in these software environments is not so suitable for
easy investigation of chemical kinetics beyond the level
of studying very simple reaction systems. The method-
ology, strengths and weaknesses of the implementation
of graphical system dynamics modeling software for
mathematical modeling of chemical kinetics are dis-
cussed in Section 3.
By thinking in terms of kinetic graph theory and by
introducing a new component in the graphical modeling
language, which handles stoichiometric relationships, a
new look of chemical reaction dynamics in the graphical
interface is achieved that is on the one hand as simple
as the associated system of coupled mathematical equa-
tions looks classically, but that is on the other hand
expected to be more accessible to students who are less
mathematically oriented or skilled. Early experiences
with prototypes and discussions with Dutch chemistry
teachers at secondary school level about the proposed
graphical modeling approach keep this prospect up-
right, but systematic research into the use and eval-
uation of the proposed method is still lacking.
In addition, the incorporation of an easy-to-use, built-
in possibility of instant change of a computer model
due to a discrete-time event or of user interaction with
a model `in real time,' by adjusting the size of an in°u-
ential variable while the model is still running, is con-
sidered an a®ordance of a modeling tool that promotes
a better understanding of the behavior of equilibrium
system when conditions change.
As examples will illustrate in Section 4, these ex-
tensions of classical graphical system dynamics based
modeling could make a quantitative approach to chem-
ical equilibrium and chemical kinetics, in which some of
the known alternative conceptions about chemical equi-
librium and chemical kinetics are directly addressed,
viable in chemistry education at an earlier level than
higher education.
In Section 5 it is brie°y illustrated that the new
graphical modeling approach has applications in other
scienti¯c areas, too. This is considered essential for a
general purpose system for mathematics, science, and
technology education, when integration of tools is high
on the list of design criteria.
Illustrative examples are given throughout the pa-
per. The modeling tool of the computer learning envi-
ronment Coach 6 is used for this purpose. Coach 6 is
a versatile computer learning and authoring environ-
ment for mathematics, science and technology educa-
tion at secondary level and beyond (Heck et al 2009).
It provides integrated tools for measurement with sen-
sors, control activities, digital image and video analysis,
modeling, simulation, and animation. It has been trans-
lated into many languages, it is used in many countries,
and the CMA Foundation (www.cma.science.uva.nl)
distributes it. For this paper it is only relevant that
the selected software environment supports the classi-
cal STELLA-like graphical system dynamics modeling
approach as well as the proposed extensions.
2 Teaching and Learning Chemical Kinetics
All over the world, chemical equilibrium and chemical
kinetics are considered di±cult topics to teach and to
learn, no matter whether a qualitative or (semi-)quanti-
tative approach has been adopted. In literature on chem-
istry education (Banerjee 1991; Cheung et al 2009; van
Driel and GrÄaber 2002; Ganares et al 2008; Justi 2002;
ÄOzmen 2008; Pedrosa and Diaz 2000; Qu¶³lez 2004a;
Qu¶³lez-Pardo and Solaz-Portol¶es 1995) it is frequently
discussed that teachers lack good subject matter knowl-
edge and pedagogical content knowledge, and that many
students have learning di±culties because of prevailing
alternative conceptions linked to macroscopic perspec-
tives, di±culties with the abstract and unobservable
particulate/submicroscopic basis of chemistry, problems
with the di®erent meanings of terms in everyday and
chemistry contexts, and insu±cient mathematical abil-
ities to cope with rate equations and computations in-
volving the equilibrium equation. A short review of
common students' alternative conceptions and instruc-
tional strategies is given in the next two subsections.
3
2.1 Alternative Conceptions
Problematic concepts of chemical equilibrium appear
to be all over the world the same and the most di±cult
ones are the dynamic and reversible nature of chemi-
cal equilibrium, the integration of several concepts con-
cerning various domains of chemistry (structure of mat-
ter, thermodynamics, kinetics, etc.) at di®erent levels
(macroscopic, submicroscopic, symbolic), the shift of
an equilibrium as a consequence of changing conditions
(concentration, temperature, pressure), the equilibrium
constant, and the e®ect of a catalyst. Numerous re-
search studies in secondary and early tertiary educa-
tion, see for example (Bergquist and Heikkinen 1990;
Garnett et al 1995; Gorodetsky and Gussarsky 1986;
Gri±ths 1994; Hackling and Garnett 1985; Huddle and
Pillay 1996; Kousathana and Tsaparlis 2002; Voska and
Heikkinen 2000), repeatedly showed the following alter-
native conceptions about the characteristics of a chem-
ical equilibrium and the involved reaction rates:
{ The rate of the forward reaction increases with time
from the mixing of the reactants until equilibrium
is established;
{ The forward reaction is completed before the reverse
reaction commences;
{ The forward reaction rate always equals the reverse
reaction rate;
{ A simple arithmetic relationship, for example con-
centrations of substances with equal stoichiometric
coe±cients are equal or [reactants] = [products],
exists between the concentrations of reactants and
products at equilibrium;
{ When a system is at equilibrium and a change is
made in the conditions, the rate of the favored re-
action increases but the rate of the other reaction
decreases (i.e., an equilibrium consists of two inde-
pendent parts rather than one whole system);
{ Chemical equilibrium involves oscillating behavior
as the concentrations of the reactants and products
°uctuate;
{ Students' prior experience of reactions that proceed
to completion appears to have in°uenced their con-
ception of equilibrium reactions. Many students fail
to discriminate clearly between the characteristics
of completion reactions and reversible reactions and
they often characterize chemical equilibrium as a
static, balanced condition;
{ Failure to distinguish between rate (how fast) and
extent (how far) of a reaction;
{ Confusion regarding amount (moles) and concentra-
tion (molarity) in an equilibrium expression or rate
equation;
{ Failure to take the stoichiometry of a reaction into
account when setting up an equilibrium expression
or rate equation;
{ Catalysts have no e®ect on or decrease the reverse
rate in an equilibrium reaction;
{ A catalyst only speeds up the forward reaction;
{ The equilibrium constant is independent of the tem-
perature, but changes when the concentration of one
the components in an equilibrium system is altered
or when the volume of a gaseous equilibrium system
is changed;
{ An increase (decrease) of temperature always means
an increase (decrease) of the value of the equilibrium
constant.
