reaction extent or advancement of the reaction
TRANSCRIPT
arX
iv:2
111.
0525
3v1
[ph
ysic
s.ch
em-p
h] 8
Oct
202
1
Reaction extent or advancement of the reaction:
A new general definition†
Vilmos Gaspar‡ and Janos Toth∗,¶,§
‡Laboratory of Nonlinear Chemical Dynamics, Institute of Chemistry, ELTE Eotvos
Lorand University, Budapest, Hungary
¶Budapest University of Technology and Economics, Department of Analysis, Budapest,
Hungary
§Chemical Kinetics Laboratory, Institute of Chemistry, ELTE Eotvos Lorand University,
Budapest, Hungary
E-mail: [email protected]
Abstract
Following the classical works by De Donder, Aris and Croce we extend the gen-
eral investigation of the concept of reaction extent to the case of arbitrary number
of species and (reversible or irreversible) reaction steps, not restricted to mass-action
type kinetics. In typical cases the reaction extent tends to infinity. However, we have
defined quantities tending to 1, as it is assumed many times. We have calculated the
reaction extent also for exotic (oscillatory and chaotic) reactions to see their meaning.
Our approach tries to adhere to the customs of the chemists and to be mathematically
correct, and avoids some generally followed bad practice—mainly in the educational
literature.
†Based on the talk given at the 2nd International Conference on Reaction Kinetics, Mechanisms andCatalysis. 20–22 May 2021, Budapest, Hungary
1
Contents
1. Introduction 3
2. The concept of reaction extent 6
2.1. Typical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Motivation and fundamental definitions . . . . . . . . . . . . . . . . . . . . . 7
2.2.1. Simple, recurring examples . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2. The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Properties of the reaction extent . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. What is it that tends to 1? 19
3.1. Scaling by the initial concentration . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1. One reaction step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.2. Stoichiometric initial condition. Excess, scarcity . . . . . . . . . . . . 23
3.2. Scaling by the ,,maximum” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3. Detailed balanced reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1. Ratio of two reaction extents . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2. Let us multiply the ratios . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4. Reactions with an attractive stationary point . . . . . . . . . . . . . . . . . . 35
3.5. The difference. Have we found the real reaction extent? . . . . . . . . . . . . 36
3.6. Reactant and product species . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4. And what if the conditions are not fulfilled? 41
4.1. Multistationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2. Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1. The Lotka–Volterra reaction . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1.1. Irreversible case . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1.2. Reversible case, detailed balanced . . . . . . . . . . . . . . . 44
2
4.2.1.3. Reversible case, not detailed balanced . . . . . . . . . . . . . 45
4.2.2. The Rabai reaction of pH oscillation . . . . . . . . . . . . . . . . . . . 46
4.3. Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5. Discussion, outlook 49
Acknowledgement 52
6. Notations 52
7. Appendix 52
7.1. On the form of the induced kinetic differential equation . . . . . . . . . . . . 52
7.2. Critical review of the literature . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2.1. Neglectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2.2. Refusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2.3. Confused . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References 59
1. Introduction
In some books on reaction kinetics one can find one of the expressions reaction extent
(extent of reaction), progress of reaction, the advancement of the reaction, con-
version, or reaction coordinate. Some other, not less important books do not even
mention these, or they may only mention without using them. (As a closure of our work we
shall present the results of a detailed literature search to prove these statements.) We have
found that it is possible and useful to give a general definition of reaction extent and system-
atically study its qualitative and quantitative properties in spite of the controversial views
in the literature. We also emphasize that such investigations should precede discussions of
possible teaching the concept.
3
After a few decades of the introduction1 of the concept of reaction extent Aris2 has shown
a few examples.
The classical textbook on thermodynamic by Callen3 also uses the term degree of reac-
tion.
Our main goal is to fit the definitions and statements into the modern theory of reaction
kinetics, also called chemical reaction network theory, thus we cannot accept the formulation
by Carst4 according to which reaction extent is a unifying basis for stoichiometry.
A large collection of related concepts has been collected by Dumon et al.5 They also
introduce the reaction advancement ratio: ∆ξ∆ξmax
without giving a clear definition of ∆ξmax.
(It will turn out that in cases when something similar can be defined one should speak of
sup instead of max.) And what is one supposed to make of the expression ,,ξ = 1 when the
reaction has advanced one go.”?
Peckham6 proposes that IUPAC should use the normalized extent of reaction ξξmax
. The
problem is that the definition ξmax :=n0m
γmcan almost never be applied.
Croce7does know that ”one degree of advancement variable has to be defined for each
elementary step”, and also that ”the degree of advancement variable is not explained for the
general case of multistep reactions”. She ”reduces” the reaction steps (that we do not think
appropriate to call elementary) to
M∑
m=1
(βm,r − αm,r)X(m) = 0 (r = 1, 2, . . . , R)
the (bad) consequences of which we shall show in the paper. She proposes to divide the
variable we call reaction extent by V and calls the result degree of advancement. One must
admit that this trick would make some of the formulas simpler. When applying the concept
to multistep reactions her most important goal is to find a differential equation for a scalar
valued variable, either exactly, or approximately, using the Quasy Steady State Hypothesis—
in the usual naive way, i.e. without rigorously applying Tikhonov’s theorem. Our goal is to
4
accept and extend some of the statements of the (most over the average) paper by Croce7
and correct other statements or offer them to the mathematician as a clearly defined open
problem.
Although Vandezande etal.8 uses the relationship n(t) = n(0)+γξ close to our definition,
they utter unacceptable statements like (p. 1179) ,,Extent of reaction can be thought of as
a one-dimensional vector...”. They also have the view that formulas do depend on how the
reaction is written meaning that one can multiply the reaction step. That is not an acceptable
view if one has in mind kinetics as tho goal. Percent completion is also introduced there.
Instead of maxima they speak about extrema—not defined more clearly than the maximum
by Dumon et al.5
Moretti9 has also vague or unacceptable sentences. He says that the x-axis must vary
from 0 to the theoretical maximum value ξmax, with the equilibrium value in the range
0 < ξeqn < ξmax. Also: Of course, ξmax = 1 is possible only when the initial conditions are
stoichiometric.
In the present paper we do not touch a few seemingly closely related topics, and these
are as follows: potential surfaces, transition states, thermodynamics. Atomic fluxes (see
e.g.10) will not be treated either, as these concepts are unrelated. Reaction coordinate as
a quantity characterizing the changes of the bonds in a transition state complex belongs to
another area. Also, we remind the reader that the expression flux of a reaction is also
used to mean the reaction rate of a step, or their sum, as used by Craciun et al.11 Of the
many possible names we shall use reaction extent as we have done above.
The structure of our paper is as follows. Section 2 introduces the concept for reaction
networks of arbitrary complexity (any number of reaction steps and species, mass action
kinetics not assumed). As it is a usual requirement that the reaction extent tends to 1
when ”the reaction tends to its end”, we try to find quantities derived from our reaction
extent having this property in Section 3. Exotic reactions (those showing oscillatory or
chaotic behaviour) are treated in Section 4. Discussions and a list of Notations follow next.
5
An Appendix contains some mathematical background that would retard the expansion
of the thoughts of chemical relevance. Also, we enumerate citations from textbooks and
monographs neglecting the concept of reaction extent, or wrongly or too specifically using
it.
2. The concept of reaction extent
Before being constructive we enumerate a few general errors or useless approaches.
2.1. Typical errors
1. Most of the existing recent definitions cannot be applied to more complex systems; it is
mainly the case of a single reaction step that is treated, see e.g. the paper by Peckham.6
(This is not so with the original definition by De Donder and Van Rysselberghe.1)
2. It is quite common and absolutely unnecessary to use differentials (denoted by δ, d,∆,
etc.) when introducing the concept. (Cf.12 p. 61.) There is nothing infinitesimal here.
3. It is not exceptional to use the concept before defining it.
4. Several authors provide an implicit definition (that does not really define the given
quantity) and cannot get out from the trap built by themselves. Again, see e.g. the
paper by Peckham.6
5. Nothing is stated about the (mathematically or chemically) relevant properties of the
reaction extent.
6. Having introduced the concept this or that way, it is never ever used any more in many
works. (E.g. Pilling and Seakins13 mentions it in the Introduction and they do not
use the concept any more; this goes against the Chekhov principle: the gun shown at
the beginning should be used during the play.)
6
Our aim is to present a treatment devoid of all these errors or failures and to study the
reaction extent more deeply as usual. We emphasize that this is not a paper on chemical ed-
ucation, the time for such a paper is only ripe when the corresponding concept is scientifically
sound.
2.2. Motivation and fundamental definitions
Before giving a general definition we take a simple example.
2.2.1. Simple, recurring examples
Let us start with an example that may be deemed chemically oversimplified but not too
trivial, still simple enough so as not to be lost in the details.
As a strong simplification of reality, assume that water formation follows the simple
reaction step 2H2 + O2=2H2O. Let the forward step be represented in a more abstract
way: 2X+Y −−→ 2 Z. This means that we do not take into consideration either the atomic
structure of the species, or the realistic details of water formation. The number of species
(that will be denoted below as M) is 3, the number of (irreversible) reaction steps (to be
denoted below as R) is 1. The 3 × 1 stoichiometric matrix γ consisting of stoichiometric
numbers (differences of the coefficients of the species on the right and on the left side of
the reaction arrow) is:
−2
−1
2
. In case this step occurs five times the vector of the number
of individual species will change as follows:
NX −N0X
NY −N0Y
NZ −N0Z
= 5
−2
−1
2
,
where N0X is the number of molecules of species X at the beginning, and NX is the number
7
of pieces of species X after five steps, and so on. If one has the reversible reaction step
2X + Y −−⇀↽−− 2 Z, and the backward reaction step takes place three times then the total
change is
NX
NY
NZ
−
N0X
N0Y
N0Z
= 5
−2
−1
2
+ 3
2
1
−2
. (1)
Note that the number of molecules and the occurrence of reaction steps are positive integers.
