Estimation of kinetic parameters of reversible chain reactions of quinoneimines with hydroquinones having self-acceleration periods

Article abstract of DOI:10.1134/S002315841005006X

A new approach is suggested for determining the kinetic parameters and rate constants of the elementary steps of reversible chain reactions having self-acceleration periods due to the long time required for the concentrations of the chain-carrier radicals to reach their steady-state values. This approach is illustrated by the example of the reversible chain reaction between N,N′-diphenyl-1,4-benzoquinonediimine and 2,5-dichlorohydroquinone in chlorobenzene. The disappearance rate of one of the initial reactants, N,N′-diphenyl-1,4-benzoquinonediimine, at the inflection point of its disappearance curve, is considered as the basic kinetic characteristic of the reaction. The empirical function y = aexp(bt ^c ) + d, where a, b, c, and d are the fitted parameters (b < 0, c > 1), is suggested for approximating the S-shaped kinetic curves and for calculating the reaction rate. The rate constants of the elementary steps are preferably derived from experimental data obtained at equal concentrations of the initial reactants, and also product additions when their effect on the reaction rate is studied. The effective rate constant of chain termination is derived from the time to reach the steady state. The results obtained in this way are compared with earlier data obtained using the "initial" reaction rates calculated by means of exponential approximation of portions of N,N′-diphenyl-1,4- benzoquinonediimine disappearance curves after the inflection point.

Full text of DOI:10.1134/S002315841005006X

ISSN 0023ꢀ1584, Kinetics and Catalysis, 2010, Vol. 51, No. 5, pp. 657–665. © Pleiades Publishing, Ltd., 2010.
Original Russian Text © V.T. Varlamov, S.Ya. Gadomsky, 2010, published in Kinetika i Kataliz, 2010, Vol. 51, No. 5, pp. 682–690.
Estimation of Kinetic Parameters of Reversible Chain Reactions
of Quinoneimines with Hydroquinones
Having SelfꢀAcceleration Periods
V. T. Varlamov and S. Ya. Gadomsky
Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia
eꢀmail: varlamov@icp.ac.ru
Received September 21, 2009
Abstract—A new approach is suggested for determining the kinetic parameters and rate constants of the eleꢀ
mentary steps of reversible chain reactions having selfꢀacceleration periods due to the long time required for
the concentrations of the chainꢀcarrier radicals to reach their steadyꢀstate values. This approach is illustrated
by the example of the reversible chain reaction between
dichlorohydroquinone in chlorobenzene. The disappearance rate of one of the initial reactants,
nylꢀ1,4ꢀbenzoquinonediimine, at the inflection point of its disappearance curve, is considered as the basic
N,N'ꢀdiphenylꢀ1,4ꢀbenzoquinonediimine and 2,5ꢀ
N,N'ꢀdipheꢀ
c
kinetic characteristic of the reaction. The empirical function y = aexp(bt ) +
d, where
a,
b,
c
, and
d
are the
fitted parameters ( < 0,
b
c > 1), is suggested for approximating the Sꢀshaped kinetic curves and for calculating
the reaction rate. The rate constants of the elementary steps are preferably derived from experimental data
obtained at equal concentrations of the initial reactants, and also product additions when their effect on the
reaction rate is studied. The effective rate constant of chain termination is derived from the time to reach the
steady state. The results obtained in this way are compared with earlier data obtained using the “initial” reacꢀ
tion rates calculated by means of exponential approximation of portions of
nediimine disappearance curves after the inflection point.
N,N'ꢀdiphenylꢀ1,4ꢀbenzoquinoꢀ
DOI: 10.1134/S002315841005006X
The reactions of quinoneimines with hydroquinoꢀ practice. These reactions are similar to the quinone +
nes proceed via a reversible chain mechanism both hydroquinone reactions, and the latter are of great sigꢀ
from the side of the initial compounds and from the nificance not only in chemistry, but also in biochemiꢀ
side of the products [1–4]. These reactions were the cal applications since they play the key role in the proꢀ
first liquidꢀphase reversible chain reactions to be disꢀ tective action of quinoneꢀtype bioantioxidants
covered. Only gasꢀphase reactions of this type had (ubiquinone, vitamin K group) [12]. Quinones and
been known earlier [5–7].
quinoneimines result from the oxidative degradation
of the most widespread phenolic (hydroquinoneꢀtype)
and aromatic amine antioxidants [12–16] in oxidation
reactions inhibited by these antioxidants. The reacꢀ
tions in which quinones and quinoneimines are
involved under these conditions exert a marked effect
on the total antioxidant efficiency of the individual
initial phenolic and amine antioxidants and on that of
their compositions [16, 17].
These reactions have a number of specific features.