Compared to chemical equilibrium, remarkably less
educational research has been reported on chemical ki-
netics. However, the main commonly identi¯ed alterna-
tive conceptions are again linked to the fact that the
introduction of reaction rate requires students to revise
their initial concepts of chemical reaction and their re-
lated corpuscular ideas (van Driel 2002; Garnett et al
1995; Justi 2002):
{ Every reaction occurs instantaneously and continues
until all reactants are exhausted;
{ The reaction rate increases as the reaction `gets go-
ing;'
{ All reaction steps are in essence rate determining;
{ Reactions between two chemical species in a solu-
tion may be analyzed without considering the e®ects
of other species present;
{ Failure to distinguish between rate (how fast) and
extent (how far) of a reaction;
{ Amount and concentration mean the same thing for
species involved in a rate equation;
{ Lack of understanding of the meaning of stoichiom-
etry in chemical kinetics. for example, the `lowest
stoichiometry' in a chemical reaction gives the lim-
iting reactant;
{ Di±culties in relating empirical data and mathe-
matical models for chemical kinetics;
{ Aspects of chemical kinetics (the rate of the reac-
tion) and thermodynamics (does reaction occur) are
separate and mutually exclusive;
{ When fast moving particles collide with each other,
it is very likely that these particles will bounce back,
without a change or reaction occurring. The molecules
simply do not have enough time to exchange atoms;
{ An increase (decrease) of temperature always means
an increase (decrease) of the reaction rate;
{ A catalyst is not consumed during a chemical reac-
tion but remains unchanged.
4
2.2 Instructional Strategies
Several e®ective ways to address and remediate stu-
dents' alternative conceptions and several qualitative
approaches to teach chemical equilibrium and chem-
ical kinetics in secondary and higher education have
been proposed and researched. Piquette (2001) iden-
ti¯ed four successful conceptual change instructional
strategies that explicitly attempt to incorporate the
four necessary conditions for conceptual change estab-
lished by Posner et al (1982), that is, the dissatisfac-
tion of the student with the currently held concept
and the intelligibility, plausibility, and fruitfulness of
the new concept from the student's point view. These
classroom strategies include use of cooperative groups,
refutational texts, analogies, and stepwise models. How-
ever, Piquette and Heikkinen (2005) concluded from
self-reported strategies of teachers to address and re-
mediate students' alternative conceptions during their
lessons in the classroom that these strategies rarely in-
cluded all four necessary conditions of (Posner et al
1982). Bilgin (2006) compared the e®ectiveness of small
group discussions and traditionally designed chemistry
instruction, and he concluded that the Turkish teacher
students in his study gained more from group discus-
sion compared to traditional, teacher-driven instruc-
tion. Canpolat et al (2006) also reported about the suc-
cess of conceptual change approach to chemical equi-
librium compared to a traditional teaching approach
for Turkish undergraduate chemistry students enrolled
in an introductory chemistry course. Locaylocay et al
(2005) reported that the use of small group discussions,
predict-observe-explain (POE) activities and other ex-
periments, simulations, and explanations through analo-
gies and metaphors, as suggested by (van Driel et al
1998, 1999) for upper secondary chemistry education
in The Netherlands, contributed positively to the con-
ceptual development on chemical equilibrium of their
17-18 years old Philippine bachelor students in chem-
istry and engineering taking a second course in general
chemistry. Qu¶³lez (2004b) proposed a historical recon-
struction in parallel with an experimental approach as
an appropriate sequence of learning in the introduc-
tion and development of chemical equilibrium. Maia
and Justi (2009) recently proposed and did empirical
research on a modeling-based approach to this topic
in which understanding of chemical equilibrium as a
process is emphasized and secondary school students
engage in making explanatory models of chemical phe-
nomena.
All previously mentioned instructional strategies try
to minimize alternative conceptions, to overcome con-
ceptual di±culties, and to facilitate conceptual change
through the creation of an authentic learning environ-
ment that promotes active engagement of students and
values `learning how' rather than `learning what'. The
qualitative nature of these approaches is prevalent. This
does not mean that there is in an introduction of chem-
ical equilibrium and chemical kinetics no role for study-
ing quantitative aspects. Hackling and Garnett (1986)
suggested that greater emphasis on the quantitative as-
pects of equilibrium through a variety of well-chosen
examples may help students gain a clearer picture of
the relationship between the concentrations of reac-
tants and products in equilibrium. Concentration-time
graphs may help students to visualize what is happen-
ing when a change is made to a system at equilibrium.
All this becomes more relevant to students when mod-
eling results can be compared with experimental data,
preferably from experiments carried out by the students
themselves in the laboratory, but otherwise from given
data sets.
The concluding words of van Driel et al (1999) in
their paper about introducing dynamic equilibrium as
an explanatory model through a conceptual change ap-
proach, which treats the topic at both macroscopic and
corpuscular level, includes assignments that challenge
students' existing alternative conceptions, and stimu-
lates active student engagement through small-group
discussions and hands-on experiments, are in the same
line of thought. The authors wrote in their discussion
to anticipate students' di±culties with the idea that
chemical reactions take place simultaneously in a state
of equilibrium the following sentences (emphasis by the
author):
\To anticipate such di±culties, teachers may
engage students in a discussion, carefully ad-
dressing students' speci¯c conceptions with re-
spect to (i) the possibility that chemical reac-
tions may occur even though this is not indicated
by observable changes; (ii) the idea that two op-
posite reactions may take place at the same time
and at equal rates; and, eventually, (iii) the no-
tion that di®erent particles of the same species
may engage in di®erent processes at the same
time. Additionally, simulations or computer an-
imations may be used to visualize the dynamic
nature of chemical equilibrium. Preferably, the
relation of these simulations or animations with
the chemical experiments the students have per-
formed is discussed explicitly."
A few examples in this paper will illustrate this point
of view about the use of computer models as a comple-
ment to an introduction of kinetic ideas about chemical
reactions in which the focus is on basic understanding of
the concept of reaction rate, at both a macroscopic and
5
a corpuscular level, as outlined by van Driel (2002). In
short, the focus of this paper is to contribute to the real-
ization of simulations of chemical kinetics that can eas-
ily be understood, used and created by students and/or
teachers. It is noted that empirical studies are needed
to investigate its impact, certainly because it is notori-
ously di±cult to counteract alternative conceptions of
students.
3 Studying Chemical Kinetics with Graphical
System Dynamics Modeling Tools
The methodology, strengths, and weaknesses of the im-
plementation of classical graphical system dynamics
modeling software for modeling chemical kinetics is dis-
cussed in the next three subsections. The following chem-
ical reactions and reaction mechanism are the main ex-
amples:
{ A unimolecular chemical equilibrium system;
{ A termolecular chemical reaction;
{ The Michaelis-Menten reaction mechanism.