Here we follow more or less Kurtz14 and Toth et al.15 We cannot rely on a discrete state
deterministic model of reaction kinetics—that would be desirable—because such a model
does not exists as far as we know.
2.2.2. The general case
Before providing general definitions we mention that Dumon et el.5 formulated a series of
requirements that should be obeyed by a well-defined reaction extent. We are unable to
accept most of these requirements. Let us mention only one: the reaction extent should be
independent of the choice of stoichiometric coefficients (invariant under multiplication), i.e.
it should have the same value for the reaction 2H2 + O2 −−→ 2H2O and for the reaction
H2 +12O2 −−→ H2O. This is not an error of the mentioned authors. Our (different) point
of view is that the reaction extent is strongly connected to kinetics, and it is not a tool to
describe stoichiometry as some further authors4 also think.
Following the books by Feinberg16 and by Toth et al.15 we consider a complex chemical
reaction, simply reaction, or reaction network as a set consisting of reaction steps as
follows:{
M∑
m=1
αm,rX(m) −−→M∑
m=1
βm,rX(m) (r = 1, 2, . . . , R)
}
; (2)
where
1. the chemical species are X(1),X(2),. . . ,X(M);
8
2. the reaction steps are numbered from 1 to R (M and R being positive integers);
3. α := [αm,r] and β := [βm,r] are M × R matrices of non-negative integer components
called stoichiometric coefficients, with the properties that ∀m∃r : βm,r 6= αm,r,
meaning that all the species take part in at least one reaction step, and ∀r∃m : βm,r 6=
αm,r, or: all the reaction steps do have some effect, and finally
4. γ := β −α is the stoichiometric matrix of stoichiometric numbers.
Instead of (2) some authors prefer writing this:
M∑
m=1
γm,rX(m) = 0 (r = 1, 2, . . . , R). (3)
This formulation immediately excludes reaction steps like X + Y −−→ 2Y (used e.g. to
describe a step in the Lotka–Volterra reaction), or reduces it to X −−→ Y. Similarly, an
autocatalytic step that may be worth being studied, see e.g. pp. 63 and 66,2 like X −−→ 2X
appears oversimplified as 0 −−→ X.
Another possibility is to exclude the empty complex, meaning that we get rid of the
possibility to simply represent in- and outflow with 0 −−→ X and X −−→ 0, respectively.
These last two examples evidently mean mass-creation and mass-destruction. If you
do not like these you should explicitly say so, because sometimes mass-creation and mass-
destruction is not so obvious as above, see the reaction
X −−→ Y + U, Z −−→ X+ U, Z −−→ U.
It may happen that one would like to exclude reaction steps with more than two (three)
molecules on the left side, such as
2MnO −4 + 6H+ + 5H2C2O4 = 2Mn +
2 + 8H2O+ 10CO2.
9
But such steps do occur e.g. on page 1236 of Kovacs et al.17 when dealing with overall
reactions. The theory and applications of decomposition of overall reactions into elementary
steps17,18 would have been impossible without this framework.
Someone may be interested in complex chemical reactions only consisting of reversible
steps. Then, (s)he will write down all the reaction steps and the corresponding ,,anti-
reactions steps.
Taking into consideration restrictions of the above kinds usually does not make the math-
ematical treatment easier. Sometimes it needs hard work to figure out what they mean, as it
is in the case of mass conservation of models containing species without atomic structure,15,19
or in the case of the existence of oscillatory reactions.20
To sum up: an author has the right to make any restriction (s)he finds to be chemically
important, but (s)he should declare these restrictions at the outset (and certainly not in the
middle of a discussion).
Finally, we mention our restriction: all the steps in (2) are really present, i.e. they
proceed with positive rate whenever the species on their left side are present.
We assume throughout that pressure, temperature and volume (V ) are constant. With
these one can write the generalized form of (1) as
N1
N2
. . .
NM
−
N01
N02
. . .
N0M
=
R∑
r=1
γ1,r
γ2,r
. . .
γM,r
Wr,
or shortly
N−N0 = γW,
where component Wr of the vector W =
[
W1 W2 . . . WR
]⊤
gives the number of occur-
rences of the rth reaction step. Note that we do not speak about infinitesimal changes.
10
(NX(m) and Nm will be used interchangeably.)
With a slight abuse of notation let W(t) be the number of occurrences of reaction steps
in the interval [0, t], a step function. Then:
N(t)−N0 = γW(t),
or turning to moles
n(t)− n0 =N(t)−N0
L= γ
W(t)
L= γξ(t),
where L is the Avogadro constant having the unit mol−1, and
n(t) :=N(t)
L,n0 :=
N0
L, ξ(t) :=
W(t)
L.
(We had to choose the less often used notation L to avoid mixing up with other notations.)
The relationship in concentrations can be expressed as
c(t)− c0 =n(t)− n0
V=
γ
Vξ(t), (4)
where V ∈ R+ is the volume of the reaction vessel, assumed as said above to be constant,
and c(t) := n(t)V
, c0 := n0
V. (Note that the component cX(m) or cm of c is traditionally denoted
in chemical textbooks as [X(m)], see e.g. Section 1.2 of13).
The concentration in Eq. (4) is again a step function; however, if the number of particles is
large, as very often it is, it may be considered to be a continuous (what is more: differentiable)
function.
Remember that the components of ξ(t) have the dimension amount of substance: mol.
Let us give a general formal explicit definition valid for any number of species and
reaction steps, and not restricted to mass action type kinetics. (However, a few
qualitative restrictions are usually made on the function rate, see15,16,21 of which we now only
11
mention continuity.) In order to do so we rewrite the induced kinetic differential equation
c(t) = γrate(c(t)) (5)
together with the initial condition c(0) = c0 into an (equivalent) integral equation:
c(t)− c0 = γ
∫ t
0
rate(c(t)) dt. (6)
and for its physical counterpart defined for a scalar variable measured in seconds and having
coordinate values measured in moldm3 .)
The component rater of the vector rate provides the reaction rate of the rth reaction
step. Note that in the mass action case (5) specializes into
c = γk⊙ cα (7)
or, in coordinates
cm(t) =
R∑
r=1
γmrkr
M∏
p=1
cαp,r
p (m = 1, 2, · · · ,M),
where k is the vector of (positive) reaction rate coefficients kr, and in (7) we used the usual
vectorial operations, see e.g. Section 13.2 in Toth et al.15 Their use in formal reaction
kinetics has been initiated by Horn and Jackson.22
In accordance with what has been said up to now we introduce the definition.
Definition 1. The reaction extent or conversion of the complex chemical reaction or
reaction network (2) is the scalar variable, vector valued function defined by the formula
ξ(t) := V
∫ t
0
rate(c(s)) ds ; (8)
its derivative ξ(t) = V rate(c(t)) is usually called the rate of conversion.
12
This reaction extent has been derived from the number of reaction steps in order to
reveal its connection to the concentrations. (In what follows we shall not use the expression
progress of reaction, and even less the reaction coordinate, but we accept the proposal
by Croce7) to call 1Vξ the degree of advancement.
One can formulate the following trivial consequences of the definition:
n = γξ, c =1
Vγξ c(t) =
1
Vγξ (9)
mentioned also by Laidler,23 sometimes as definitions, sometimes as statements (!). Note
that neither the rate of the reaction rate(c(t)) = ξ
V, nor the reaction extent ξ (fulfilling
thus one of the requirements formulated by5), nor the rate of conversion ξ depends on the
stoichiometric matrix γ.
What is wrong with the almost ubiquitous implicit ”definition” (4)? We show an example
to enlighten this.
Example 1. Consider the reaction
Xk1−−→ Y, X+ Y
k2−−→ 2Y
(expressing the fact that the same step occurs parallelly, with and without autocatalysis)
with
γ =
−1 −1
1 1
.
Then (6) specializes into
cX(t)− cX(0) = −∫ t
0
k1cX(s) ds−∫ t
0
k2cX(s)cY(s) ds = −ξ1(t) + ξ2(t)
V
cY(t)− cY(0) =
∫ t
0
k1cX(s) ds +
∫ t
0
k2cX(s)cY(s) ds =ξ1(t) + ξ2(t)
V,
and these two relations do not determine ξ1 and ξ2 individually, only their sum (even if one
13
utilizes cY(t) = cX(0) + cY(0)− cX(t)). The problem originates in the fact that the reaction
steps are not linearly independent as reflected in the singularity of the matrix γ.
Example 2. If the reaction steps of a complex chemical reaction are independent the situ-
ation is better.
In some special cases there is a way of making the ”definition” (4) into a real, explicit
definition. Assume that R ≤ M , and that the stoichiometric matrix γ is of the full rank, i.e.
the reaction steps are independent. Then, one can rewrite (4) in two steps as follows:
γ⊤(c(t)− c0) =1
Vγ⊤γξ(t)
ξ(t) = V (γ⊤γ)−1γ⊤(c(t)− c0).
Now one can accept as a definition ξ(t) := V (γ⊤γ)−1γ⊤(c(t) − c0). Nevertheless, in this
special case (6) implies
γ⊤(c(t)− c0) = γ⊤γ
∫ t
0
rate(c(s)) ds
and
(γ⊤γ)−1γ⊤(c(t)− c0) =1
Vξ(t) =
∫ t
0
rate(c(s)) ds,
thus this definition is the same as the one in (8). This derivation can always be done if
R = 1—a not-so-interesting case. Also, R ≤ M does not happen very often. In case of
combustion reactions Law’s law24 (see page 11) states that R ≈ 5M.
2.3. Properties of the reaction extent
The usual assumptions on the vector valued function rate are as follows, see.15,21
1. All of its components are continuously differentiable functions defined on RM taking
only non-negative values. This is usual, e.g. in the case of mass action kinetics, but—
14
with some restrictions—also in the case when the reaction rates are rational functions
as in the case of Michaelis–Menten or Holling type kinetics, see e.g.23,25–27
2. The value of rater(c) is zero if and only if some of the species needed for the rth
reaction step to take place is missing, i.e. for some m : αm,r > 0 and cm = 0 (see p.