They have no rateꢀlimiting chain propagation step [1,
3
, 8, 9]. It has recently been demonstrated by the
example of the reaction between 'ꢀdiphenylꢀ1,4ꢀ
N,N
benzoquinonediimine (QDI) and 2,5ꢀdichlorohydroꢀ
quinone (Ar(OH) ) in chlorobenzene [10] that the
2
reactions of quinoneimines with hydroquinones genꢀ
erally have an induction period. In the case of the
chain reaction between QDI and Ar(OH) (initial
2
4
5
chain length of
ν
= 10 –10 units), the induction
EXPERIMENTAL
period is 3–5 min, depending on the experimental
conditions [10, 11]. Determination of the rate conꢀ
stants of the elementary steps of a complex reaction
with a selfꢀacceleration period involves serious diffiꢀ
culties. Here, using the chain reaction between QDI
The synthesis and purification of QDI, Ar(OH)2,
N,N'ꢀdiphenylꢀ1,4ꢀphenylenediamine (H QDI), 2,5ꢀ
dichloroquinone (Q), tetraphenylhydrazine (TPH)
initiator) and chlorobenzene as the solvent were
described earlier [18]. Reaction kinetics was studied as
quinoneimine disappearance. Experiments were carꢀ
2
and Ar(OH) as an example, we suggest a new
2
approach to this problem.
Detailed investigation of the reversible chain reacꢀ ried out at
T = 298 ± ±0.1 in a temperatureꢀcontrolled
tions of quinoneimines with hydroquinones is stimuꢀ quartz bubbled cell built in a Specord UVꢀVIS specꢀ
lated by the high significance of these reactions in trophotometer (cell volume of 8.5 ml, optical path
657

6
58
VARLAMOV, GADOMSKY
[QDI]20
5
5
[
QDI]
×
10 , mol/l
×
10 , mol/l
9
8
.0
.4
6
3
8
1
1
8
.0
.6
2
3
2
3
7
0
500
1000
1500
tdelay, s
0
800
1600
Time, s
Fig. 1. QDI disappearance kinetics in the chain reaction
between QDI and Ar(OH)2 ) in the absence and ( ) in
(
1
2, 3
the presence of an initiator for different delay times (tdelay
)
Fig. 2. Initial concentration [QDI]20 as a function of the
delay time tdelay between combining the Ar(OH)2 solution
between combining the Ar(OH) solution with the initiaꢀ
2
tor and adding QDI. In all runs, [QDI] = [Ar(OH) ] =
0
2 0
with the initiator and the reaction onset (QDI addition).
–
5
11
9
1
.0
.41
×
×
10 mol/l.
w
= (
.
1
) 0 (uninitiated reaction) and
i
–5
–
–1 –1
In all runs, [QDI] = [Ar(OH) ] = 9.0 × 10 mol/l. wi ×
10
mol l
s
tdelay = ( ) 100 and ( ) 950 s. The
2
3
0 2 0
1011 = ( ) 2.11 mol l–1 s–1. Chloꢀ
1) 0.352, (2) 1.41, and (3
robenzene, 298 K, bubbling argon.
points represent experimental data, and the lines are the
fits to formula (4). Chlorobenzene, 298 K.
length of l = 2.0 cm, argon bubbling). The spectrophoꢀ data presented in Fig. 2 demonstrate that [QDI]20 iniꢀ
tometer was coupled to a computer. Absorbance was
measured at a certain absorption band of QDI in the
visible region, specifically, 22260, 20000, or
tially falls rapidly as tdelay in increased, but for tdelay
>
400–600 s this effect is not observed any longer. This is
due to the fact that, before the onset of the chain reacꢀ
tion, semiquinone radicals buildup in the solution
containing Ar(OH)2 and the initiator, their concentraꢀ
–1
1
9000 cm , depending on the initial QDI concentraꢀ
tion, at intervals not longer than 1 s. The reaction
product Q also absorbs, though weakly, in this region,
but its absorption was neglected because of its small
tion exceeding the steadyꢀstate concentration
•
[
Ar(OH)O ] for the chain reaction between QDI and
2
Q
2260
st
–1
–1
molar extinction coefficient (
ε
≈ 20 l mol cm
Ar(OH)2 (which starts after QDI addition) [11]. This
inference is consistent with the simulated kinetics of
this reaction [11].
2
2260
3
–1
–1
versus
ε
~ 7 ×10 l mol cm ).
QDI
For technical reasons (time needed to pour the
solution, to stir the contents of the reactor, to close the
cell compartment, etc.), we began recording the QDI
disappearance kinetics some time (on the average,
No such dependence of wQDI and [QDI]20 on tdelay
is observed when the QDI and initiator solutions are
initially combined in the reactor, the mixed solution is
held for the time tdelay, and the reaction is then started
2
0 3 s) after the addition of the second component
±
and the onset of the reaction. This time was taken into
account in the determination of induction periods.
by adding an Ar(OH) solution. For this reason, it is
2
this order of reactant mixing that was used in the invesꢀ
tigation of the reaction in the presence of the initiator,
It was established by specialꢀpurpose experiments
that the observed kinetic parameters of the reaction—
initial quinoneimine concentration [QDI]20 (meaꢀ unless otherwise specified.
sured after 20 s), reaction rate wQDI, and induction
Significant data distortion might arise from the
period —depend on the experimental procedure,
τ
capability of the semiquinone radicals to react with O2
including the order in which the reactants and initiator
were combined and the reactor charging time. If a
solution of Ar(OH)2 and the initiator is prepared in the
reactor and then a QDI solution is added after some
delay time (tdelay), the kinetics recorded immediately
after that will depend on tdelay. It is possibly due to this
specific feature that the QDI disappearance rate
[
19–23]. For this reason, argon was bubbled though
the solution both before and during the reaction.