The last two examples are also used in Section 4 to
exemplify the new approach.
3.1 Basics of Graphical System Dynamics Modeling
System dynamics modeling environments like STELLA
and Coach 6 are examples of so-called aggregate-focused
modeling tools that allow students to construct exe-
cutable models of dynamics systems. Such tools use
aggregated amounts, i.e., quantities (commonly called
levels or stocks) that change over time through physi-
cal in°ows and out°ows, as the core components of a
speci¯c system. Not only °ow of material, but also in-
formation °ow determines the system's behavior over
time. Information °ow is best understood as an indi-
cation of dependencies or in°uences between variables
in the model. These relations are made explicit in the
form of mathematical formulas and graphical or tabu-
lar relationships. The variables involved can be levels,
°ows, parameters, and auxiliary variables.
The level-°ow modeling language has a graphical
representation in which a user can express his or her
thoughts about the behavior of a dynamic system and
these ideas are then translated into more formal math-
ematical representations. An example of a graphical
model, implemented in the modeling tool of Coach 6,
is depicted in Fig. 1. It represents the chemical kinetics
of the isomerization
cis-Mo(CO)4[P(n-Bu)3]2
k1f
¡)*¡
k1r
trans-Mo(CO)4[P(n-Bu)3]2:
Four types of variables are present in this graphical
model and they are di®erently iconi¯ed:
1. A parameter (temperature T);
2. Auxiliary variables (reaction rate constants k1f , k1r);
3. Levels (concentrations [cis], [trans]);
4. Flows (rates of change of concentrations r1f , k1r).
Information arrows indicate dependencies between these
variables: For example, the arrows from T to k1f and
k1r indicate that the author of the model wanted to
express that the forward rate constant k1f and the re-
verse rate constant k1r both depend on the tempera-
ture T at which the reaction takes place. The follow-
ing mathematical expressions have been derived from
(Bengali and Mooney 2003) and used in the simulation:
k1f = T ¢ 108:87¡5195
T ; k1r = T ¢ 108:78¡5394
T ; (1)
where temperature T has been speci¯ed here in Kelvin
(instead of ±C) and rate constants are in s¡1.
Fig. 1 Screen shot of the graphical model of cis-trans isomer-
ization, and concentration-time graphs in a simulation starting
from pure cis-Mo(CO)4[P(n-Bu)3]2 at 85±C.
The model window in the upper part of the screen-
shot in Fig. 1 illustrates what graphical modeling is
all about: an author (curriculum designer, teacher, or
student) literally `draws' variables representing phys-
ical quantities or mathematical entities and the rela-
tions between them. The graphical model can be con-
sidered as a representation at conceptual level of the
system dynamics, where physical °ows represent rates
of changes and information arrows indicate dependen-
cies between quantities. Once the sketch of the model
6
had been made, the details of a model, that is, the alge-
braic formulas needed to build up the system of equa-
tions, can be ¯lled in by clicking on the icons and be
hidden again. The general picture of the model is con-
sidered most important for understanding. In this par-
ticular example, the graphical model almost literally
presents a chemical equilibrium.
A graphical system dynamics model corresponds in
mathematical terms with a system of di®erential equa-
tions or ¯nite di®erence equations. Under the assump-
tion that only elementary, unimolecular reaction steps
are involved, the graphical model of Fig. 1 represents
the following coupled di®erential equations for the rate
of change in the concentrations of the three species in-
volved:
d [cis]
dt
= ¡r1f + r1r;
d [trans]
dt
= r1f ¡ r1r ; (2)
where r1f = k1f ¢ [cis] ; r1r = k1r ¢ [trans]. But the graph-
ical model represents in fact more: it also represents an
automatically generated computer program that solves
this system numerically and allows the user to simu-
late the behavior of the modeled reaction system and
to interpret the modeling results.
3.2 Strengths of Classical Graphical Modeling Tools
Research indicates the following:
{ Despite the apparent di±culties of computer mod-
eling in an inquiry approach, students can over-
come the problems in a modeling task (Jacobson
and Wilensky 2006; Sins 2006; Stratford et al 1998);
{ A graphical modeling tool supports novice model-
ers better in constructing their own models and in
understanding other people's models than model-
ing tools that require their users to work with tex-
tual representations in which they have to explicitly
write down a sort of equation or a piece of program-
ming code (LÄohner 2005).
{ \Creating dynamic models has great potential for
the use in classrooms to engage students in thought
about science content, particularly in those thinking
strategies best fostered by dynamic modeling: anal-
ysis, relational reasoning, synthesis, testing and de-
bugging, and making explanations." (Stratford et al
1998, p. 229) Through modeling, students can ac-
quire model-based scienti¯c reasoning skills (Mil-
rad et al 2003) and learn about the speci¯c domain
(Ergazaki et al 2005; Schecker 1998);
{ Students can get more insight in the behavior of
a dynamic system by running an executable model
(Wilensky and Resnick 1999; Westra 2008);
{ Valid, feasible, and e®ective learning and teaching
strategies about dynamic behavior using modeling
and systems thinking in authentic practices can be
realized (Spector 2000; Westra 2008).
In the context of chemical kinetics, students are imme-
diately confronted in a simulation of a reaction system
with potential alternative conceptions. In the ¯rst ex-
ample of cis-trans isomerization, a student can for in-
stance observe in the graphs of the lower part of Fig. 1
that (i) it takes time before the equilibrium is reached;
and (ii) at equilibrium, the concentrations of cis- and
trans-complexes are not necessarily equal.
More alternative conceptions about chemical equi-
librium, which were listed in Subsection 2.1, can be ad-
dressed when one looks at rate-time and net rate-time
graphs of the chemical equilibrium shown in Fig. 2.
Fig. 2 Rate-time and net rate-time graphs for cis-trans isomer-
ization at 85 ±C.
Some of the points that a student could notice in
the graphs displayed in Fig. 2 are:
{ The rate of the forward reaction decreases with time
until equilibrium is reached (and not to completion);
{ The rate of the reverse reaction increases with time
until equilibrium is reached;
{ The forward and reverse reaction rates are not al-
ways the same;
{ The forward and reverse reaction start at the same
time;
7
{ A system in equilibrium does not mean that the
reactions ceased;
{ A system in equilibrium means that the net rate
of concentrations is zero and that the forward- and
reverse-reaction rates are equal.