613, Condition 1 in21). We shall say in this case that reaction step r cannot start
from the concentration vector c.
The second assumption implies—even in the general case, i.e. without restriction to the mass
action type kinetics—that rater(c) > 0 if all the necessary (reactant, see below) species are
present initially: αm,r > 0 =⇒ cm > 0.
Let us sum up the relevant qualitative characteristics of the reaction extent. In case
there is no misunderstanding, we shall use the same notation for a physical
quantity (expressed by its numerical value and unit) and for its numerical value,
as it is customary.
Theorem 1.
1. The domain of ξ is the same as that of c.
2. Both c and ξ ∈ C2(J,RR); with some open interval J ⊂ R such that 0 ∈ J.
3. ξ obeys the following initial value problem:
ξ(t) = V rate(c0 +1
Vγξ(t)), ξ(0) = 0. (10)
4. The velocity vector of the reaction extent (also called the rate of conversion) points
into the interior of the first orthant at the beginning, and this property is kept for all
times in the domain of ξ.
5. The components of ξ are either positive, strictly monotonously increasing functions,
or constant zero. The second case implies that some of the reaction steps cannot start
15
at the beginning.
Proof.
1. This follows from (8).
2. As c is the solution of a differential equation with a continuously differentiable right-
hand-side, it is twice continuously differentiable. The definition (8) implies ξ ∈
C2(J,RR).
3. Take the derivative of (8) and use (6). The initial condition also comes from (8).
4. The derivative of ξr is non-negative.
5. If the derivative of ξr is positive, then ξr is strictly monotonously increasing.
If it is zero at some time t0, then for some m : αm,r > 0 and cm(t0) = 0. However,
cm(t0) = 0 can only hold if cm(0) = 0 held at the beginning, because an initially
positive concentration cannot turn to zero, see Theorem 1 on p. 617 in the book by
Volpert and Hudyaev.21 But then cm(t) = 0 for all t ∈ R+0 . Thus in this case for all
t ∈ R+0 : ξr(t) = 0, this, however, together with the initial condition ξr(0) = 0 implies
that for all t ∈ R+0 : ξr(t) = 0; therefore ξr is (not strictly) monotone, it is constant
zero in this case.
Let us make a few remarks.
The last property mentioned in the Theorem can be realized with limt→+∞ ξ(t) = +∞
(the simplest example for this being 0 −−→ X), or with a finite positive value of limt→+∞ ξ(t),
see the example X −−→ Y −−→ Z below.
Eq. (10) shows that we would have simpler formulas if we used ξ(t)V
as proposed by Aris2
on p. 44, but we keep the more complicated form, as is is more often used.
In the mass action case both c and ξ are infinitely many times differentiable.
16
If one uses a kinetics different from the mass action type not fulfilling Conditions 1 and
2, or—as Pota28 has shown—if one applies an approximation, then it may happen that some
of the initially positive concentrations turn to zero.
The condition in the last sentence of Theorem 1 is only necessary but not sufficient as the
example below shows. But before that we need a technical remark. We label the first axis
(usually: horizontal) in the figures with t/s, to emphasize that the figure shows a relation
between pure numbers and not between physical quantities. Other axis labels are formed in
a similar way.
And now, let us start the consecutive reaction Xk1−−→ Y
k2−−→ Z from the initial vector of
concentrations
[
c0X 0 0
]⊤
, and suppose k1 6= k2. Then the second step cannot start at the
beginning, yet the second reaction extent is positive for all positive times as the solution of
the equations
ξ1 = V k1
(
c0X − ξ1V
)
, ξ2 = V k2
(
0 +ξ1 − ξ2
V
)
(11)
being
ξ1(t) = V (c0X − e−k1t),
ξ2(t) = V(k1(c
0X − e−k2t) + k2(−c0X + e−k1t))
k1 − k2.
shows.
Positivity also follows from the fact that the velocity vector of the differential equation
(11) points inward, into the interior of the first quadrant.
Note that ξ2/mol in Fig. 1 has an inflection point, because its second derivative is zero at
some positive time T/s for all choices of the reaction rate coefficients, and the third derivative
is not zero at T/s. ξ1/mol in Fig. 1 is concave from below no matter what the reaction rate
coefficients are.
One should take into consideration that although in the practically interesting cases
the number of equations in (10) is larger than those in (5), the equations are of a simpler
17
2 4 6 8 10t /s
0.2
0.4
0.6
0.8
1.0
reaction
extents
/mol
ξ1(t)
ξ2(t)
Figure 1: Reaction extents in the simple consecutive reaction Xk1−−→ Y
k2−−→ Z.k1 = 1 1
s, k2 = 2 1
s, c0X = 1 mol
dm3 , c0Y = c0Z = 0 moldm3 , V = 1 dm3.
structure.
Monotonicity mentioned in the Theorem implies that all the components of the reaction
extent do have a finite or infinite limit as t tends to sup(Dom(c)) =: t∗; a finite or infinite
time. It is an interesting open question: when does a coordinate of the reaction extent vector
tend to infinity?
As the emphatic closure of this series of remarks we mention that strict monotonicity of
the reaction steps shows that the reaction steps do not cease to proceed during equilibrium,
if they have started at all. This is the meaning of dynamic equilibrium taught also in
high schools. This important fact
• is independent on the form of kinetics,
• is a property of the deterministic model of reaction kinetics,
• and no reference to thermodynamics or statistical physics is needed to prove
it.
Example 3. In case when the domain of c is a proper subset of the non-negative real
numbers, ξ has the same property. Let us consider the induced kinetic differential equation
18
c = kc2 of the quadratic autocatalytic reaction 2Xk−−→ 3X with the initial condition
c(0) = c0 > 0. Then, c(t) = c01−kc0t
,(
t ∈[
0, 1kc0
[
( [0,+∞[)
. Now (10) specializes into
ξ = V k(c0 + ξ/V )2, ξ(0) = 0 having the solution ξ(t) = V c0kt
1−kc0t,(
t ∈[
0, 1kc0
[)
, thus ξ
blows up at the same time (t∗ := 1kc0
) when c.
Definitions and a few statements about blow-up in kinetic differential equations are given
in the works by Csikja et al.29,30
As the right hand side of (10) is a polynomial (vector-vector) function, the following
problem naturally arises.
Problem 1. What are the necessary or sufficient conditions that (10) is Hungarian,31
or kinetic (to use the original name)? (For the definition and characterization of kinetic
differential equations in the class of polynomial equations see the Appendix, Lemma 1, for
related results see also32,33)
In most of the examples below the differential equations for the reaction extent are
Hungarian, thus it will suffice to show a case when it is not.
Example 4. Consider the equations for the reaction extents in the case of the reaction
Xk1−−→ 0, X
k2−−→ 2X:
ξ1 = V k1(c0 +ξ2 − ξ1
V), ξ2 = V k2(c0 +
ξ2 − ξ1V
) = V k2c0 −k2ξ1 + k2ξ2. (12)
The boxed term shows that (12) is not Hungarian.
3. What is it that tends to 1?
Our interest up to this point was the number of occurrences of reaction steps. However,
many authors think it to be useful and visually attractive that the ”reaction extent tends to
1 when the reaction tends to its end,” see e.g. Fig. 1.34 Peckham6 also argues for [0,1].
19
Another approach is given by Moretti,9 Dumon et al.5 and others to introduce the re-
action advancement ratio ξξmax
and state that this ratio is always between 0 and 1. It is
really instructive that Atkins35 pp. 272–276 shows a figure ofG versus ∆ξ, where the first
axis is labeled from zero to one. The next edition36 however on pp. 216–217 have changed
their graphs to leave the first axis without 1 as an upper bound.
Being loyal to the usual belief,6,37 we are looking for quantities (pure numbers) tending
to 1 as e.g. t → +∞. Scaling might help find such quantities.
3.1. Scaling by the initial concentration
3.1.1. One reaction step
Now we are descending from the height of generality by considering a single irreversible step
(R = 1), assuming that the kinetics is of the mass action type. Thus, the reaction is
M∑
m=1
αmX(m)k−−→
M∑
m=1
βmX(m) (13)
therefore one has for the reaction extent
ξ(t) = V k
c01
c02
. . .
c0M
+
γ1
γ2
. . .
γM
ξ(t)
V
α
= V k
M∏
m=1
(
c0m + γmξ(t)
V
)αm
, ξ(0) = 0; (14)
with γm := βm − αm.
Theorem 2.
1. If the reaction step (13) cannot start then ξ(t) = 0 for all non-negative real times t:
∃m : (αm 6= 0 & c0m = 0) =⇒ ∀t ∈ R : ξ(t) = 0. (15)
20
2. If the reaction step (13) can start and all the species are produced, then ξ(t) tends to
infinity (blow-up included):
∀m : γm := βm − αm > 0 =⇒ limt→t∗
ξ(t) = +∞, (16)
where t∗ := sup(Dom(ξ)).
3. If some of the species decreases: ∃m : γm < 0, then
limt→+∞
ξ(t) = min
{
−V c0mγm
; γm < 0
}
. (17)
Proof.
1. The condition in (15) implies that ∀t ∈ R+0 and for all such m ∈ M for which αm 6= 0
one has cm(t) = 0, thus the right-hand-side of (14) is zero, and this, together with the
initial condition ξ(0) = 0 implies the statement.
2. The condition in (16) implies that the right-hand-side of (14) is positive, and
∀t ∈ Dom(ξ) : ξ(t) = k
M∑
m=1
αmγmc0m + γmξ/V
M∏
p=1
(c0p + γpξ(t)/V )αp > 0,
thus ξ is not only monotonously increasing but also strictly convex from below implying
the statement.