RESULTS AND DISCUSSION
(wQDI) values measured in replica experiments differ
The reaction between QDI and Ar(OH)
2
proceeds
by a factor of 2–3 (Fig. 1, curves ). Likewise, the according to the stoichiometric equation
2, 3
KINETICS AND CATALYSIS Vol. 51
No. 5
2010

ESTIMATION OF KINETIC PARAMETERS OF REVERSIBLE CHAIN REACTIONS
Cl
659
N
N
+ HO
OH
Cl
Ar(OH)2
Cl
QDI
H
N
H
N
+ O
O
(s, –s)
Cl
H QDI
Q
2
In this reaction, QDI is reduced to H QDI owing to reversibility of the reaction can be neglected unless
2
the oxidation of Ar(OH)2 into Q. The equilibrium the products H QDI and Q are added. The mechaꢀ
2
constant of reaction (s, –s) in chlorobenzene at
nism of reaction (
s
, –
s
) includes the following eleꢀ
2
98
2
98 K is
K
= 92
±
13 [18]. In view of this, the mentary steps [1, 10]:
eq
Ar(OH)2
•
Ar(OH)O•
HQDI + Ar(OH)O
(
(
(
(
(
(
(
(
i
)
TPH
1, –1) QDI + Ar(OH)2
Ph N
2
initiation
•
•
chain initiation
•
•
2, –2) QDI + Ar(OH)O
•
HQDI + Q
H QDI + Ar(OH)O
chain propagation
•
3, –3) HQDI + Ar(OH)2
•
''
2
•
4, –4) HQDI + HQDI
•
H QDI + QDI
2
chain termination
•
•
–1, 1) HQDI + Ar(OH)O
•
QDI + Ar(OH)2
H QDI + Q
''
''
''
5, –5) HQDI + Ar(OH)O
2
•
•
6, –6) Ar(OH)O + Ar(OH)O
Ar(OH) + Q
2
Scheme.
Here, HQDI• and Ar(OH)O• are the 4ꢀanilinoꢀ
diphenylaminyl and 2,5ꢀdichloroꢀ4ꢀhydroxophenoxyl equal to the negative derivative of function (1a) at the
semiquinone radicals, which result from H QDI and point , at which it intersects the horizontal line indiꢀ
Ar(OH)2, respectively, via the abstraction of a hydroꢀ cating the preset initial concentration of the quinone
gen atom. diimine, [QDI] = [QDI]0
a2b2exp{–lna2 – ln([QDI] – c2)}.
The “initial” reaction rate wQDI was taken to be
τ
2
:
Figure 1 (curve
pearance kinetics is characterized by selfꢀacceleration
1
) indicates that the QDI disapꢀ
wQDI
=
0
The parameters of function (1a) were determined
periods. It was mentioned above that, in the earlier by fitting, to formula (1a), portions of the QDI disapꢀ
studied reactions of quinoneimines with hydroquinoꢀ pearance curve after the inflection point (medium and
nes [1, 3, 8, 9], these periods were negligibly short and large extents of the reaction). Obviously, this way or
it was possible to derive kinetic parameters from initial processing kinetic data could provide only estimates of
rate (wQDI) data. In order to determine wQDI, the the parameters.
quinone diimine disappearance curves were fitted to
the exponential function
In this work, the
to the empirical function
[QDI] =
exp(btc) +
where a, b, c, and are the parameters fitted by iteraꢀ
tions; < 0 > 1. Function (2) does not allow the iniꢀ
tial reaction rate to be determined, because
[QDI]/dt = 0, However, starting at fairly small
extents of the reaction, this function provides a good
fit to the observed curves up to the completion of the
reaction (Fig. 1). Table 1 lists a, b, c, and d values for a
Sꢀshaped kinetic curves are fitted
[
QDI] = a1exp(–b1t) + 1
b1, and c1 are the parameters fitted by iteraꢀ
tions; wQDI a b
c
,
(1)
a
d,
(2)
d
where a1
,
b
; c
=
.
1 1
In our earlier study [10], kinetic data for the reacꢀ
tion between QDI and Ar(OH)2, which has selfꢀaccelꢀ
eration periods, were fitted to a modification of funcꢀ
tion (1):
(d
)|
t
= 0
[QDI] = a2exp{–b2(t –
τ
)} + c2
.
(1a) number of initial reactant concentrations.
KINETICS AND CATALYSIS Vol. 51
No. 5
2010

6
60
VARLAMOV, GADOMSKY
Table 1. Numerical values of the parameters a, b, c, and d of function (2) in several runs
4
4
104 [Q]0
104
[QDI]0
×
10 [Ar(OH) ]
×
10 [H QDI]
2
×
×
104,
mol/l
105,
mol/l
2 0
0
a
×
d
×
4
–c
b
×
10 , s
c
mol/l
0
1
2
0
0
0
3
3
3
3
.9
0.9
1.8
2.7
0.9
0.9
0.9
3.6
3.6
3.6
3.6
0
0
0
0
0
0
0
0
0.70
1.47
1.82
0.50
0.36
0.36
±
±
±
±
±
±
±
±
±
±
0.01 –0.46
0.01 –0.38
0.01 –2.29
0.01 –11.6
0.01 –5.5
0.01 –5.85
0.01 –15.7
0.02 –32.5
0.01 –39.0
0.01 –53.2
±
±
±
±
±
±
±
±
±
±
0.01 1.41
0.01 1.51
±
±
±
±
±
±
±
±
±
±
0.01 2.23
0.01 4.00
0.01 5.8
0.01 4.28
0.02 5.2
±
±
±
±
±
±
±
±
±
±
0.01
.8
0.01
0.1
.7
.9
.9
.9
.6
.6
.6
.6
0
0.1
0.5
0.5
0.2
0.3
1.9
2.2
2.8
1.33
1.09
1.22
1.17
1.08
1.04
1.05
1.05
9
0.04
0.1
27
36
0
0.01 4.69
0.003 5.82
0.01
0.01
0.1
3.6 2.51
0
27
54
108
2.19
1.76
1.53
0.01
9.5
0
0.01 11.4
0.01 16.4
0.1
0
0.02
An advantage of function (2) is that its parameters
с
с−1
wQDI = −abcexp(bt ) × t
.