Executable models o®er students the opportunity to
observe the e®ects of changing the model or, less dra-
matically, of changing the parameter values and initial
conditions. In fact, this has already been anticipated in
the model shown in Fig. 1: The introduction of tem-
perature T, the change of which has been simpli¯ed by
the incorporation of a corresponding slider in the activ-
ity, is motivated by the wish to investigate the e®ect of
temperature on the reaction system. Fig. 3 shows the
results of a simulation of the isomerization at a lower
temperature, namely at 80 ±C. The graphs of the previ-
ous simulation at a temperature of 85 ±C are shown in
gray at the background to support easy comparison.
Fig. 3 concentration-time, rate-time, and net rate-time graphs
for cis-trans isomerization at 80 ±C.
A student could discover from the graphs in Fig. 3
that changing the temperature
{ does not necessarily mean that the concentrations
at equilibrium are a®ected. In other words, changing
the temperature does not mean that the equilibrium
constant is a®ected.
{ may change the magnitudes of the reaction rate con-
stants without changing their ratio. In such case it
only a®ects the time needed for the system to reach
equilibrium;
{ may change the absolute magnitudes of the forward
and reverse rates, also at equilibrium.
It must be emphasized that these conclusions only hold
because the isomerization is almost thermoneutral. It is
actually a misconception to believe that the equilibrium
constant is independent of temperature. When a chem-
ical equilibrium is chosen in which activation energies
of the forward and backward reactions di®er substan-
tially, a change in temperature will lead to a noticeable
shift of the equilibrium.
By changing the initial concentrations of the cis-
and trans-complex it can easily be veri¯ed that the sys-
tem will always reach the same equilibrium concentra-
tions, no matter what the starting concentrations are.
By playing with the forward and reverse reaction rate
constants, a student could discover that while the ab-
solute magnitudes of the forward and reverse rate con-
stants do not control the ¯nal equilibrium, equilibrium
concentrations are controlled by the ratio of the rate
constants.
3.3 Weaknesses of Classical Graphical Modeling Tools
The graphical modeling of chemical kinetics illustrated
by the example of cis-trans isomerization is rather sim-
ple. Other examples of reaction systems that can be
dealt with in this way are unimolecular. Any other type
of reaction system would lead for stoichiometric reasons
to a disconnected, from chemical point of view incom-
prehensible graphical model. The basic example used
in the paper to illustrate this is the gas-phase oxida-
tion of nitric oxide: 2NO + O2 ! 2NO2. This example
of a third-order rate reaction system has been chosen
because it is a classical illustration of the fact that re-
action rate data alone are not su±cient to determine
the underlying reaction mechanism. The following three
mechanisms have been identi¯ed by Tsukuhara et al
(1999), which all lead to third-order reaction kinetics:
{ A termolecular reaction, i.e., two molecules of NO
and one O2 collide and form a transient complex,
which in a single step forms two molecules of NO2;
8
{ A pre-equilibrium mechanism with dimer of NO as
an intermediate;
{ A pre-equilibrium mechanism with NO3 as an inter-
mediate.
The graphical model in Fig. 4 represents the ter-
molecular reaction mechanism through the following
coupled di®erential equations for the rate of change in
the concentrations of the three species involved:
d[NO]
dt
= ¡2r;
d [O2]
dt
= ¡r;
d [NO2]
dt
= 2r ; (3)
where r = k ¢ [NO]2 ¢ [O2] and the rate constant is given
by the Arrhenius-type equation k = 1:2£103£10230=T .
It also gives information about the units used.
Fig. 4 Screen shot of the graphical model and simulation of the
termolecular reaction 2NO + O2
k ¡!
2
N
O
2
.
Although the °ow arrows, which represent the rate
of change of concentrations, have been drawn in the
graphical model such that the reader is given the im-
pression of a chemical reaction in the form of a chemi-
cal network or a metabolic network, all of a sudden the
icons that represent concentrations have become dis-
connected. The reason that one cannot directly draw
physical °ow arrows from reactants toward products is
that the meaning of the graphical modeling tool, which
is based on the level-°ow model in which the sum of in-
°ows in a level variable is by de¯nition equal to the sum
of out°ows of this level variable (the so-called `principle
of °ow balance'), does not lead to the correct coupled
di®erential equations. In other words, if both an arrow
from [NO] toward [NO2] and an arrow from [O2] toward
[NO2] were drawn, this would mean that the increase in
concentration of NO2 over time is equal to the sum of
the decrease in concentration of NO over time and the
decrease in concentration of O2 over time. This is from
chemical point of view incorrect for the given reaction.
Moreover, the implication that both °ows can be inde-
pendently regulated is not true in chemical kinetics.
In fact, due to the selected graphical modeling ap-
proach of level-°ow diagrams, which is based on a meta-
phor of water tanks and valves, the diagrams for bi- and
termolecular chemical reactions are inevitably discon-
nected. Forrester, the founder of the system dynam-
ics and level-°ow modeling approach in the context of
socio-economic systems, was aware of this limitation
and wrote (Forrester 1961, p. 70):
\It should be noted that °ow rates transport
the content of one level to another. Therefore,
the levels within one network must all have the
same kind of content. In°ows and out°ows con-
necting to a level must transport the same kind
of items that are stored in the level. Items of
one type must not °ow into levels that store an-
other type. For example, the network of mate-
rials deals only with material and accounts for
the transport of the material from one inventory
to another. Items of one type must not °ow into
levels that store another type."
Clearly, chemical reactions do not meet this `princi-
ple of material consistency' in the structure of a graph-
ical model that is written in terms of levels intercon-
nected by rates of °ow: In a bimolecular reaction, two
molecules may react to result in one molecule, that's
chemistry! On the other hand, it must be stressed that
the problem only lies in the translation of the graph-
ical model into the coupled di®erential equations that
describe the kinetics of the chemical reaction.