3. First of all, if one can show that
sup(Dom(c)) = +∞, (18)
then sup(Dom(ξ)) = sup(Dom(c)) = +∞ follows. If all the species decrease then (18)
21
is true. If X(m) decreases and X(p) increases then − cmγm
+ cpγp
= 0 implies
−cm(t)
γm+
cp(t)
γp= −cm(0)
γm+
cp(0)
γp,
which—together with the non-negativity of the concentrations see pp. 613–615 in21—
proves this part of the statement.
The last case of the Theorem includes the situation when the number of species is 1. The
simplest example to illustrate this is X −−→ 0.
The example X −−→ Y with c0X = 0, c0Y > 0 shows that a species can have positive
concentration for all positive times in a reaction where ”none of the steps” can start.
To have a dimensionless quantity, Aris2 proposes the division by the sum of the initial
concentrations in a special case on p. 47.
The table below shows a series of examples illustrating the different possibilities.
Table 1: Different cases of reaction extent
Step c0 ξ = ξ(t) = Case
X1−−→ 2X 0 V ξ 0 (15)
X1−−→ 2X 1 V (1 + ξ) V (et − 1) (16)
2X1−−→ 3X 1 V (1 + ξ/V )2 V t
1−t(16)
2X + Y1−−→ 2 Z V (c0X − 2ξ/V )2(c0Y − ξ/V ) (17)
Example 5. Here we give the details of the last example. In the case of the reaction
2X +Y1−−→ 2 Z mimicking water formation one has the following quantities: R := 1,M :=
3,X := H2,Y := O2,Z := H2O. Furthermore,
α =
2
1
0
, β =
0
0
2
, γ =
−2
−1
2
.
22
The initial value problem to describe the time evolution of reaction extent is
ξ = V k
c0X
c0Y
c0Z
+
−2
−1
2
ξ
V
2
1
0
= V k(c0X − 2ξ
V)2(c0Y − ξ
V), ξ(0) = 0. (19)
At the moment we can only provide the inverse of the solution to (19). Different initial
conditions lead to different results, see Figs. 2–4.
Obviously, the last part of the theorem is of main practical use. For this case one has
the following statement.
Corollary 1. In the third case of the Theorem above dividing ξ by the initial concentration
scaled by the quantity −γm/V we obtain a (pure) number tending to 1, as t tends to infinity:
limt→+∞ξ(t)
V c0m/(−γm)= 1.
3.1.2. Stoichiometric initial condition. Excess, scarcity
Alas! Before studying the mentioned figures, however, we need some definitions in order to
avoid the sin of using a concept without having defined it. The concepts of stoichiometric
initial condition and initial stoichiometric excess are elsewhere often used and never
defined.
Definition 2. Consider the induced kinetic differential equation (5) of the reaction (2) with
the initial condition c(0) = c0 6= 0. This initial condition is said to be a stoichiometric
initial condition (and c0 is a stoichiometric initial concentration), if for all such m ∈
M, r ∈ R for which γm,r < 0 the ratios c0m−γm,r
are independent from m and r. If the ratios are
independent of r, but for some p ∈ M the ratioc0p
−γp,ris larger than the others, then X(p)
is said to be in initial stoichiometric excess, or it is in excess initially. If the ratios are
independent of r, but for some p ∈ M the ratioc0p
−γp,ris smaller than the others, then X(p) is
23
said to be in initial stoichiometric scarcity, or it is in scarcity initially. The last notion
is mathematically valid, but in such cases one prefers saying that all the other species are in
excess.
Note that in combustion theory the expressions stoichiometric, fuel lean and fuel rich
are used in the same sense, see page 115 of.10
Although the above classes of initial conditions are disjoint, they do not cover all the
possible cases. A series of examples will shed some light on the fine points of the definition.
Example 6.
1. 2X + Y −−→ 0: here γX = −2, γY = −1, thus the stoichiometric initial concentrations
are of the form
[
2c0Y c0Y
]⊤
with arbitrary positive concentration c0Y.
2. 2X+Y −−→ 2 Z: here γX = −2, γY = −1, thus the stoichiometric initial concentrations
are of the form
[
2c0Y c0Y c0Z
]⊤
with arbitrary non-negative concentrations c0Y > 0, c0Z ≥
0.
3. 2X −−→ X or X −−→ 0: here c0X > 0 is always a stoichiometric initial concentration,
because in this case the number of species is 1, therefore the constancy criterion is
automatically fulfilled. (We admit that this case is trivial or uninteresting from the
chemist’s point of view.)
4. X −−→ 2X, X+Y −−→ 2Y, Y −−→ 0: any initial concentration would be stochiometric
for the second step, and also for the third one, considered separately. However, for the
first step the definition cannot be applied.
5. 5X+Y −−→ 3X+ 4Y: as γY > 0, the definition only applies to X, and c0X > 0 can be
arbitrary to be stoichiometric, if c0Y > 0 also holds.
6. 2X + 3Y −−→ X + Y:
[
c0X 2c0X
]⊤
are the stoichiometric initial concentrations with
arbitrary c0X > 0.
24
7. X + 2Y −−→ Z, 2X + Y −−→ Z: the conditionsc0X
−1=
c0Y
−2and
c0X
−2=
c0Y
−1can be
fulfilled with no positive initial concentration vector, therefore no stoichiometric initial
condition can be found. Funny to realize, ifc0Y
2< c0X < 2c0Y, then X is present in
both stoichiometric excess and scarcity at the same time. As it would be desirable
to eliminate such cases, the problem arises: determine those reactions for which this
cannot happen.
8.∑M
m=1 αmX(m) −−⇀↽−−∑M
m=1 βmX(m) : If the species are ordered in such a way that
γ1, γ2, . . . , γK < 0; γK+1, γK+2, . . . , γM ≥ 0,
(for some 1 ≤ K ≤ M) then
[
−γ1d0 . . . −γKd0 c0K+1 . . . c0M
]⊤
is a stoichiomet-
ric initial concentration vector for any numbers d0 > 0; c0K+1 ≥ 0, . . . , c0M ≥ 0.
9. It does not seem to be easy to give a similar general representation of the stoichiometric
initial concentrations in the general case (2).
0 20 40 60t /s
0.2
0.4
0.6
0.8
1.0
ξ(t)/mol
Figure 2: k = 1.0 dm6
mol2s, c0X/2 = 0.7 mol
dm3 < c0Y = 1.2 moldm3 , c0Z = 3 mol
dm3 , V = 1 dm3. Y isin stoichiometric excess.
25
0 100 200 300 400t /s
0.2
0.4
0.6
0.8
1.0
ξ(t)/mol
Figure 3: k = 1.0 dm6
mol2s, c0X/2 = 0.6 mol
dm3 = c0Y, c0Z = 3 mol
dm3 , V = 1 dm3. Stoichiometricinitial condition; slow convergence.
0 20 40 60t /s
0.2
0.4
0.6
0.8
1.0
ξ(t)/mol
Figure 4: k = 1.0 dm6
mol2s, c0X/2 = 0.6 mol
dm3 > c0Y = 0.5 moldm3 , c0Z = 3 mol
dm3 , V = 1 dm3. X is instoichiometric excess.
At this point it may not be obvious how to generalize Corollary 1 for more complicated
cases. We shall see this below.
Peckham6 also argues for [0,1], and under stoichiometric initial conditions he gives a
definition of ξmax and proposes to use ξ(t)ξmax
—in extremely special cases.
Note that instead of saying that the scaling factor is some initial concentration in the
26
0 20 40 60t /s
0.2
0.4
0.6
0.8
1.0
ξ (t)
VcX0 2
0 20 40 60t /s
0.2
0.4
0.6
0.8
1.0
ξ (t)
VcX0 2
0 20 40 60t /s
0.2
0.4
0.6
0.8
1.0
ξ (t)
VcY0
Figure 5: Scaled reaction extents tend to 1. The reaction rate coefficient and the initial dataare the same as in Figs. 2–4.
third case of Theorem 2 one can equally well say that the divisor is the stationary value of the
reaction extent, as it is in the cases in the figures: V c0X/2, V c0X/2 = V c0Y, V c0Y, respectively.
This result will come handy below.
3.2. Scaling by the ,,maximum”
In cases when ξ∗ := limt→+∞ ξ(t) is finite, then ξ∗ = sup{ξ(t); t ∈ R}, thus ξ∗ may be
identified (mathematically incorrectly) with ξmax, and surely limt→+∞ξ(t)ξ∗
= 1. That is the
27
procedure applied by most authors.5,8,9
3.3. Detailed balanced reactions
Definition 3. The complex chemical reaction
M∑
m=1
αm,rX(m)kr−−⇀↽−−k−r
M∑
m=1
βm,rX(m) (r = 1, 2, . . . , R); (20)
endowed with mass action kinetics is said to be conditionally detailed balanced at the
positive stationary point c∗ if
kr(c∗)α.,r = k−r(c
∗)β.,r (21)
holds. It is (unconditionally) detailed balanced if (21) holds for any choice of (positive)
reaction rate coefficients.
Note that all the steps in (20) are reversible. Furthermore, in such cases the reaction steps
are indexed by r and −r, and it is our choice in which order the forward and backward steps
are written, expressing the fact that ”backward” and ”forward” has no physical meaning.
3.2.1. Ratio of two reaction extents
Suppose we have a single reversible reaction step:
M∑
m=1
αmX(m) −−⇀↽−−M∑
m=1
βmX(m). (22)
(This reaction is unconditionally detailed balanced because the number of the forward and
backward pairs is 1.) Then the initial value problem for the reaction extents is as follows.
ξ1 = V k1
M∏
m=1
(c0m + γm(ξ1 − ξ−1)/V )αm , ξ1(0) = 0,
ξ−1 = V k−1
M∏
m=1
(c0m + γm(ξ1 − ξ−1)/V )βm, ξ−1(0) = 0,
28
where γm := βm − αm.
