Research: Science and Education
Detailed Balanced and Unbalanced Triangle Reactions
Ilona Nagy* (corresponding author)
Department of Mathematics, Szent István University, Gödöllő, HUNGARY, H-2103;
ilona.nagy@gek.szie.hu
Tel.: 36-28-522-041, Fax: 36-28-410-804
György Póta
Institute of Physical Chemistry, University of Debrecen, Debrecen, HUNGARY, H-4010;
potagy@delfin.unideb.hu
Tel.: 36-52-512-900, Fax: 36-52-512-915
János Tóth
Department of Mathematical Analysis, Budapest University of Technology and
Economics, Budapest, HUNGARY, H-1111; and
Reaction Kinetics Laboratory, Eötvös Loránd University, Budapest, HUNGARY; H1111; jtoth@math.bme.hu
Tel.: 36-1-463-2314, Fax: 36-1-463-3172
Abstract
A recent paper by Alberty in the present Journal underlined the importance of
detailed balance in chemical kinetics, and used Mathematica, a tool useful for teachers
and researchers of chemistry. However, there are some major points where we disagree
with the author, the source of disagreement being his neglecting some important research
papers on detailed balance and overshoot-undershoot phenomena in chemical kinetics.
JCE Keywords
Audience:
Upper-Division
Undergraduate,
Graduate
Education/Research
Domain: Physical Chemistry Pedagogy: Misconceptions/Discrepant Events Topic:
Computational Chemistry, Equilibrium, Kinetics, Mathematics/Symbolic Mathematics,
Professional Development, Reactions, Theoretical Chemistry, Area: Research: Science
and Education
The number of words in the manuscript is approx. 1700. We should like to
publish our manuscript in print, and we also have a Supplemental Material: a
Mathematica notebook to be distributed online. We think the paper might fit into the
Addressing Misconceptions Feature Column (if this is a feature, because
http://jchemed.chem.wisc.edu/AboutJCE/Features/index.php does not seem to work.)
2
Detailed Balanced and Unbalanced Triangle Reactions
Ilona Nagy* (corresponding author)
Department of Mathematics, Szent István University, Gödöllő, HUNGARY, H-2103;
ilona.nagy@gek.szie.hu
Tel.: 36-28-522-041 Fax: 36-28-410-804
György Póta
Institute of Physical Chemistry, University of Debrecen, Debrecen, HUNGARY, H-4010;
potagy@delfin.unideb.hu
Tel.: 36-52-512-900, Fax: 36-52-512-915
János Tóth
Department of Mathematical Analysis, Budapest University of Technology and
Economics, Budapest, HUNGARY, H-1111, and
Reaction Kinetics Laboratory, Eötvös Loránd University, Budapest, HUNGARY; H1111; jtoth@math.bme.hu
Tel.: 36-1-463-2324, Fax: 36-1-463-3172
It was a pleasure to read the paper by Alberty (1) for several reasons. First, it
underlined the importance of detailed balance in chemical kinetics, a notion important
both from theoretical and practical points of view. Second, we also think that the tool he
used, Mathematica is extremely useful for the chemist as it can (also) be used as a
powerful symbolic and numerical calculator without needing much programming effort.
Third, it presented an occasion for us to call the attention of the reader to recent
developments in modern formal reaction kinetics.
3
However, there are some major points where we disagree with the author.
The model he investigates is the triangle reaction describing the interactions of
three isomers, A, B and C:
Figure 1: The triangle reaction with all the possible reaction steps included
with three different sets of the reaction rate coefficients
Case I: k1=k3=k5=1 and k2=k4=k6=0;
Case II: k1=k2=k3=k4=k5=k6=1;
Case III: k1=2 and k2=k3=k4=k5= k6=1.
Transient Behavior in General
The induced kinetic differential equation system is
d[A]
 (k1  k6 )[A]  k 2 [B]  k5 [C]
dt
d[B]
 k1[A]  (k 2  k3 )[ B]  k 4 [C]
dt
d[C]
 k6 [A]  k3 [B]  (k 4  k5 )[C]
dt
(1)
which is studied under the initial conditions [A](0)  1, [B](0)  0, [C](0)  0.
Since
d[A] d[B] d[C]