The fact that the conventions of a classical graph-
ical system dynamics modeling tool, which state how
the coupled di®erential equations or di®erence equa-
tions are to be generated from the graphical representa-
tion, are inconvenient for chemical kinetics comes even
more to the fore when complex chemical reaction net-
works are modeled instead of elementary reactions. The
following example, which is the simplest (`Michaelis-
Menten' and `Briggs-Haldane') mechanism for a two-
step enzyme-catalyzed reaction, will illustrate this:
E + S
k1f
¡)*¡
k1r
ES k2f ¡¡! E + P
9
where E, S, ES, and P are the unbound enzyme, sub-
strate, intermediate enzyme-substrate, and product,
respectively. One of the things students learn from or
need to accept in this mechanism is that a species can si-
multaneously be involved in more than one reaction: the
intermediate enzyme-substrate can both form a product
as well as the original substrate. All reaction steps are
considered as elementary reactions. See (Bruist 1998)
and (Halkides and Herman 2007) for simulations of the
reaction system with a spreadsheet, and (Mulquiney
and Kuchel 2003) for simulations with a computer al-
gebra system. A steady-state approximation is used in
most cases to simplify the algebraic and computational
work. The graphical model that represents this enzyme-
catalyzed reaction without using this approximation is
shown in Fig. 5.
Fig. 5 Screen shot of the graphical model and simulation of the
E + S
k1f
¡)*¡
k1r
ES
k2f ¡¡! E + P reaction system.
The graphical model represents the following cou-
pled di®erential equations for the rate of change in the
concentrations of the four species involved and it also
gives the reader information about the units used for
concentration and rate of change of concentration:
d[S]
dt
= ¡r1f + r1r;
d [E]
dt
= ¡r1f + r1r + r2f ;
d[ES]
dt
= r1f ¡ r1r ¡ r2f ;
d[P]
dt
= r2f ; (4)
where r1f = k1f ¢[E]¢[S] ; r1r = k1r ¢[ES] ; r2f = k2f ¢[ES].
Values of kinetic parameters have been taken from
(Halkides and Herman 2007), which presents in fact a
toy model of enzyme kinetics. Its unrealistic character
is taken for granted because the graphical model of the
reaction mechanism can still be used to convince stu-
dents that the steady-state approximation makes sense.
For example, using the computed time course of the
enzyme-catalyzed reaction one can apply the regression
tool in Coach 6 to ¯nd values of the Michaelis-Menten
constant Km and of the maximum velocity Vmax by
nonlinear regression and one can verify that these values
are close to the theoretical values Km = (k1r + k2f)/k1f
and Vmax = k1f ¢ [E]jt=0. Values of kinetic parameters
found by nonlinear regression can also be compared
with values obtained from a Lineweaver-Burk plot. Fur-
thermore, the simulation tool can be used to discuss the
validity of the steady-state approximation in the kinetic
model.
This example makes clear that a standard, rather
simple chemical reaction network already leads to a
disconnected graphical model in which the chemical re-
action mechanism is obscured by the spaghetti (and
meatballs) tangle of arrows and boxes. When the reac-
tion mechanism of the enzyme-catalyzed reaction be-
comes more complicated, the corresponding graphical
model that represents the chemical kinetics readily gets
snarled up, to put it mildly. This happens, for exam-
ple, when the urea cycle (Mulquiney and Kuchel 2003)
is studied, a reaction network in which more than one
substrate is available for the enzyme, more than one
product is formed, and more than one enzyme may be
involved.
In summary, the following weaknesses have been
identi¯ed and exempli¯ed in using the classical graphi-
cal system dynamics modeling and simulation approach
to chemical kinetics:
² Except for simple unimolecular reaction systems,
the graphical models based on the traditional level-
°ow metaphor do not present a clear overview of the
chemical reaction mechanism, but instead they have
often an incomprehensible spaghetti (and meatballs)
tangle of arrows and boxes. Especially the number
of information arrows can be overwhelming.
10
² In most graphical models of chemical reaction sys-
tems levels represent concentrations of chemical spe-
cies and °ows represent rates of change of the species.
Because the principle of °ow balance holds in the
level-°ow metaphor, this means that graphical mod-
els of chemical reactions must be predominantly mod-
els in which levels are disconnected. Such a graph-
ical model does not give any indication anymore of
which species are reactants and which species are
products of chemical reactions or reaction steps. The
reaction mechanism is not clearly revealed in the
graphical model.
² Although many graphical modeling tools o®er user
interface elements such as knobs and sliders to set
parameter values and initial conditions, not all of
them allow their users to change values during a
simulation run. Thus, many modeling tools do not
o®er much to investigate external e®ects on the ki-
netics of a chemical reaction system such as addition
of extra reactants, depletion of products, and so on,
in an exploratory approach.
These di±culties in graphical modeling of chemical ki-
netics with level-°ow based system dynamics modeling
and simulation software are known and suggestions for
improvement have been made. For example, the key
ideas of chemical kinetics and thermodynamics have
been expressed in a bond graph approach (Cellier 1991,
ch. 9) and the level-°ow metaphor has been replaced in
(Elhamdi 2005; LeFµevre 2002, 2004) by the so-called ki-
netic process metaphor, which was inspired by graphical
models of biochemical reaction networks and metabolic
pathway systems. But these alternatives for and ex-
tensions of the traditional level-°ow metaphor are at
the level of system dynamics specialists and they are
too complicated for use in chemistry education at high
school level or ¯rst-year undergraduate level. In the
next section, a much simpler graphical approach to mod-
eling of chemical reactions is presented that covers the
basics of chemical kinetics.
4 Improved Graphical Modeling of Chemical
Reaction Systems
A solution to most of the previously identi¯ed problems
with classical graphical system dynamics modeling is
presented in the form of an improved approach of chem-
ical reactions based on a graph theoretic description of
reaction kinetics. To this end, a new icon, viz. the Er-
lenmeyer °ask symbol, is added to the graphical mod-
eling tool. After a formal underpinning of the proposed
extension, examples will illustrate the new approach to
chemical kinetics. In Section 5, other applications of the
Erlenmeyer icon will be presented and this motivates
the more general reference to this icon as a `process' in-
stead of a more chemistry related name like `reaction.'
In Subsection 4.3, the usefulness of adding interactivity
elements such as sliders, buttons, and event controls to
the graphical model tool will be illustrated.