infl
(4)
infl
are derived almost from the entire kinetic curve,
including the initial portions of the curve before the
inflection point. The coordinated of the inflection
points are determined by differentiation:
To calculate [QDI] , we used Eq. (3), and the
infl
concentrations of the other compounds were calcuꢀ
lated from their initial concentrations taking into
account the [QDI]infl value and the stoichiometry of
the reaction.
Under the assumption that, at t = tinfl, the reaction
between QDI and Ar(OH) proceeds in the steadyꢀ
1
c
1 − c
bc
c
tinfl = exp ln
),
(
(3)
[
QDI]infl = aexp(bt ) + d.
2
infl
state regime, the expression for wQDI appears as
The QDI disappearance rate wQDI at the inflection
point of the curve defined by (2), with coordinates
given by Eq. (3), will be used as the kinetic characterꢀ
istic of the reaction. It was established experimentally
that the QDI concentration at the inflection point is
typically not lower than ~80% of [QDI]0. At comparaꢀ
bly large rate constants, this allows one to neglect the
effect of the products and to consider wQDI an anaꢀ
i 2
k1
[
QDI
]
[
Ar
(
OH 2 + 0.5w = k [HQDI ]
)
]
i 4
(5)
i 2
i
i
+
(
k + k
)
[HQDI ][Ar
(
OH
)
O ] + k [Ar
(
OH O ] .
)
−
1
5
6
Let us express [HQDI•] and [Ar(OH)O•] from the
equation
i
wQDI = k2
[
QDI
]
[Ar
(
OH
)
O ]
logue of the initial reaction rate at t = 0, but at the conꢀ
(6)
i
centrations corresponding to the inflection point. The
expression for wQDI at the inflection point is obtained
by differentiation of function (2):
=
k [HQDI ]
[
Ar
(
OH
)
]
3
2
and substitute them into Eq. (5). We will obtain
2
2
2
2
3
2
2
2
k
k
[
QDI] [Ar(OH)2
]
(
k
1
[
QDI][Ar(OH)2
]
+ 0.5wi
)
w
=
2,
(7)
QDI
2
2
k4k2
[
QDI
]
+ k k k Ar(OH)2][QDI
[
]
+ k6k3
[
Ar(OH)2
]
V
2 3
where kV
=
k–1
+
k5. The complexity of expression (7) under these conditions. At [QDI]infl = [Ar(OH) ]
,
2 infl
Eq. (7) can be written as
for wQDI is due to the fact that, on the rightꢀhand side
of Eq. (5), it is necessary to keep all of the four terms
because of the absence of a rateꢀlimiting chain propaꢀ
gation step.
k2
wQDI
=
2
2
1/ 2
{k (k /k ) + k (k /k ) + k }
4
2
3
V
2
3
6
1/2
2
infl
×
[QDI]infl k1[QDI]
+ 0.5wi
)
(8)
(
The expression for wQDI is much simpler in the case
of equal reactant concentrations [3, 4, 10]. For this
reason, most experiments in this study were performed
k2
1/ 2
=
[QDI] w .
infl iΣ
term
1/ 2
i)
(
2k
Ar(OH)O
KINETICS AND CATALYSIS Vol. 51
No. 5
2010

ESTIMATION OF KINETIC PARAMETERS OF REVERSIBLE CHAIN REACTIONS
661
w2
13
−1
−
1 2
7
−1 −1
×
10 , (mol l s )
wQDI × 10 , mol l s
QDI
3
2
6
3
2
1
2
1
0
3
1
0
0
0
0.5
1.0
QDI]inflwi
1.5
10 , mol l s
0.5
[QDI]infl × 10 , (mol l )
1.0
−1 2
2
18
3
−
3
−1
7
[
×
Fig. 4. wQDI versus squared [QDI]infl at equal reactant
concentrations: ( ) no products added, [QDI]0
Ar(OH)2]0; ( [QDI] = [Ar(OH) ] = [H QDI]0; (3)
2
and[QDI]2 wi
Fig. 3. Linear relationships between
w
QDI
infl
1
=
in the runs with equal initial reactant concentrations:
× 10 and (2) 1.8 ×
0⎯ mol/l. Chlorobenzene, 298 K, bubbling argon.
[
2)
0
2 0
2
–
5
[
QDI] = [Ar(OH) ] = (
1)
9.0
0
2 0
[QDI]0 = [Ar(OH) ] = [Q]0. Chlorobenzene, 298 K,
2 0
4
1
bubbling argon.