Proposition 1. Under the above conditions one has limt→+∞ξ1(t)ξ−1(t)
= 1.
Proof. First of all, let us note that the concentrations (and therefore the reaction extents)
do not blow up,38 i.e. sup(Dom(c)) = +∞.
Next, as limt→+∞ ξ1(t) = +∞ and limt→+∞ ξ−1(t) = +∞ and limt→+∞ ξ1(t) = V k1(c∗)α
and limt→+∞ ξ−1(t) = V k−1(c∗)β = V k1(c
∗)α one can apply l’Hospital’s Rule to get the
desired result.
Note that initially one only knows that it is the derivatives of the reaction extents that
have the same value at equilibrium.
Example 7. Consider the reversible reaction 2X + Y −−⇀↽−− 2 Z for ”water formation” with
the data
k1 = 1dm6
mol2s, k−1 = 1
dm3
mol s, c0Y = 1
mol
dm3 , c0Z = 1
mol
dm3 . (23)
• If c0X = 3 moldm3 , then X is in excess initially;
• if c0X = 2 moldm3 , then one has a stoichiometric initial condition;
• if c0X = 1 moldm3 , or c0X = 1
2moldm3 , then Y is in excess initially.
Note that it is not the excess or scarcity that is relevant, see Conjecture 1 below. The initial
rates of the forward and backward reactions are as follows:
1 · 9 · 1 > 1 · 1,
1 · 4 · 1 > 1 · 1,
1 · 1 · 1 = 1 · 1,
1 · 1/4 · 1 < 1 · 1.
The results are in accordance with Conjecture 1 below and can be seen in Figs. 6 and 7.
29
0.0 0.2 0.4 0.6 0.8t /s
0.5
1.0
1.5
2.0
2.5
ξ1(t)/ξ-1(�)
l����
ratio
0.0 0.2 0.4 0.6 0.8t /s
0.5
1.0
1.5
2.0
2.5
ξ1(t)/ξ-1(t)
limit
ratio
Figure 6: The ratio ξ1(t)ξ−1(t)
tending to 1 from above: X is in excess at the top and the initialcondition is stoichiometric at bottom. Data are given in the text.
30
0.0 0.2 0.4 0.6 0.8t /s
0.2
0.4
0.6
0.8
1.0
1.2
ξ1(t)/ξ-1(t)
limit
ratio
0.0 0.2 0.4 0.6 0.8t /s
0.2
0.4
0.6
0.8
1.0
1.2
ξ1(t)/ξ-1(t)
limit
ratio
Figure 7: The ratio ξ1(t)ξ−1(t)
being constant and tending to 1 from below. Y is in excess in bothcases. Data are given in the text.
31
Now we formulate our experience collected on several models as a conjecture. Consider
reaction (22).
Conjecture 1. If k1(c0)α > k−1(c
0)β (or k1(c0)α < k−1(c
0)β) then the ratio ξ1(t)ξ−1(t)
tends to
its limit in strictly monotonously decreasing (increasing) way. If k1(c0)α = k−1(c
0)β, then
the ratio is constant 1, and c0 = c∗.
Convergence has been proved above. The ratio at t = 0 is not defined, but the limit of
the ratio when t → +0 can be calculated using the l’Hospital Rule as
limt→+0
ξ1(t)
ξ−1(t)= lim
t→+0
ξ1(t)
ξ−1(t)=
k1k−1
(c0)α−β,
and k1k−1
(c0)α−β < 1 is equivalent to saying that k1(c0)α < k−1(c
0)β, and the remaining cases
can be similarly treated.
What is left to prove is that k1(c0)α < k−1(c
0)β implies that ξ1ξ−1
is strictly monotonously
increasing.
The meaning of the above conjecture is quite obvious: if the forward reaction proceeds
slower at the beginning than the backward reaction, then the limit 1 of the ratio is approached
from below etc. The concept of stoichiometric initial condition seem to play no role here.
Instead of studying other simple reactions we generalize the above result.
3.2.2. Let us multiply the ratios
Theorem 3. If the complex chemical reaction (20) is detailed balanced, i.e. (21) is fulfilled,
then limt→+∞∏R
r=1ξr(t)ξ−r(t)
= 1.
Proof. It is similar to that of Proposition 1: all the separate factors tend to 1 as t → +∞.
Example 8. Consider the example in Fig. 8 that is not unconditionally detailed balanced.
In this case the condition of detailed balancing to hold is
k−1k−2k−3 = k1k2k3, k3k5 = k−3k−5, k3k4 = k−3k−4
32
k1
k-3
k-2
k4
k
k5
k-5
X +Y
2 ZY +Z
2 X
3 XX +Z
Figure 8: A triangle coupled with a Wegscheider-type reaction
33
as applying either the circuit conditions and the spanning forest conditions39 or use
the algebraic condition coming from the Fredholm alternative theorem (see p. 133 of15)
gives. (The tedious calculations providing the above equalities can be carried out using the
function DetailedBalancedQ of the package ReactionKinetics, supplement to the book
by Toth et al.15)
0 1 2 3 4t /s
0.5
1.0
1.5
2.0
r=1
R
r (t)/ -r (t)
Figure 9: Convergence of the product of ratios from above in case of reaction in Fig. 8 withdata implying detailed balancing, as follows: k1 = 1 dm6
mol2s, k−1 = 2 dm3
mol·s , k2 = 3 dm3
mol·s ,
k−2 = 1 dm3
mol·s , k3 = 13
dm3
mol·s , k−3 = 12
dm6
mol2s, k4 = 3 dm3
mol·s , k−4 = 2 1s, k5 = 3 dm6
mol2s,
k−5 = 2 dm3
mol·s , c0X = c0Y = c0Z = 1 mol
dm3 .
Even if the reaction is not detailed balanced, it will not blow up,38 the stationary point
exists and is unique, and the product of the ratios will converge; although the limit will be
different from 1, see Fig. 10. Let us use l’Hospital’s Rule to show this in the case of a single
factor:
limt→+∞
ξr(t)
ξ−r(t)= lim
t→+∞
ξr(t)
ξ−r(t)=
ξr(+∞)
ξ−r(+∞)=
krk−r
cα(.,r)∗
cβ(.,r)∗
.
One can form quite different and complicated expressions that tend to 1, e.g. 1+ ξ2−ξ3ξ1
is
also a possible choice in case of R = 3. We may as well start from the pseudo-Helmholtz
34
0 50 100 150 200t /s
0.5
1.0
1.5
2.0
r=1
R
r (t)/ -r (t)
Figure 10: Convergence of the product of ratios in case of reaction in Fig. 8 with datanot fulfilling detailed balance, as follows: k1 = 1 dm6
mol2s, k−1 = 2 dm3
mol·s , k2 = 3 dm3
mol·s ,
k−2 = 1 dm3
mol·s , k3 = 1 dm3
mol·s , k−3 = 1 dm6
mol2s, k4 = 3 dm3
mol·s , k−4 = 2 1s, k5 = 3 dm6
mol2s,
k−5 = 2 dm3
mol·s , c0X = c0Y = c0Z = 1 mol
dm3 .
function introduced on p. 98 by Horn and Jackson22
H(c) := c⊤ · (ln(c)− ln(c∗)− 1)
(see also Liptak et al.40) and rewrite it in terms of the reaction extent H(ξ) := H(c0+γξ/V ),
finally divide it with the stationary value: Ψ(ξ(t)) := H(ξ(t))
H(ξ∗). The function Ψ has been defined
with a non-zero denominator and is tending to 1 as t → +∞ (at least in the case of weakly
reversible zero deficiency reactions endowed with mass action kinetics). As this function uses
all the reaction extents we shall often use it below.
3.4. Reactions with an attractive stationary point
All our observations can be summarized in the trivial proposition below. Before stating it
we need a definition for general ordinary differential equations.
35
Let M ∈ N, f ∈ C1(RM ,RM) and consider the initial value problem
x(t) = f(x(t)),x(0) = x0 (∈ RM). (24)
Definition 4. The stationary point x∗ of (24) is said to be attractive, if all the solutions
starting from a neighbourhood of x∗ are defined for all positive time, and tend to it as
t → +∞.
Note that attractiveness is less then being asymptotically stable and different from being
stable. The neighbourhood mentioned in the definition is the domain of attraction.
As a trivial reformulation of the definitions we arrive at our general statement.
Proposition 2. Let x∗ an attractive stationary point of (24), and assume that x0 is located
in the attracting domain of x∗. Assume furthermore, that for some g ∈ C(RM ,R) : g(x∗) 6= 0,
then limt→+∞g(x(t))g(x∗)
= 1.
We applied this statement when mentioning the expression g(ξ) := 1 + ξ2−ξ3ξ1
, and also
when calculating the ratios of reaction extents.
3.5. The difference. Have we found the real reaction extent?
Example 9. Let us try to fit the example on p. 204 of the book by Atkins and De Paula41
N2(g) + 3H2(g) −−→ 2NH3(g)
into the framework presented here (using abstract notations, aligning with the book). The
reaction is of the form
A + 3Bk−−→ 2C,
36
thus the induced kinetic differential equation for the concentrations of species (assuming
mass action kinetics) is as follows.
cA = −kcAc3B, cB = −3kcAc
3B, cC = 2kcAc
3B,
therefore for the amounts we have
nA = − k
V 3nAn
3B, nB = −3
k
V 3nAn
3B, nC = 2
k
V 3nAn
3B.