4.1 Adding a Process Element to the Modeling Tool
The improved graphical modeling approach of chem-
ical reactions is based on a graph theoretic descrip-
tion of reaction kinetics that is similar to the oriented
species-reaction graph introduced in (Craciun and Fein-
berg 2006) and the directed bipartite graph of a reac-
tion network developed by Vol'pert and Ivanova (1987)
[ see also (Vol'pert and Hudjaev 1985, ch. 12) ], more
thoroughly analyzed in (Ermakov and Goldstein 2002;
Mincheva and Roussel 2007), and for example imple-
mented in a computer simulation and visualization en-
vironment for metabolic engineering (Qeli 2007). The
graphical approach will be exempli¯ed by the termolec-
ular gas-phase oxidation of nitric oxide, which was also
discussed in Subsection 3.3, 2NO + O2
k¡
! 2NO2, with
the associated system of di®erential equations:
d[NO]
dt
= ¡2r;
d [O2]
dt
= ¡r;
d [NO2]
dt
= 2r ; (5)
where for a given reaction rate constant k holds
r = k ¢ [NO]2 ¢ [O2] : (6)
For a thorough description of the improved graphi-
cal approach it is wise to linger over chemical notation
of reactions and reaction networks and its meaning. A
chemical reaction network is formally de¯ned (Feinberg
1979) as a triple (S, C, R) that consists of a ¯nite set of
chemical species (reactants and products of the reaction
steps) S, a ¯nite set of complexes (the objects before
and after the reaction arrows) C, and a set of reactions
R ½ C £ C with the properties that (y; y) =2 R for any
y 2 C and that for each y 2 C there exists a y0 2 C
such that (y; y0) 2 R or such that (y0; y) 2 R. In plain
words this means that each species must appear on the
left- or right-hand side of at least one reaction step,
that there are no super°uous species, that no complex
reacts to itself, and that no complex is isolated. Hence-
forth the more suggestive y ! y0 is used in place of
(y; y0) when (y; y0) 2 R and equilibrium reactions are
considered as two separate irreversible reaction steps.
In case of the reaction 2NO + O2
k¡
! 2NO2 the sets are
equal to S = fNO;O2;NO2g, C = f2NO + O2;2NO2g,
and R = f2NO + O2
k¡
! 2NO2gg. The stoichiometric
coe±cient of a species s in a complex y is the positive
11
integer in front of the species if it is contained in the
complex and it is equal to zero otherwise. For example,
in the complex 2NO+O2 the stoichiometric coe±cients
of NO, O2, and NO2 are 2, 1, and 2, respectively. The
kinetic graph of the chemical reaction network (S, C,
R) is a directed graph in which the set of vertices is
partitioned into two sets, namely, a set of species nodes
and a set of reaction nodes. There is one species node
for each species in the network and one reaction node
for each (irreversible) reaction in the network. Each di-
rected edge of the kinetic graph joins a species node to
a reaction node or a reaction node to a species node (so
the kinetic graph is a directed bipartite graph) accord-
ing to the following prescription: Consider some reac-
tion y ! y0 in R. There is one directed edge toward
the reaction node y ! y0 coming from each node of a
species present in the complex y of the reaction. There
is one directed edge from the reaction y ! y0 toward
each node of a species present in the complex y0. Oth-
erwise stated, arrows are drawn for each reaction in the
network from the reactants toward the reaction node
and from the reaction node toward the products cre-
ated in the reaction. The kinetic graph of the reaction
2NO + O2
k¡
! 2NO2 is shown in Fig. 6.
Fig. 6 The kinetic graph of the reaction 2NO + O2
k ¡!
2
N
O
2
.
Here, a species node is represented by a square and a
reaction node is suggestively represented by an Erlen-
meyer °ask symbol. The directed bipartite graph is
called a kinetic graph because it also incorporates by
de¯nition information about the kinetics of the chem-
ical reaction. This information is about (contributions
to) rates of change of species involved in the given re-
action, based on the stoichiometric coe±cients associ-
ated with the reaction: In this particular example, For-
mula (6) holds. In general, the elementary jth reaction
in a reaction network
®j1R1 + ¢ ¢ ¢ + ®jmRm
kj ¡! ¯j1P1 + ¢ ¢ ¢ + ¯jnPn
has a reaction rate
rj = kj ¢ [R1]®j1 ¢ ¢ ¢ ¢ ¢ [Rm]®jm ; (7)
where kj is the kinetic coe±cient and [Ri] is the concen-
tration of reactant Rj . Normally
P
m ®jm, which is the
number of reactants involved in the jth reaction step,
is a natural number less than or equal to 3. The time
course of the concentrations depends on the reaction
rates rj : For example, the dynamics of [Ri] and [Pi] in
the above reaction depends on rate rj in the form
d
dt
[Ri] = ¡®jirj ;
d
dt
[Pi] = ¯jirj : (8)
In the kinetic graph, there exists a directed edge from
the species node Rk toward the jth reaction node if
®jk > 0 and similarly a directed edge from the jth
reaction node toward the species node Pk if ¯jk > 0.
Note that this formalism does not exclude the situation
that reactants and products involve the same species
(for example, in auto-catalytic reactions).
The kinetic graph of a chemical reaction network
clearly suggests how the classical level-°ow formalism of
graphical system dynamics modeling tools could be ex-
tended to function well for chemical reaction networks:
A graphical icon for a reaction, say an Erlenmeyer °ask
symbol, must be added to the formalism and then lev-
els can represent concentrations of species involved in
the reaction network, provided that °ows are between
level icons and Erlenmeyer symbols. In°ows of an Erlen-
meyer symbol originate from reactants and out°ows of
an Erlenmeyer symbol point at products in the chemi-
cal reaction that is symbolized by the Erlenmeyer °ask.
The Erlenmeyer symbol also represents the dynamics
of the levels connected with it via the stoichiometry
of the reaction: The Erlenmeyer symbol is linked to a
formula for the reaction rate, which depends on the
kinetic coe±cient, the concentrations of reactants and
their stoichiometric coe±cients, and the stoichiometric
coe±cients determine the formulas for the in°ows and
out°ows of the Erlenmeyer symbol.
The improved graphical modeling of chemical re-
action, based on kinetic graphs, leads to much clearer
visual representations of chemical reaction networks for
the following reasons:
² Levels, °ows, and process elements give a visual
overview of the reaction mechanism;
² The stoichiometry of a reaction already determines
the formulas for the in°ows and out°ows so that
there is no need to use information arrows from the
reaction node toward these °ows.
The examples in the next subsection will illustrate that
the new graphical models resemble more the pictures
that chemists already draw for ages to illustrate reac-
tion mechanisms. This is also the reason that students
do not need to be introduced the kinetic graphs in the
same formal way as was done in this subsection to un-
derpin the proposed approach; a more informal intro-
duction su±ces to work with it in a sensible way.