Here, it is taken into account that the total radical forꢀ
mation rate in the system is
According to Eq. (9), in the absence of an initiator,
1
1
/2
k k
2
2
wQDI
=
/2[QDI]infl,
(10)
2
1
wi = 2k1
[QDI]
Σ
+
wi,
term
infl
(k
•
)
Ar(OH)O
2
2
2
3
term
and k (k /k ) + k (k /k ) + k = k
•
is the effecꢀ
4
V
2
3
6
Ar(OH)O
that is, the reaction rate wQDI is proportional to the
product of the equal concentrations [QDI] and
tive rate constant of chain termination on the semiꢀ
quinone radical Ar(OH)O . Clearly, the equation for
•
[
Ar(OH)2] (Fig. 4). From the slope of straight line 1 in
wQDI at [QDI]infl = [Ar(OH) ] has the same form as
2 infl
Fig. 4, we find
the rate equation for nonbranched chain reactions
with quadraticꢀlaw chain termination and a rateꢀlimꢀ
iting chain propagation step [24, 25].
1
1
/ 2
k k
2
–1 –1
=
/ 2
5.0 ± 0.2 l mol s .
1
term
(k
•)
Ar(OH)O
Equation (8) can be represented as
From the above parameters, we obtain k1 = (9.7
±
2.6)
×
–5
–1 –1
2
2
2
1
0
l mol s , which is almost one order of magniꢀ
2
k k
4
infl
k
2
2
1
w
=
[QDI]
+
[QDI] w
infl i
QDI
term
term
tude larger than the earlier reported value of k1 = 1.1
×
k
•
2k
•
(9)
–5
–1 –1
Ar(OH)O
Ar(OH)O
10 l mol s [10].
2
infl
=
A + B ×[QDI] w .
In the absence of an initiator (
reduces to
w
i
= 0), formula (7)
i
Here, [QDI]infl stands for the equal QDI and Ar(OH)2
concentrations at the inflection point of the QDI disꢀ
appearance curve, calculated via Eq. (3). Since
on the rightꢀhand side of
Eq. (9) can be neglected because it is small as comꢀ
pared to the second term. In addition, this makes it
3
[QDI] [Ar(OH) ]
k4
infl
2 infl
=
2
2
w
k k
1 3
QDI
[
QDI]infl
Ӷ
1, the first term
А
(11)
2
kV [Ar(OH)2]infl
k ⎛[Ar(OH) ] ⎞
6
2 infl
+
+
.
2
⎜
⎟
⎠
k k k [QDI]infl
k k
⎝
[QDI]infl
1
2
3
1
2
possible to take
А
to be constant, so there must be a
2
2
Figure
5
plots
the
dependence
of
linear relationship between
w
and [QDI] w . As is
QDI
infl i
3
2
[
QDI] [Ar(OH) ] /w
on [Ar(OH) ] /[QDI]infl,
2 infl
demonstrated in Fig. 3, this inference is corroborated
infl
2 infl
QDI
which can be approximated by a straight line. Thereꢀ
fore, the last term on the rightꢀhand side of Eq. (11)
can be neglected. Because of the large error in the
determination of the ordinate intercept, only the
slopes of the straight lines (Fig. 5) were used in further
by experimental data. From the slope
lines in Fig. 3, we derive
B of the straight
1
/2
term
–1 –1 1/2
k k
= 500 ± 50 (l mol s ) .
2
(
•
)
Ar(OH)O
KINETICS AND CATALYSIS Vol. 51
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2010

6
62
VARLAMOV, GADOMSKY
3
2 infl QDI
2
−
1 2
1/(4 2
)
10 , s−
4
2
[
QDI] [Ar(OH) ] /
infl
w
, (mol s l )
τ
×
0
.12
.08
.04
6
4
2
0
1
2
0
0
0
2
Ar(OH) ] /[QDI]
4
infl
0
2
4
6
8
[
11
−1 −1
2
infl
wi × 10 , mol l s
Fig. 5. Fitting experimental data obtained in the absence of
an initiator to Eq, (11). Chlorobenzene, 298 K, bubbling
argon.
Fig. 6. 1/(4 2
tant concentration of [QDI] = [Ar(OH) ] = (
τ
)
versus wi for two series of runs at equal reacꢀ
9.0
1)
×
0
2 0
–5
–4
10 and (
2)
1.8 × 10 mol/l.
−
1/2; that is,
calculations. This made it possible to determine the wi
)
τ
= tch [24, 25]. The time τ will be estiꢀ
Σ
parameter
mated as the abscissa of the intersection point between
exp
kV
the horizontal line [QDI] = [QDI]0 and the tangent
line
–2
–1 2
=
(2.70
±
0.14)
×
10 (mol s l ) .
k k k
2 3
1
–1
–1
Setting kV to be
into account that k1 = (9.7
3
×
108 l mol s [10] and taking
[QDI] = Mt + N
10 l mol s (see for the QDI disappearance curve (2) at its inflection
above), we obtain, from this equality,
point, whose coordinates are given by Eq. (3). Here,
M = is derived from the equation
(12)
–5
–1 –1
±
2.6)
×
1
4
–1 –1 2
−
wQDI and N
k2k3 = (1.1
±
0.3) 10 (l mol s ) .