The authors provide nA(0) = 10 mol, but they do not specify either nB(0) or nC(0). A tacit
assumption may be nB(0) = −γBnA(0) = 30 mol (the stoichiometric initial condition), but
our arguments work for any
nB(0) ≥ 3nA(0), (25)
and for any nC(0). (If, however, nB(0) < 3nA(0), then limt→+∞ ξ(t) = nB(0)3
< nA(0), thus
one can in no sense speak about the fact that ξ(T ) = 10 for some T ∈ R+.) As
nA(t)− nA(0) = −ξ(t),
for some T ∈ R+ : ξ(T ) = 1mol implies nA(T ) = 9mol, and we also have nB(T )− nB(0) =
−3mol, nC(T ) − nC(0) = 2mol. For no finite T ∈ R+ we can have ξ(T ) = 10mol; this
can only be understood in such a way that limt→+∞ ξ(t) = 10mol (or, to use the short
notation ξ(+∞) = 10mol) and, certainly, limt→+∞ nA(t) = 0mol (if one has enough H2 at
the beginning as assumed in (25)). Note that the above results are independent on the value
of k and V , and—as far as (25) holds—on the value of nB(0) = nH2(0).
Example 10. Let us consider the reversible version of the previous example, Problem 7.25
of41 with a rich moral of the fable. The problem set by the authors is as follows. ”Express
the equilibrium constant of a gas-phase reaction A+3B −−⇀↽−− 2C in terms of the equilibrium
value of the extent of reaction, ξ, given that initially A and B were present in stoichiometric
37
proportions.”
A well-prepared student in classical physical chemistry is supposed to proceed in the
following way.
Initially, one had 1, 3 and 0 mols of A, B, and C. At equilibrium the changes of the species
(in mols) are as follows: −ξ∗,−3ξ∗,+2ξ∗; (where ξ∗ is the reaction extent at equilibrium),
thus at equilibrium we have the following amounts of the species A, B, and C: 1− ξ∗, 3(1−
ξ∗), 2ξ∗; therefore the total amount of the species at equilibrium is ntotal = 1 − ξ∗ + 3(1 −
ξ∗) + 2ξ∗ = 2(2− ξ∗). The equilibrium constant (having the unit mol−2) expressed in terms
of mols is
Kn=
M∏
m=1
nγmX(m) =
(2ξ∗)2
(1− ξ∗)(3(1− ξ∗)3)=
4(ξ∗)2
27(1− ξ∗)4. (26)
The molar fractions in equilibrium are
1− ξ∗
ntotal
=1− ξ∗
2(2− ξ∗),
3(1− ξ∗)
ntotal
=3(1− ξ∗)
2(2− ξ∗),
2ξ∗
ntotal
=2ξ∗
2(2− ξ∗);
thus the equilibrium constant (being a pure number) expressed in terms of molar ratios is
Kx=
M∏
m=1
xγmX(m) =
( 2ξ∗
2(2−ξ∗))2
1−ξ∗
2(2−ξ∗)(3(1−ξ∗)2(2−ξ∗)
)3=
2(ξ∗)2(2− ξ∗)2
27(1− ξ∗)4.
Now let us try to fit the solution of the problem into the general framework presented
here, and en route let us discern the tacit assumptions in the classical derivation. The
differential equations for the reaction extents are
ξ1 = V k1
(
n0A − (ξ1 − ξ−1)
V
)(
n0B − 3(ξ1 − ξ−1)
V
)3
,
ξ−1 = V k−1
(
n0C + 2(ξ1 − ξ−1)
V
)2
,
38
Assuming V = 1dm3, n0A = 1mol, n0
B = 3mol, n0C = 0mol we get
ξ1 = k1(
n0A − (ξ1 − ξ−1)
) (
n0B − 3(ξ1 − ξ−1)
)3,
ξ−1 = k−1
(
n0C + 2(ξ1 − ξ−1)
)2,
Let us introduce ζ := ξ1 − ξ−1 (that will turn out to be the advancement of the reaction
in this special case), then one can write
cA(t) = 1− ζ(t), cB(t) = 3− 3ζ(t), cC(t) = 2ζ(t),
because of (9). If t → +∞, then limt→+∞ c(t) exists and is positive, thus limt→+∞ ζ(t) =: ζ∗
also exists and 0 < ζ∗ < 1.
The evolution equation for ζ is
ζ = k1 (1− ζ) (3− 3ζ)3 − k−1 (2ζ)2 .
If t → +∞, then the limit of the right-hand-side exists, therefore limt→+∞ ζ(t) also exists
and is zero, therefore 0 = k1 (1− ζ) (3− 3ζ)3 − k−1 (2ζ)2 , implying
k1k−1
=(2ζ∗)2
(1− ζ∗) (3− 3ζ∗)3=
4(ζ∗)2
27(1− ζ∗)4, (27)
cf. (26).
Be careful, (27) does not mean that the equilibrium constant depends on the stationary
(equilibrium) concentrations. It is the equilibrium value of ζ∗ that depend on them so that
the equilibrium constant is really constant.
It would be comforting to see that ζ∗ is a(n asymptotically) stable stationary point.
We have seen that the cited author gave no definition whatsoever for the advancement
of the reaction. It turned out that he uses the difference of the two reaction extents useful
39
only for a single pair of reactions. Our definition works for any initial concentration vectors,
whereas he used one possible stoichiometric initial concentration for the irreversible
step A+ 3B −−→ 2C.
Let us mention that Aris2 also uses the trick turning to the difference of two reaction
extents on p. 86.
3.6. Reactant and product species
Favorite expressions in the chemists’ vocabulary are reactant species and product species.
In order to give a general definition of these concepts we can say that a species is a reactant
species if it occurs at the left side of a reaction arrow (or reaction step), and it is a product
species if it occurs at the right side of a reaction arrow.
Table 2 below shows that their use is much more limited than thought.
Table 2: Reactants and products
Reaction Reactant Product Bothstep(s) species species
only onlyX −−→ Y X Y —X+ 2Y −−→ 3 Z X, Y Z —X+Y −−→ 2Y X — YX+ 2Y −−⇀↽−− 3 Z — — X, Y, ZX −−→ Y −−→ Z X Z YX −−→ Y — —2Y −−→ 2X — — X, YX −−→ Y −−→ Z — —2Z −−→ 2Y X — Y, Z
One should be very careful, however. Although the above definition is in no contradiction
with the common use of these words, still we cannot provide a formal refinement of the
classification above although it would be useful.
40
4. And what if the conditions are not fulfilled?
What happens if one takes an exotic reaction, such as one having multiple stationary points,
oscillation or chaos?
4.1. Multistationarity
1
ε
1
ε
� X
� + 2 Y � �
+ �
Figure 11: Reaction having three stationary points with 0 < ε < 1/6
Horn and Jackson,22 (see p. 110) has shown that the complex chemical reaction in Fig. 11
has three (positive) stationary points in every stoichiometric compatibility class if 0 < ε < 16.
To be more specific, let us choose ε = 110, and let c0X = c0Y = 1 mol
dm3 . Then, easy calculation
shows that in the stoichiometric compatibility class defined by {(x, y); x+ y = 2}
c∗1 :=
[
1−√33
1 +√33
]
41
is an attracting stationary point with the attracting domain
{(x, y); 0 < x < 1−√3
3, 1 +
√3
3< y < 1), x+ y = 2},
and
c∗3 :=
[
1 +√33
1−√33
]
is an attracting stationary point with the attracting domain
{(x, y); 1 +√3
3< x < 1, 0 < y < 1−
√3
3, x+ y = 2},
and c∗2 := (1, 1) (a special case of the stoichiometric initial concentration
[
c0X c0X
]
with
c0X > 0) is a non-attracting stationary point. Let us start the reaction from the attracting
domain of the first stationary point and let us calculate the value of the normed pseudo-
Helmholtz function Ψ as a function of time. The result can be seen in Fig. 12. This example
0.5 1.0 1.5t /s
0.9980
0.9985
0.9990
0.9995
Ψ
Figure 12: Convergence to 1 in reaction of Fig. 11.
fits into the validity region of Proposition 2, it is only interesting because of the presence of
multiple stationary points.
Let us mention that criteria of multistationarity have been collected by Joshi and Shiu.42
42
4.2. Oscillation
We mention here two oscillatory reactions: the often used theoretical Lotka–Volterra reaction
and the experimentally based Rabai reaction43 aimed at describing pH oscillations. (One
may say that Brusselator would be a more realistic choice, as it leads to a limit cycle, but
on the other hand it has a third order step. The calculations to be carried out here would
be almost the same.)
4.2.1. The Lotka–Volterra reaction
4.2.1.1. Irreversible case It is known20,44 that under some mild conditions the only two-
species reaction to show oscillation is the (irreversible) Lotka–Volterra reaction Xk1−−→ 2X,
X +Yk2−−→ 2Y, Y
k3−−→ 0. (Cf. also the paper by Toth and Hars.45) It has a single positive
stationary point that is stable but not attractive, therefore one cannot apply Proposition 2.
2 4 6 8t /s
0.55
0� �
����
����
����
0.80
Ψ (t)
Figure 13: Time evolution of the function Ψ in the irreversible Lotka–Volterra reaction withk1 = 3 1
s, k2 = 4 dm3
mols, k3 = 5 1
sand c0X = 1 mol
dm3 , c0Y = 2 mol
dm3 .
What we see in Fig. 13 is that the pseudo-Helmholtz function is oscillating. The question
is if it can be used for some purposes. Note, however, that the individual reaction extents
do not oscillate, they are ”pulsating” (and tending to infinity), they have an oscillatory
derivative, and the zeros of their second derivative clearly show the endpoints of the periods,
43
see Fig. 14. It may be a good idea to calculate any kind of reaction extent or the pseudo-
Helmholtz function for a period in case of oscillatory reactions. We shall return to this
point in a following paper.
2 4 6 8t /s
10
20
30
ξ (t)/m���
ξ1
ξ2
ξ3
2 4 6 8t /s
2
4
6
8
10
12
ξ ' (t)/�� !/s
ξ1'
ξ2'
ξ3'
2 4 6 8t /s
-40
-20
20
40
ξ '' (t)/"#$%/s^2
ξ1''
ξ2''
ξ3''
Figure 14: The individual reaction extents and their first and second derivatives of theirreversible Lotka–Volterra reaction with k1 = 3 1
s, k2 = 4 dm3
mols, k3 = 51
sand
c0X = 1 moldm3 , c0Y = 2 mol
dm3 .