 0, [A](t )  [B](t )  [C](t )  const.  1 in all the cases,
dt
dt
dt
system 1 reduces to
4
d[A]
 (k1  k 5  k 6 )[ A]  (k 2  k 5 )[ B]  k 5
dt
d[B]
 (k1  k 4 )[ A]  (k 2  k 3  k 4 )[ B]  k 4
dt
(2)
Instead of being afraid what "thermodynamics allows" and what are its
"expectations" we introduce a few definitions. By oscillation we mean the existence of
more than one local extrema on any of the concentration vs time curves. This may mean
either

periodicity in the strict mathematical sense, or

the presence of an infinite number of local extrema, or

the existence of a finite number (more than one) local extrema.
In this last case we speak about oligo-oscillation (2). Finally, we speak about

overshoot or undershoot phenomenon (3), when there is a stationary point, but the
concentrations do not tend to them in a monotone way.
The divergence of the right hand side of eq 2 being (k1  k2  k3  k4  k5  k6 ) is
always negative if at least one of the reaction steps do take place. Therefore no strictly
periodic solution can exist as Dulac's criterion is satisfied (see e.g. page 137, (4), and cf.
also (5)). This is true in all the Cases I, II and III.
Stationary Behavior in General
Constant solutions to the system of differential equations 1 or 2 will be called
stationary solutions, and we try to avoid ambiguous expressions such as steady state and
equilibrium filled with thermodynamic reminiscences. The stationary solution (unique in
the present case) can be calculated from the following equations
5
 (k1  k6 )a  k 2b  k5 (1  a  b)  0
k1a  (k 2  k3 )b  k 4 (1  a  b)  0
and it is
a
k 2 k 4  k 2 k5  k3 k5
k1k 3  k 5 k 3  k 6 k 3  k1k 4  k 2 k 4  k1k 5  k 2 k 5  k 2 k 6  k 4 k 6
(5)
b
k1k 4  k 6 k 4  k1k 5
k1k 3  k 5 k 3  k 6 k 3  k1k 4  k 2 k 4  k1k 5  k 2 k 5  k 2 k 6  k 4 k 6
(6)
Stationary solutions can be asymptotically stable, stable or unstable. The fact that
the stationary point given in 5,6 above is an asymptotically stable constant solution to eq
2 comes from the fact that all the eigenvalues of the Jacobian of eq 2 have negative real
parts, again, if at least one of the reaction steps takes place. (The corresponding stationary
point of
eq 1 is only relatively asymptotically stable. This means that if a small
perturbation changes the total mass, then the trajectories will tend to the equilibrium close
to the original one corresponding to the perturbed total mass. In other words, the
trajectories must remain in the same stoichiometric compatibility class (see (6), p. 99).
We shall also need below the ratios of the stationary concentrations:
a k3k5  k 2 k 4  k 2 k5

b k1 k 4  k1 k 5  k 4 k 6
b k1 k 4  k1 k 5  k 4 k 6

c k1 k 3  k 2 k 6  k 3 k 6
c k1 k 3  k 2 k 6  k 3 k 6

a k3k5  k 2 k 4  k 2 k5
(7)
6
Remarks to the Individual Cases
Case I
In the light of the general statement based on Dulac's criterion above, strictly
periodic solutions cannot be found. Here the eigenvalues of the coefficient matrix are the
complex numbers 0, 32i 3 , 32i 3 , thus the number of local extrema of the individual
concentration vs time curves i.e. the presence of oligo-oscillation cannot be decided
based on existing theorems. In the present special case, however, simple considerations
are sufficient. According to the theory of first-order differential systems ((9), p. 69), the
solution of eq 1 belonging to the given rate coefficients and initial concentrations is

