arXiv:2111.05253v1 [physics.chem-ph] 8 Oct 2021 Reaction extent or advancement of the reaction: A new general definition † Vilmos G´ asp´ ar ‡ and J´ anos T´ oth ∗,¶,§ ‡Laboratory of Nonlinear Chemical Dynamics, Institute of Chemistry, ELTE E¨ otv¨ os Lor´ and University, Budapest, Hungary ¶Budapest University of Technology and Economics, Department of Analysis, Budapest, Hungary §Chemical Kinetics Laboratory, Institute of Chemistry, ELTE E¨otv¨ os Lor´ and 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 and Catalysis. 20–22 May 2021, Budapest, Hungary 1
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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,
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
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