×
с
The k2 and k3 values were earlier derived by comꢀ
bining data obtained in the presence and in the
absence of an initiator [8–10]. Here, we suggest
another procedure for this purpose.
[QDI]infl = aexp(btinfl) + d = −wQDIτ + N,
whence it follows that
с
N = aexp(btinf l) + d + wQDIτ.
(13)
(14)
To determine k2, we will use the above value of Therefore,
term
exp
–1
–1 1/2
k2/(
k
•
)1/2 = 500
±
50 (l mol s ) . The necesꢀ
N −[QDI]0
Ar(OH)O
τ =
.
term
wQDI
sary quantitative estimate of
k
•
can be derived
Ar(OH)O
In view of this,
from the selfꢀacceleration period length as a function
of the initiation rate wi by performing a series of experꢀ
iments at equal QDI and Ar(OH)2 concentrations.
The selfꢀacceleration periods (Fig. 1) are largely due
to the long time required for the establishment of the
−
1/2
term
τ = t = 2k
i
wiΣ)
ch
(
Ar(OH)O
−
1/2
term
2
infl
=
(2k
×(2k [QDI] + wi))
i
1
Ar(OH)O
•
•
steadyꢀstate HQDI and Ar(OH)O concentrations in or
the system [10]. The concentrations of these radicals
are related by expression (6). It is, therefore, possible
to take into account only the variation of the concenꢀ
1
term
2
term
= k
2
τ
k
1
•
[QDI]infl + 0.5k w
•
.
(15)
i
Ar(OH)O
Ar(OH)O
4
•
Figure 6 illustrates the fit between the data of two
series of measurements at equal reactant concentraꢀ
tions and formula (15). From the slope of the straight
line in Fig. 6, we obtain
tration of one of them, e.g., Ar(OH)O . At the early
stages of the reaction without an initiator, immediately
after combining the reactants, the concentration of
this radical increases from 0 to its steadyꢀstate value
term
•
7
–1 –1
[
Ar(OH)O ]st, and this is accompanied by an increase
k
i
= (1.6
±
0.1)×10 l mol s .
Ar(OH)O
in the reaction rate. Assume that the time it takes for
term
From this
k
i
value and the above
•
the steadyꢀstate concentration [Ar(OH)O ] to be
Ar(OH)O
st
term
)1/2 value, we obtain
established,
τ
, is equal to the mean lifetime of the k2/(
k
i
Ar(OH)O
term
6
–1 –1
t
reaction chain in the steady state, ch
= (2
k
•
×
Ar(OH)O
k2 = (2.0
±
0.2)
×
10 l mol s .
KINETICS AND CATALYSIS Vol. 51
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2010

ESTIMATION OF KINETIC PARAMETERS OF REVERSIBLE CHAIN REACTIONS
663
term
lated rate constants k2, k3, kV, and k4 because of the
These
k
i
and k2 values differ from the earlier
Ar(OH)O
large error in the resulting estimate. We determined k6
by a new method based on the unsteadyꢀstate kinetics
of the initiated reaction between QDI and Ar(OH)2
term
–1
–1
determined values
k
i
= 3.9
107 l mol s and
–1
×
Ar(OH)O
–1
k2 = (9.7
±
3.4)
106 l mol s [10]. The causes of this
×
discrepancy were noted above. Firstly, the data
obtained in the previous study and in this work were
fitted to different functions—Eqs. (1) and (2). The
[
11]. For this purpose, an Ar(OH)2 solution with the
initiator was initially prepared in the reactor. This
second, most important cause is that, in the previous solution was held for 7–10 min under bubbling argon
kinetic study of the reaction in the presence of the iniꢀ (Fig. 2). Thereafter, we added QDI, and, after 20 s,
tiator, we attached no significance to the order of comꢀ
bining the reactants. Knowing k2 and k2k3, we can
obtain
began to record the QDI disappearance kinetics. The
k6 value was derived from the dependence of the decꢀ
rement of the quinoneimine concentration in a given
short time (t = 20 s) from the onset of the reaction
upon the addition of QDI (unsteadyꢀstate stage of the
reaction) on the initiation rate:
7
–1 –1 2
k3 = (5.5
±
0.3)
×
10 (l mol s ) .
Unfortunately, it seems impossible to estimate k6
term
from the
k
i
value obtained and from the calcuꢀ
Ar(OH)O
[
QDI]0
QDI]t
•
ln
= −k [Ar(OH)O ] t
2
st
[
term
i
(16)
4
k
i[Ar(OH)O ]stt
•
•
Ar(OH)O
•
•
k
([Ar(OH)O ] + [Ar(OH)O ] )e
− ([Ar(OH)O ] − [Ar(OH)O ] )
0 st
2
0
st
+
ln
,
term
•
2
k
•
2[Ar(OH)O ]
st
Ar(OH)O
where [Ar(OH)O ] = (wi/(2k6))1/2 and [Ar(OH)O ] = on the rate constants of these reactions. Nearly all of
•
•
0
st
the data reported in [27–29] were obtained in polar
solvents (water–alcohol mixtures, acetonitrile), while
our k6 value pertains to the reaction in chlorobenzene.
term
(wi/(2
k
i
))1/2
.