 3t 


2


A(t )  1 1  2e 3t / 2 cos
3 

 3t 
 3t 
  3e 3t / 2 sin 


 2 
2




B(t )  1 1  e 3t / 2 cos
3 

 3t 
 3t 
  3e 3t / 2 sin 


 2 
 2 


C(t )  1 1  e 3t / 2 cos
3 
with the derivatives

 3t 
 3t 
dA 
1
  3 sin 

(t )   e 3t / 2 3 cos

 2 
dt
3
2






 3t 
 3t 
dB
1
  3 sin 

(t )  e 3t / 2 3 cos

 2 
dt
3
2





dC
2 3 3t / 2  3t 
.
(t ) 
e
sin 

dt
3
2


(8)
Elementary considerations show that each of the periodic factors in eq 8 has an infinite
number of positive zeros. According to the uniqueness of the solutions of autonomous
differential systems ((9), p. 163), the three derivatives cannot vanish at the same time.
7
Since the sum of the three derivatives is 0, it follows that two derivatives cannot vanish
at the same time, either. Thus, from eq 1,
d 2 A
dB
( ) 
( )  0
2
dt
dt
at the value  where the first derivative of [A] with respect to t is zero. This means that
[A] has a local extremum whenever its first derivative vanishes, so it has infinitely many
local extrema. The same argumentation yields the same result for the other two
concentrations too.
The case is different if the eigenvalues are real, then we have the Póta–Jost
theorem, see below, under Case III. This is the case e.g. if k1  k3  0.1 and k5  1.0
because then the eigenvalues are 0,
15 3
15 3
 and 
 . Thus, all the concentrations
10
5
10
5
can have no more than one ( 3  2) local extremum. Therefore in such cases it is not
true that "these oscillations continue at longer times", and "the circulation around the
cycle never stops".
Figure 2. Irreversible triangle reaction
In a system in which the reaction steps are irreversible, the classical equilibrium
constants have no meaning. However, in the case of weakly reversible networks Horn
and Jackson ((6), p. 96−98), proposed a generalization which applied here gives for the
8
 B in the network of Fig. 2 the finite value
step A 
value
k3
k5
 A the value
 1, and for the step C 
k5
k1
k1
k3
 C the
 1, for the step B 
 1. These ratios — no wonder —
correspond to the ratios of the stationary concentrations as calculated from eqs 7.
Case II
Here we mention that the equality of the product of the reaction rate constants
along cycles is only a necessary condition of detailed balancing, see the definitive paper
(7) by Feinberg on detailed balancing in chemical kinetics which we think important for
chemical educators.
The lack of oscillations found by the author is in accordance with the general
statements of Wei and Prater (10), acoording to which the eigenvalues of the detailed
balanced first-order reversible reaction systems are non-positive real numbers.
Case III
Now, let us turn to Case III. The author states on page 1209 and also shows (see
his figure reproduced here as Figure 3a) that it is possible to have a large number of local
extrema (oligo-oscillation, in our terms) in some of the concentration vs time curves.
However, Póta (3) has formulated and rigorously proved the theorem we can rightly call
Póta–Jost theorem, according to which no more than n  2 local extrema can be present
on any of the concentration vs time curves of a closed system of first-order reactions
with real eigenvalues of the coefficient matrix, if the number of species is n. Since in
Case III a first-order reversible triangle reaction ( n  3 ) with the eigenvalues -4, -3 and 0
is investigated, the Póta—Jost theorem applies and gives 1 for the maximal number of
local extrema on any of the concentration vs. time curves of the system (cf. Case I).
9
The cause of the misunderstanding might come from the fact that the author used
an earlier version of Mathematica. Figure 3 below shows his results and ours, the second
one using also his code but calculated with a more recent version (5.2) of Mathematica.
Figure 3a,b. Fake oligo-oscillation and the exact curve
In Case III the author characterizes the violation of the principle of
detailed balance with the fact that the ratio [B]/[A] tends to 5/3 instead of the
formal equilibrium constant (k1/k2)=2. This problem remains also in the case of
the correct curve calculated by us. For the sake of completeness we remark that
complex eigenvalues of the coefficient matrix (see Case I) can also occur here. If,
for example, k1 = 1.1, k3 = k5 = 1 and k2 = k4 = k6 = 0.1, the eigenvalues will be 0,

17
65i
17
65i

and  
. This raises the possibility of the temporal behavior
10
10
10
10
outlined in Case I.
The remarks above show that although some results of thermodynamics
must also be valid for chemical kinetics, it is well-worth developing “standalone”
kinetic methods. This may help us to describe some, thermodynamically less
tractable, biological systems too. For example, the kinetic formalism of the firstorder reaction systems applied here is equally useful in the exploration of the
compartmental systems (see (8), p. 30).
10
Supplemental Material
The Mathematica programs used for calculations and figures for this article are
available in this issue of JCE Online.
Acknowledgements
The present work has partially been supported by the National Science
Foundation, Hungary under No. T047132.
Literature Cited
1. Alberty, R. A.: Principle of detailed balance in kinetics, J. Chem. Educ. 2004, 81 (8),
1206–1209.
2. Beck, M. T., Rábai, Gy.: Oscillations and oligo-oscillations in hydrogen ion
concentration, J. of Phys. Chem. 1985, 89, 3907–3910.
3. Póta Gy.: On a theorem of overshoot-undershoot kinetics, React. Kinet. Catal. Lett.
1981, 17, 35–39.
4. Farkas M.: Periodic Motions; Springer-Verlag, New York, Berlin, Heidelberg,
London, Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1994.
5. Tóth, J.: Bendixson-type theorems with applications, Zsch. Angew. Math. und Mech.
1987, 67, (1) 31–35.
6. Horn, F. J. M., Jackson, R.: General mass action kinetics, Arch. Ratl. Mech. Anal.
1972, 47, 81–116.
7. Feinberg, M.: Necessary and sufficient conditions for detailed balancing in mass
action systems of arbitrary complexity, Chem. Engng. Sci. 1989, 44, 1819–1827.
11
8. Érdi, P., Tóth, J.: Mathematical Models of Chemical Reactions. Theory and
Applications of Deterministic and Stochastic Models, Princeton University Press,
Princeton, 1989.
9. Hirsch, M. W., Smale, S.: Differential Equations, Dynamical systems and Linear
Algebra, Academic Press, New York, 1974.
10. Wei, J. D., Prater, C. D.: The Structure and Analysis of Complex Reaction Systems,
Advances in Catalysis, Vol. 13, Academic Press, New York, 1962, pp. 203-392.
12
List of captions to the figures
Figure 1: The triangle reaction with all the possible reaction steps included
with three different sets of the reaction rate coefficients
13
Figure 2. Irreversible triangle reaction
14
Figure 3a,b. Fake oligo-oscillation and the exact curve
15