4.2.1.2. Reversible case, detailed balanced The reversible Lotka–Volterra reaction
Xk1−−⇀↽−−k−1
2X, X + Yk2−−⇀↽−−k−2
2Y, Yk3−−⇀↽−−k−3
0 is also worth studying. First, let us note that for all
values of the reaction rate coefficients it has a single, positive stationary point, because the
reaction is reversible, therefore it is permanent,46,47 the trajectories remain in a compact set.
If the trajectories remain in a compact set, then they are either tending to a limit cycle, or
the stationary point is asymptotically stable. The first possibility is excluded by the above
mentioned theorem by Pota, thus, it is only the second possibility that remains.
Let us note that both existence and uniqueness of the stationary state also follow from
the Deficiency One Theorem (p. 106 in Feinberg,16 or p. 176 in Toth et al.15)
44
Next, if the reaction is detailed balanced holding if and only if
k1k2k3 = k−3k−2k−1 (28)
is true, then our Proposition 2 implies that limt→+∞Ψ(ξ(t)) = 1. Based on Theorem 3) we
0.5 1.0 1.5t /s
-1.5
-1.0
-0.5
0.5
1.0
Ψ (t)
Figure 15: Time evolution of the function Ψ in the reversible detailed balanced Lotka–Volterra reaction with k1 = 1 1
s, k2 = 2 dm3
mols, k3 = 3 1
sk−1 = 4 dm3
mols, k−2 = 5 dm3
mols,
k−3 = 158
1s, and c0X = 1 mol
dm3 , c0Y = 2 moldm3 .
can state the same about the product of the ratios of the reaction extents.
4.2.1.3. Reversible case, not detailed balanced If (28) does not hold, the reaction
still has an attracting stationary point, what is more, it has an asymptotically stable
stationary point as we have seen above. Fig. 17 shows the behavior of the individual reaction
extents.
One would expect that the behaviour of the pseudo-Helmholtz function is similar to the
irreversible case if the backward rate coefficients are small. However, this situation cannot
be realized, because Condition (28) shows that it is impossible to decrease the backward
rates without changing the forward ones.
45
2 4 6 8t /s
5
10
15
ξ (t)/mol
ξ1
ξ-1
ξ2
ξ-2
ξ3
ξ-3
0.5 1.0 1.5t /s
5
10
15
ξ ' (t)/mol/s
ξ1′
ξ-1′
ξ2′
ξ-2′
ξ3′
ξ-3′
0.5 1.0 1.5t /s
-15
-10
-5
5
10
ξ '' (t)/mol/s^2
ξ1′′
ξ-1′′
ξ2′′
ξ-2′′
ξ0&′
ξ-3′′
Figure 16: The individual reaction extents and their first and second derivatives of thereversible, detailed balanced Lotka–Volterra reaction with k1 = 1 1
s, k2 = 2 dm3
mol s,
k3 = 3 1sk−1 = 4 dm3
mol s, k−2 = 5 dm3
mol s, k−3 = 15
81s, and c0X = 1 mol
dm3 , c0Y = 2 moldm3 .
4.2.2. The Rabai reaction of pH oscillation
Here we include a reaction proposed by Rabai43 to describe pH oscillations. This reaction has
a much more direct contact to chemical kinetic experiments and is much more challenging
from the point of view of numerical mathematics than the celebrated Lotka–Volterra reaction.
Rabai43 starts with the steps
A− +H+ k1−−⇀↽−−k−1
AH,
AH+ H+ + {B} k2−−→ 2H+ + P−,
where {B} is an external species, i.e. one considered to have constant concentration. This
46
0.2 0.4 0.6 0.8t /s
0.0
0.2
0.4
0.6
0.8
1.0
Ψ (t)
Figure 17: Time evolution of the function Ψ in the reversible, not detailed balanced Lotka–Volterra reaction with k1 = 1 1
s, k2 = 2 dm3
mol s, k3 = 3 1
sk−1 = 4 dm3
mol s, k−2 = 5 dm3
mol s,
k−3 = 158
1s, and c0X = 1 mol
dm3 , c0Y = 2 mol
dm3 .
reaction has a single stationary point
c∗A− = 0, c∗H+ = c0H+ + c0AH, c∗AH = 0, c∗P = c0A− + c0AH + c0P
specializing into c∗A−
= 0, c∗H+ = c0
H+, c∗AH = 0, c∗P = c0A−
with the natural restriction on the
initial condition c0AH = 0, c0P = 0.
Putting the reaction into a CSTR (continuously stirred tank reactor) means in the terms
of formal reaction kinetics that all the species can flow out and some of the species flow in,
i.e. in the present case the following steps are added
A− k0−−→ 0, (29)
0k0c0A−−−−→ A−, (30)
H+ k0−−→ 0, (31)
0k0c0H+−−−→ H+, (32)
AHk0−−→ 0. (33)
47
0.5 1.0 1.5t /s
2
468
10
ξ (t)/mol
ξ1
ξ-1
ξ2
ξ-2
ξ3
ξ-3
0.5 1.0 1.5t /s
2
&'8
10
ξ ' (t)/mol/s
ξ1′
ξ-1′
ξ2′
ξ-2′
ξ3′
ξ-3′
0.5 1.0 1.5t /s
-15
-10
-5
5
10
ξ '' (t)/mol/s^2
ξ1′′
ξ-1′′
ξ2′′
ξ-2′′
ξ0&′
ξ-3′′
Figure 18: The individual reaction extents and their first and second derivatives of thereversible, not detailed balanced Lotka–Volterra reaction with k1 = 1 1
s, k2 = 2 dm3
mol s,
k3 = 3 1sk−1 = 4 dm3
mol s, k−2 = 5 dm3
mol s, k−3 = 15
81s, and c0X = 1 mol
dm3 , c0Y = 2 moldm3 .
As a result of adding these steps, multistability (existence of more than one stationary points)
may occur with appropriately chosen values of the parameters.
When the reaction step
H+ + {C−} k3−−→ CH (34)
is also added, one may obtain periodic solutions having appropriate parameter values, see
Fig. 19.
Let us remark that neither the Lotka–Volterra reaction nor the Rabai reaction is mass-
conserving. However, we could have chosen a mass-conserving oscillatory reactions, as e.g.
the Ivanova reaction: X + Y −−→ 2Y,Y + Z −−→ 2 Z,Z + X −−→ 2X. Finally, we remark
that the connection between multistationarity and oscillation is far from being as simple as
part of the literature asserts, see.48
4.3. Chaos
Here we use a version of the Rabai reaction that can numerically be shown to exhibit
chaotic behaviour, see Fig. 20 and is a good model for experimental pH oscillators also show-
48
400 600 800t /s
()*
5.0
5.5
+,-
./5
789
pH(t)
Figure 19: Time evolution of the pH and the projection of the first three coordinates ofthe trajectory in the oscillating Rabai reaction with k1 = 1010 dm3
mol s, k−1 = 103 1
s,
k2 = 106 dm3
mol s, k3 = 1 dm3
mol s, k0 = 10−3 1
s, and c0
A−= 5
1000moldm3 , c0H+ = 1
1000moldm3 ,
c0AH = 0 moldm3 , c
0P = 0 mol
dm3 .
ing chaotic behaviour. (Let us mention that rigorously verified kinetic differential equation
is not known to exist.)
Including the removal of CH
CHk11−−→ 0 (35)
and using appropriate parameters and favorable input concentrations chaotic solutions are
obtained. Fig. 20 illustrates (does not prove!) this fact.
The reaction extents tend to +∞ in such a way that their derivative is chaotic, as
expected, see Fig. 21.
5. Discussion, outlook
We have introduced the concept of reaction extent for reaction networks of arbitrary com-
plexity (any number of reaction steps and species) without assuming mass action kinetics.
This reaction extent gives the advancement of each individual irreversible reaction step; in
case of reversible reactions we have a pair of reaction extents. In all the practically important
cases the fact that the reaction extent is strictly monotonous, implies that the reaction steps
do not cease to proceed even during equilibrium. Thus, we provided a meaning of dynamic
49
2500 3000 3500t /s
:;<
=>?
@AB
6.0
CDE
7.0
pH(t)
FGH IJK LMN 6.0 OPQ 7.0pAH
4.0
RST
UVW
Z[\
6.0
]^_
7.0
pH
Figure 20: Time evolution of the pH and the projection of the logarithm of the first twocoordinates of the trajectory in the oscillating Rabai reaction with the same parameters asin Fig. 19 and k11 = 5
1001s.
equilibrium usually taught also in high schools. Note that we made no allusion to either
thermodynamics or statistical mechanics.
After a few statement on the qualitative behaviour of the reaction extent we made a
few effort to connect the notion with the traditional ones. Thus, we have shown that if the
number of reaction steps is one, as in all the practically important cases, the reaction extent
in the long run (as t → +∞) tends to 1 if appropriately scaled.
Then, for an arbitrary number of reversible detailed balanced reaction steps the product
of the ratios of the reaction extents are shown to tend to 1 as t → +∞.
Our most general statement follows for arbitrary reactions having an attracting stationary
point and with a function not vanishing on the stationary point: in this case the value of the
chosen function along the time dependent concentrations divided by the value of the given
function at the stationary concentration tends to 1. Thus, this statement is true not only
for the reaction extent but also for any appropriate functions.
Recalculations of some examples by Atkins and DePaula41 make clear the vague meaning
of some further concepts and statements including the concept of reactant, intermediate and
product species.
50
400 600 800 1000 1200t /s
`abcde
fghijk
nopqrs
ξ11(t)
Figure 21: Time evolution of the reaction extent of the step (34) in the chaotic Rabai reactionwith the same parameters as in Fig. 20.
We have numerically studied a few examples of reactions where our statements cannot
be applied: multistationary, oscillatory and chaotic reactions.