Ar(OH)O
By fitting the data of two series of experiments to
formula (16), we determined the rate constant of the
The rate constants of reverse reactions (–2), (–3,),
disproportionation of the 2,5ꢀdichlorosemiquinone (–4), and (–6) were determined by experiments in
6
–1 –1
radical: k6 = (3.0
±
0.5)
×
10 l mol s .
which the products were added to the reaction system.
The disproportionation rate constants of structurꢀ The addition of H QDI makes significant the contriꢀ
2
ally similar semiquinone radicals [27–29] are larger bution from reactions (–3) and (–4). In this case,
than this k6 value by one order of magnitude or more. based on the above scheme, we arrive at the following
This large difference is likely due to the solvent effect expression for wQDI [30]:
1
/2
2
2
2
3
2
3
⎛
k k [Ar(OH) ] [QDI] × (k [Ar(OH) ] + k [H QDI])
⎞
2
1
2
−4
2
wQDI
=
.
(17)
(18)
⎜
⎝
2
2
3
2
⎟
k (k [QDI] + k [H QDI]) + k k [Ar(OH) ](k [QDI] + k [H QDI]) + k k [Ar(OH) ] ⎠
4
2
−3
2
3
V
2
2
−3
2
6
2
In the presence of Q added, it is necessary to take into consideration reactions (–2) and (–6):
1
/2
2
2
2
3
3
2
⎛
k k [Ar(OH) ] [QDI] × (k [QDI] + k [Q])
⎞
2
1
−6
wQDI
=
.
⎜
⎝
2
2
2
⎟
k k [QDI] + k k [QDI](k− [Q] + k [Ar(OH) ]) + k (k− [Q] + k [Ar(OH) ]) ⎠
4 2 V 2 2 3 2 6 2 3 2
The concentrations of all compounds appearing in The points in Figs. 7 and 8 represent the experimental
Eqs. (17) and (18) refer to the inflection point of the data that were fitted to formula (17) or (18) by iterative
QDI consumption curve, and they should be calcuꢀ methods for calculating the rate constants k–2
, k–3, k–4,
lated using the [QDI]infl values determined via forꢀ and k–6 (Table 2).
mula (3) with the stoichiometry of the reaction taken
According to Eqs. (17) and (18), the reaction rates
into account. The dependences of wQDI on the addiꢀ at the wQDI inflection point at equal concentrations of
tion concentrations [H QDI] and [Q]infl in a series the three compounds are proportional to the squared
2
infl
of runs with preset initial reactant concentrations concentration. For example, it follows from Eq. (17)
QDI]0 and [Ar(OH)2]0 are shown in Figs. 7 and 8. that, at [QDI] = [Ar(OH) ] = [H QDI]infl,
[
infl
2 infl
2
KINETICS AND CATALYSIS Vol. 51
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2010

6
64
VARLAMOV, GADOMSKY
7
−
1
−
1
7
−1 −1
wQDI
×
10 , mol l s
wQDI × 10 , mol l s
2
8
2
3
2
4
1
1
1
0
0
0
0
2
4
6
4
8
[Q]infl 10 , mol l
3
−1
3
−1
[H QDI]
2
×
10 , mol l
infl
×
Fig. 7. wQDI as function of the
N,N'ꢀdiphenylꢀ1,4ꢀpheꢀ
nylenediamine concentration [H QDI]infl. The initial
2
reactant concentrations (
each particular experimental series: (
[
×
104, mol/l) are invariable in
[QDI]0
Ar(OH)2] = 0.90; (2) [QDI] = 0.90, [Ar(OH)2]0 = 1.80;
0
Fig. 8. wQDI as a function of the 2,5ꢀdichloroquinone conꢀ
centration [Q]infl. The initial reactant concentrations
1
)
=
104, mol/l) are invariable in each particular experimenꢀ
tal series: ( [QDI]0 [Ar(OH)2]0 = 0.90; ( ) [QDI] =
[Ar(OH) ] = 3.60. The points represent experimental
0
(×
(3
) [QDI] = [Ar(OH)2]0 = 1.80. The points represent
1
)
=
2
0
0
experimental data, and the lines represent the iterative calꢀ
culation via formula (17) at seriesꢀaverage [QDI]infl and
2
0
data, and the lines represent the iterative calculation via
104, mol/l) of (
1) 0.82, 0.82; (2)
formula (18) at seriesꢀaverage [QDI]infl and [Ar(OH) ]
2 infl
[Ar(OH)2]infl values (
×
1.74, 0.84; (3) 1.45, 1.45. Chlorobenzene, 298 K, bubbling
values (
Chlorobenzene, 298 K, bubbling argon.
× 1) 0.78, 0.78 and (2) 2.94, 2.94.
104, mol/l) of (
argon.
1
/2
QDI disappearance curve. However, the results of
experiments at equal initial concentrations lead to
⎛
k1 + k−4
⎞
wQDI = k2k
3
⎜
⎝
2
2
3
⎟
⎠
k
(k
+ k−
) + k
k
(k
+ k−
) + k
k
6
(19) quite reliable semiquantitative conclusions. Figure 4
4
2
3
3
V
2
3
2
2
infl
presents experimental data obtained at [QDI]
0
=
×
[QDI] = α[QDI]
.