12
4.2 Some Illustrative Examples
The improved graphical modeling approach has been
implemented in Coach 6 and the ¯rst example in this
subsection is the same as the last example discussed in
Subsection 3.3, namely, the two-step enzyme-catalyzed
reaction E+S
k1f
¡)*¡
k1r
ES k2f ¡¡! E+P. This o®ers the reader
the opportunity to compare the graphical model pre-
sented in Subsection 3.3 (Fig. 5) with the model based
on the improved formalism (Fig. 7). The reader can
also compare the graphical model of enzyme kinetics in
Fig. 7 with the equivalent graphical model in Fig. 8,
taken from (Lee and Yang 2008), that has been imple-
mented in Powersim (www.powersim.no), another clas-
sical graphical modeling tool, and that does not re°ect
anymore the underlying reaction mechanism.
Fig. 7 Screen shot from the improved graphical model of the
E + S
k1f
¡)*¡
k1r
ES
k2f ¡¡! E + P network.
Fig. 8 A model of E + S
k1f
¡)*¡
k1r
ES
k2f ¡¡! E + P in Powersim.
Because one of the goals was to make graphical sys-
tem dynamics modeling of chemical kinetics viable in
cases of more complicated reaction mechanisms, a sec-
ond example is shown in Fig. 9, which would not be as
comprehensible in a classical system dynamics graphi-
cal approach. It is the following enzyme-catalyzed reac-
tion mechanism, taken from (Kuchel et al 1977; Maher
et al 2003; Mulquiney and Kuchel 2003), for the hydrol-
ysis of arginine to ornithine and urea catalyzed by the
hydrolytic enzyme arginase, which is only one step of
the urea cycle:
E + A
k1f
¡)*¡
k1r
EA k2f ¡¡! EO + U; EO
k3f
¡)*¡
k3r
E + O
where A, U, and O denote arginine, urea, and ornithine,
respectively. The graphical model represents the follow-
ing coupled di®erential equations for the rate of change
in the concentrations of the species involved:
d[A]
dt
= ¡r1f + r1r;
d[E]
dt
= ¡r1f + r1r ¡ r3r + r3f ;
d[EA]
dt
= r1f ¡ r1r ¡ r2f ;
d[EO]
dt
= r2f ¡ r3f + r3r;
d[U]
dt
= r2f ;
d[O]
dt
= r3f ¡ r3r; (9)
where
r1f = k1f ¢ [E] ¢ [A] ; r1r = k1r ¢ [EA] ; r2f = k2f ¢ [EA] ;
r3f = k3f ¢ [EO] ; r3r = k3r ¢ [E] ¢ [O] :
Values of kinetic parameters can be taken from (Kuchel
et al 1977). The graphical model, the construction of
which is envisioned to be doable by upper secondary
school chemistry students or ¯rst-year undergraduate
chemistry students, is very informative about the reac-
tion mechanism. Thus, chemical kinetics of more real-
istic reaction mechanisms is not expected to be beyond
the level of students anymore.
Fig. 9 Screen shot of the graphical model of E+A
k1f
¡)*¡
k1r
EA
k2f ¡¡!
EO + U, EO
k3f
¡)*¡
k3r
E + O network.
4.3 Interactivity in Chemical Kinetics Modeling
Le Ch^atelier's Principle is often used in textbooks to
explain how a system in equilibrium responds to an
external perturbation such as addition of a reactant,
depletion of a product, change in pressure or tempera-
ture, and so on. Many research studies (Canpolat et al
2006; Cheung et al 2009; Qu¶³lez 2004a; Qu¶³lez-Pardo
and Solaz-Portol¶es 1995; Tyson et al 1999; Voska and
13
Heikkinen 2000) reported that teachers and students
have di±culties in applying this principle appropriately
and accurately. A common mistake is to reason that in-
creasing the concentration of one of the reactants will
result in an increase of the forward rate and a de-
crease of the reverse rate, because the forward reac-
tion is favored over the reverse one. Such misinterpre-
tations and misapplications of Le Ch^atelier's Principle
have brought Cheung et al (2009) and others (Allsop
and George 1984; Qu¶³lez 2004a) to question the ap-
propriateness of this principle in chemistry education
for predicting the direction in which a chemical equi-
librium will shift when it is disturbed. In a qualita-
tive or semi-quantitative approach to chemical equilib-
rium phenomena there is hardly any other instructional
strategy than applying Le Ch^atelier's Principle or rea-
soning with the Equilibrium Law. But then one better
resists the temptation to combine this thermodynamic
approach to chemical equilibrium, which does not make
statements about forward or backward reactions, with
a kinetic approach to chemical equilibrium based on
the `law of mass action.' A quantitative approach seems
more suitable for discussing how chemical equilibrium is
reached or how it changes when conditions change. This
holds especially when a modeling and simulation envi-
ronment o®ers tools to interactively change conditions
during a simulation and/or allows an easy implemen-
tation of event-handling such as response to a sudden
change in concentration, temperature, and so on. Fur-
thermore, Solomonidou and Stavridou (2001) pointed
at the potential of computer simulations and anima-
tions to help students construct appropriate concep-
tions about Le Ch^atelier's Principle and the equilibrium
constant law.
Interactive change of initial conditions as well as
event-handling of sudden changes during a simulation
run have been implemented in the Coach 6 environ-
ment and are exempli¯ed with the equilibrium shift of
a gas mixture of hydrogen, iodine, and hydrogen as a
response to a sudden change in hydrogen concentration
and temperature. The reaction system under considera-
tion is H2+I2
k1f
¡)*¡
k1r
2HI, where second-order rate kinetics
is assumed given by the following coupled di®erential
equations for the rate of change in the concentrations
of the three species involved:
d [H2]
dt
= ¡r1f + r1r;
d [I2]
dt
= ¡r1f + r1r;
d [HI]
dt
= r1f ¡ r1r ; (10)
where rf = kf ¢ [H2] ¢ [I2] ; rr = kr ¢ [HI]2, and the
Arrhenius equations for the rate constants are given
(Graven 1956) for temperature T (in ±K) by kr =
7:18£1012£e¡24775=T and kf = 1:23£1012£e¡20646=T .
It follows from these equations that the forward gas
phase reaction is exothermic. Figure 10 is a screenshot
of a simulation run based on this kinetic model.
Fig. 10 Screen shot of the graphical model of the H2+I2
k1f
¡)*¡
k1r
2HI
equilibrium reaction and a simulation with user interaction and
an event during execution of the model.