One should take into consideration that although in the practically interesting cases the
number of equations in (10) is larger than those in (5), the equations for the reaction extents
are of a much simpler structure, as the right hand side of the differential equations describing
them only consist of a single term.
As a by-product we have given an exact definition of stoichiometric initial concentration
and initial concentration in excess. We have also shown that reactant and product can be
defined in full generality—contrary to the literature.
In the Appendix we cited a few examples proving that peer review of textbooks are not
less important than that of papers.
One can say that the concept of reaction extent can be usefully applied to a larger class
of reactions than usually, but in some (exotic) cases its use need further investigations.
However, it is the reader to decide if we succeeded in avoiding all the traps mentioned at the
beginning.
Quite a few authors treats the methodology of teaching the concept; we think this ap-
proach will only have its raison d’etre when the scientific background will have been clarified.
51
Acknowledgement
The present work has been supported by the National Office for Research and Development
(2018-2.1.11-TET-SI-2018-00007 and SNN 125739). JT is grateful to Dr. J. Karsai (Bolyai
Institute, Szeged University) and for Daniel Lichtblau (Wolfram Research) for their help.
Members of the Working Party for Reaction Kinetics and Photochemistry, especially Prof.
T. Turanyi and Dr. Gy. Pota, made a number of useful critical remarks.
6. Notations
Some of the readers may appreciate that we have collected the used notations.
7. Appendix
7.1. On the form of the induced kinetic differential equation
It is relatively easy to characterize those polynomial differential equations that can emerge
as the mass action type induced kinetic differential equation of a complex chemical reaction.
Lemma 1 (32). Suppose the right hand side f of a differential equation c(t) = f(c(t)) is a
polynomial. It is the induced kinetic differential equation of a reaction endowed with mass
action type kinetic if and only if its mth coordinate function is of the form
fm(c) = gm(cm)− cmhm(c), (36)
where all the coefficients of the polynomials gm and hm are non-negative, and gm does not
depend on the scalar cm : cm :=
[
c1 c2 . . . cm−1 cm+1 . . . cM
]⊤
.
A right-hand-side of the form (36) is sometimes said to be Hungarian. One would think
52
Table 3: Notations I
Notation Meaning Unit Typical value=⇒ implies∈ belongs to∀ universal quantor ”for all”∃ existential quantor ”there is”⊙ coordinate-wise product of vectors
cm, cX concentration of X(m) and X moldm3
c vector of concentrationsc0m initial concentration of X(m) mol
dm3
c∗m stationary concentration of X(m) moldm3
Ci(A,B) i times continuouslydifferentiable functions
Dom(u) the domain of the function uJ ⊂ R an open interval
k, kr, k−r reaction rate coefficient 1s( moldm3 )1−
∑Mm=1 αm,r
M the number of chemical species ∈ N
nm the quantity of species X(m) mol ∈ N
N the set of positive integersN0 the set of
non-negative integersn0m the initial quantity mol ∈ N
of species X(m)n the vector of
the quantity of speciesn0 the vector of
the initial quantity of species
rater rate of the rth reaction step moldm3s
rate vector of reaction ratesR the set of real numbersR+ the set of
positive real numbersR+
0 the set ofnon-negative real numbers
t time ∈ R
53
Table 4: Notations II
Notation Meaning Unit Typical value
V volume dm3
Wr number of occurrences ∈ N0
of the rth stepW vector of
number of occurrencesxm mth dependent variable in a
differential equationx vector of variables in a
differential equationX,Y,X(m) chemical speciesαm,r, αm stoichiometric coefficient 1 0, 1, 2, 3
in the reactant complexα matrix of
reactant complex vectorsβm,r, βm stoichiometric coefficient 1 0, 1, 2, 3
in the product complexβ matrix of
product complex vectorsγm,r, γm stoichiometric number 1 −3,−2, . . . , 2, 3
in the reactant complexγ stoichiometric matrixξr reaction extent mol
of the rth stepξ vector of
reaction extents
54
that this property is equivalent to
fm(c1, c2, · · · , cm−1, 0, cm+1, · · · , cM) ≥ 0, (37)
but it is not true, as the following example shows.
c1 = c22 −2c2c3 + c23 + · · · , c2 = · · · , c3 = · · · .
The first equation fulfils (37), but is not Hungarian because of the boxed term.
Note that any polynomial equation can be transformed into a Hungarian one (as far as
the trajectories in the open first order are concerned) by multiplying the right-hand-side
with the product of the variables, an idea of Samardzija49 generalized by Gh. Craciun.
7.2. Critical review of the literature
The authors dealing with reaction kinetics seem to show three different kinds of behaviour.
1. Totally or practically neglect the concept, some of them do not even mention any of the
names reaction extent (extent of reaction), advancement/progress of reaction, reaction
flux. (It is not clear what the reaction coordinate means but it is surely different from
the previous concept(s).)
2. Strongly refuse the use of the concept.
3. Utter scientifically unacceptable confused, ambiguous sentences.
Let us starts with remarks on two classics. The concept comes from De Donder and Van
Rysselberghe,1 although almost always it is only the first author who is mentioned. Here
the general case is considered, the definition is implicit and the properties are not studied.
The definition of the degree of reaction on page 169 of the classic textbook by Callen3
commits most of the faults enumerated above.
Now let us see the point of view of more recent authors.
55
7.2.1. Neglectors
It is an acceptable standpoint at the present status of things.
• Fig. 17.15 of the popular textbook50 by Dillard and Goldberg is typical (first, horizon-
tal coordinate: reaction coordinate; second, vertical coordinate: energy). Fig. 17.17
pretends to show the effect of a catalyst. The figure actually shows a potential energy
surface, and this concept is not related to the one treated here.
• Feinberg16 is not interested.
• Pilling and Seakins13 on p. 14 show the typical figure in the Introduction (!), and never
return to it.
• On page 90 Toth et al.15 provided an initiative of the general investigations. In a later
chapter, see p. 268, they show a similar approach for the stochastic model. Historically
(for us) it was Kurtz14 the first to give a general clear definition, but that author was
not interested in the details of the topic.
• Turanyi and Tomlin10 are not interested.
• The introduction of the degree of reaction in the classic textbook3 (see p. 169)also
commits some of the faults but one may say that the author is not really interested in
the use of the concept.
7.2.2. Refusers
• The opinion of Lente51 as expressed on p. 96 is: The x axis is often called ”reaction
coordinate” but typically lacks any meaningful definition.
• Lente in a paper with Schmitz52 emphasizes the restricted use of the concept: ”At
this point, it should be emphasized that the extent of reaction only makes sense if the
studied reaction is a reasonably simple one that does not have any intermediates in
significant concentrations.”
56
7.2.3. Confused
Our standpoint is that if an author seems to be mistaken then it is not enough to express
our disapprovement, we have to find out their unrevealed errors and tacit assumptions, etc.
• The view shared by most of the chemists is summarized in a few key words by Laidler23
We make a few remarks on those key words using our notations.
Extent of Reaction (or Extent of Conversion) Under the usual assumptions the
extent of reaction at any time is defined by ξ(t) := nm(t)−n0m
γmwhere n0
m is the initial
amount of the reactant or product entity X(m), n0m is the amount at time t (both
measured in moles), and γm is the stoichiometric number for that species in the
reaction equation as written. The extent of reaction defined in this way
depends on how the reaction equation is written, but it is independent of
which entity in the reaction equation is used in the definition.
For the general reaction (13) the extent of reaction is
ξ :=∆n1
γ1=
∆n2
γ2= · · · = ∆nM
γM.
As the author (without mentioning this fact) excludes steps of the form X+Y −−→
X + Z, we had to essentially change his equation to be meaningful, also avoiding
to speak about reactant, product and signs of stoichiometric coefficients. Note
that general here means a single, irreversible step.
Rate of Conversion The rate of conversion is defined as the time derivative of the
extent of reaction, ξ = nm
γm.
Rate of Reaction The rate of reaction is the rate of conversion per unit volume:
rater := ξV, thus it is a derived quantity (?!). For the species X(m) the rate of
reaction is thus given by 1V γm
nm. If the volume does not change during the course
of reaction the rate of reaction may be expressed in terms of concentrations as
57
rater =1γm
cX(m).
• Let us scrutinize the ”Bible” of Physical Chemistry,41 published in eleven editions up
to the writing of the present paper. (We do not have access to all the editions.)
1. On page 201 they give a definition for the reaction A −−⇀↽−− B.
2. On page 201 they provide a general (?) definition: ”The general case of a
reaction.—Consider the reaction 2A + B −−→ 3C + D. A more sophisticated
way of expressing the chemical equation is to write it in the symbolic form 0 =
3C + D − 2A − B by subtracting the reactants from both sides (and replacing
the arrow by an equals sign).” This is a less sophisticated way as one loses part
of the information in case a species occurs on both sides.
3. Let us see p. 233 of the cited work: ”Checklist of key ideas—The extent of reaction
ξ is defined such that, when the extent of reaction changes by a finite amount ∆ξ,
the amount of A present changes from nA,0 to nA,0 −∆ξ.” This sentence should
be taught what a definition is not.
4. On p. 233, after the nebulous quasi-definitions the rate of reaction—that is a
clearly understood, measurable quantity—it is now defined as a derived quantity
using the extent of reaction.
5. P. 809 clearly illustrates why it is an even heavier task to incorporate the concept
reaction coordinate into the area of rational science: ”The reaction coordinate
is the collection of motions, such as changes in inter-atomic distances and bond
angles, that are directly involved in the formation of products from reactants.
(The reaction coordinate is essentially a geometrical concept and quite distinct
from the extent of reaction.)”
6. The concept is applied on p. 238: ”Express the equilibrium constant of the
gas phase reaction A + 3B −−⇀↽−− 2C in terms of the equilibrium value of the
58
extent of reaction ξ, given that initially A and B were present in stoichiometric
proportions.” This example has been treated in detail above.
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63
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