[
Ar(OH) ] = [H QDI] (line ) and at [QDI] =
2
2
0
2
0
0
From Eq. (18) at [QDI]infl = [Ar(OH) ] = [Q] , it
2 infl
infl
[Ar(OH) ] = [Q] (line 3). In both cases, [QDI] calꢀ
2 0 0 infl
follows that
culated via formula (3) was used as the compound
concentrations at the inflection point. From the slopes
of the lines plotted in Fig. 4, we determine the followꢀ
1/2
⎛
k1 + k−6
(k− + k ) + k
⎞
wQDI = k2k3
⎜
⎝
2
2
2
⎟
k
k
4
+ k
k
2
(k−
+ k
) ⎠ (20) ing quantities (l mol s ):
–1 –1
2
V
2
3
6
2
3
2
infl
α
exp
= 6.85
±
0.58,
β
exp
= 6.35 ± 0.24.
×
[QDI] = β[QDI]
.
By substituting the rate constants determined in
this work into Eqs. (19) and (20) (without errors taken
into account), we obtain
Experimental verification of expressions (19) and
20) meets difficulties because it is difficult to establish
equivalent concentrations of the initial compounds
and reaction products at the inflection point of the
(
αcalc = 7.09,
βcalc = 6.04.
Table 2. Rate constants of reverse steps obtained by fitting experimental wQDI data as a function of the added product concenꢀ
tration to formulas (17) and (18)
k–3
×
10–5
k–4
×
104
k–2
×
10–6
k–6
×
105
Curve number
l mol–1 s–1
1
2
3
1
2
,
,
,
,
,
Fig. 7
Fig. 7
Fig. 7
Fig. 8
Fig. 8
7.5
9.5
9.2
±
±
±
4.5
1.4
2.9
1.8
1.2
2.1
±
1.3
0.2
0.7
–
–
–
–
±
±
–
–
±
±
–
–
±
–
–
±
4.5
8.9
6.7
2.5
7.2
4.7
8.9
6.0
7.5
6.1
3.9
4.3
±
±
Average
8.7
2.8
1.7
1.1
±
±
KINETICS AND CATALYSIS Vol. 51
No. 5
2010

ESTIMATION OF KINETIC PARAMETERS OF REVERSIBLE CHAIN REACTIONS
665
Clearly, the experimental and calculated data are in 10. Gadomskii, S.Ya. and Varlamov, V.T., Kinet. Katal.
,
2
009, vol. 50, no. 3, p. 368 [Kinet. Catal. (Engl. Transl.),
satisfactory agreement in spite of the error in the
QDI]infl values used in the determination of and
from the data presented in Fig. 4 and in spite of the 11. Gadomskii, S.Ya and Varlamov, V.T., Zh. Fiz. Khim.
vol. 50, no. 3, p. 348].
[
α
exp
β
exp
,
estimating character of the calculation of
and calc.
α
calc
2009, vol. 83, no. 10, p. 1888 [Russ. J. Phys. Chem. A
Engl. Transl.), vol. 83, no. 10, p. 1704].
(
β
1
2. Denisov, E.T. and Afanas’ev, I.B., Oxidation and Antioxꢀ
idants in Organic Chemistry and Biology, Boca Raton,
Fla.: Taylor & Francis, 2005.
Thus, using the reversible chain reaction between
N,N'ꢀdiphenylꢀ1,4ꢀbenzoquinonediimine and 2,5ꢀ
dichlorohydroquinone as an example, we have
described a new approach to determination of the rate
constant of elementary steps for chain reactions havꢀ
ing a selfꢀacceleration period due to the long time necꢀ
1
3. Pospisil, J., Polym. Degrad. Stab., 1991, vol. 34, p. 85.
4. Pospisil, J., Angew. Makromol. Chem., 1990, vols. 176–
1
177, p. 347.
essary for the reaction to come to the steady state. The
1
5. Roginskii, V.A., Fenol’nye antioksidanty: reaktsionnaya
aktivnost’ i effektivnost’ (Phenolic Antioxidants: Reacꢀ
tivity and Effectiveness), Moscow: Nauka, 1988.
c
empirical function y = aexp(bt ) +
d is suggested for
approximating the Sꢀshaped kinetic curves. This funcꢀ
tion provides a good fit to experimental data up to a
large extent of the reaction. The kinetic parameters of
the reaction (initial quinoneimine concentration,
reaction rate, induction period, calculated rate conꢀ
stants) depend strongly on the experimental proceꢀ
dure, including the order in which the reactants and
initiator are introduced into the system and the reactor
charging time.
1
6. Denisov, E.T. and Azatyan, V.V., Ingibirovanie tsepnykh
reaktsii (Inhibition of Chain Recations), Chernogolovka,
Moscow oblast: Inst. Khi. Fiz., 1997, chs. 3, 5.
17. Denisov, E.T., Usp. Khim., 1996, vol. 65, no. 6, p. 547.
18. Gadomskii, S.Ya. and Varlamov, V.T., Izv. Akad. Nauk,
Ser. Khim., 2007, no. 12, p. 2296.
19. Lisa, E. and Pospisil, J., J. Polym. Sci., Polym. Symp.
,
1973, no. 40, p. 209.
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KINETICS AND CATALYSIS Vol. 51
No. 5
2010