Figure 10 shows a simulation run starting with only
a nonzero concentration of HI at a temperature T =
721±K. After 6000 seconds the concentration of H2 is
suddenly raised by 0.002 M, which has an immedi-
ate e®ect on the concentration time course. This sud-
den change is realized in the graphical model by in-
troduction of an event (iconized by the thunderbolt
symbol). The code behind this event icon is very sim-
ple: Once t>6000 then [H2] := [H2] + 0.002. The
e®ect is that the equilibrium which was almost estab-
lished is shifted right to less dissociation of hydrogen
iodine. After a new equilibrium has been established es-
tablished equilibrium, the user has pressed about 12000
seconds after the start of the reaction the button in the
control panel to cause a sudden raise in temperature of
50±K. The e®ect is that the equilibrium shifts to the
left, that is, more hydrogen iodide dissociates again.
This is in agreement with the Le Ch^atelier Principle
that states that increasing the temperature will shift
the equilibrium to the left because the forward reac-
tion is exothermic. Although the kinetic and thermo-
dynamic approaches to chemical equilibrium phenom-
ena are of di®erent nature, results obtained by either
method complement each other.
14
5 Other Applications
Graphical modeling and simulation tools have other ap-
plications in chemistry education, for example in mod-
eling and simulating acid-base titration curves (Heck
et al 2010), and in other science ¯elds [see for example
(van den Berg et al 2008)]. Although no attention has
been paid to it in this article so far, it is worth men-
tioning that the improved graphical modeling approach
also has applications beyond chemical kinetics. This as-
pect is important in education because it would most
probably not be worth the e®ort to add new elements
to a general purpose graphical modeling tool if they
were only relevant for a small part of the science cur-
riculum. Students and teachers have to use their time
e®ectively and economically. Much is won when stu-
dents and teachers can use one and the same modeling
environment for many science subjects. Then they have
ample opportunities to grow into their roles of knowl-
edgeable and skilled modelers of natural phenomena.
We discuss two examples of usage of the new graph-
ical formalism that are conceptually rather close to the
chemical context of this paper. But one must realize
that examples of completely di®erent nature, such as
for instance the modeling of the height of beer foam
(Heck 2010), could be presented as well.
The ¯rst example illustrates that quantitative phar-
macokinetic models can be conveniently treated through
the new graphical formalism. Fig. 11 shows a graphi-
cal model and simulation of the pharmacokinetics of
the metabolism of ecstasy in the human body [taken
from high school lesson materials \Swilling, Shooting,
and Swallowing," see also (Heck 2007)]: The improved
graphical modeling approach provides a connected di-
agram that indicates the °ow of the pharmacon in the
body over time.
The second example is the classical SIR (susceptible-
infected-recovered) epidemic model, also known as the
Kermack and McKendrick (1927) model:
S0 = ¡¯IS; I0 = ¯IS ¡ ®I; R0 = ®I : (11)
The origin of the above system of di®erential equations
can be a description of an epidemic though a compart-
mental model. Alternatively, the SIR model can be con-
sidered, like a chemical reaction network, as a two-step
process with the following mechanism:
S + I ¯ ¡! 2I; I ® ¡! R
The ¯rst step in the process is linked with contact be-
tween healthy and infected persons, which leads to two
infected persons. When an infected person has on av-
erage ¯ contacts per day and infected persons are on
Fig. 11 Screen shot of a graphical model of pharmacokinetics of
ecstasy in the human body and a simulation run with real data
in the background.
average 1=® days ill, then the above system of di®eren-
tial equations follows from probabilistic considerations.
Fig. 12 shows a graphical model based on a process
network (the non-default stoichiometric coe±cient has
been added in the graphical model as an annotation).
Fig. 12 Screen shot of a process network based graphical model
of the Kermack-McKendrick model of epidemics.
6 Conclusion
A central aspect of inquiry learning is that the learners
must develop their own models and in particular their
own executable computer models of real world phe-
nomena. Classroom experience and case studies indi-
cate that this is possible at secondary school level when
graphical system dynamics based software is used. Sub-
sections 3.1 and 3.2 showed the potentiality of graphi-
cal system dynamics based modeling environments like
STELLA in the context of chemical equilibrium and
chemical kinetics. However, as was illustrated in Sub-
section 3.3, level-°ow based modeling tools are of lim-
ited use in studying chemical kinetics when bi- or tri-
molecular reactions or chemical reaction networks come
into play. It is tempting to associate in these graphical
15
models levels with concentrations of species in a reac-
tion and physical °ows with chemical reaction arrows,
representing at the same time the kinetics of the re-
actions. But stoichiometry and plurimolecular reaction
types are a spoil-sport. The graphical representation of
a chemical reaction network in the form of a kinetic
graph, as it has been introduced in Subsection 4.1, is
more suitable. Only one thing is needed for this in the
graphical level-°ow formalism, namely, inclusion of a
new icon for a reaction step. Then, levels can indeed
represent concentrations of species and °ows can repre-
sent changes in concentrations of species provided that
these °ows are between levels and reaction icons (the
Erlenmeyer °ask symbols in this paper). Each reaction
icon is linked with a formula that describes the reaction
rate and the stoichiometry of the reaction determines
the formulas for the in°ows and out°ows of the reac-
tion icon. In this way the graphical model gives a clear
overview of the reaction mechanism (as was exempli¯ed
in Subsection 4.2).
Another improvement comes from the addition of
user interaction tools like sliders and button to in°uence
simulation run while they are going on and of a special
icon for discrete time event handling. This o®ers stu-
dents the opportunity to explore \what if?" questions.
An example of such an investigation of the in°uence of
external e®ects on a chemical equilibrium was given in
Subsection 4.3.
Both improvements have been exempli¯ed in this
paper by the computer implementation in Coach 6.
Furthermore, the examples in Sections 3 and 4 illus-
trated that alternative conceptions of students about
chemical equilibrium and chemical kinetics, which were
reviewed in Section 2, can be directly addressed with
the (extended) graphical modeling approach. Section 5
contained some examples to brie°y illustrate that this
improved approach has applications beyond chemical
kinetics and helps to clarify the dynamics of all kinds
of real world phenomena.
Acknowledgements The author would like to thank his col-
leagues Leendert van Gastel andWolter Kaper for the fruitful dis-
cussions on the improved graphical modeling approach and their
suggestions. Special thanks go to his colleague Martin Beugel for
implementing and beta testing the new approach in the Coach 6
computer learning environment.
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