Table Of ContentDISPERSIVE KINETICS
Dispersive Kinetics
by
Andrzej Plonka
Institute of Applied Radiation Chemistry,
Technical University of Lodz. Po/and
Springer-Science+Business Media, B.V.
A c.I.p. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5754-9 ISBN 978-94-015-9658-9 (eBook)
DOI 10.1007/978-94-015-9658-9
Printed on acidjree paper
All Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint ofthe hardcover 1st edition 2001
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To my wife Ewa
Table of contents
1. Chapter 1. Introduction 1
2. Chapter 2. Phenomenological approach to dispersive kinetics 12
2.1 Kinetics in renewing environments 12
2.2 Temperature dependence ofrate parameters 44
2.3 Time-scale invariance of dispersive rate processes 52
3. Chapter 3. Approximation of c1assical kinetics 67
3.1 Viscosity dependence of specific reaction rates 69
3.2 Compensation law 71
4. Chapter 4. Kinetics in condensed media 76
4.1 Solvation 76
4.2 Reaction course in fluids 79
4.3 Fluid-solid transition 106
4.4 Reaction course in solids 118
4.5 Some heterogeneous systems 196
5. References 211
6. Subject index 233
Preface
The purpose of the book is to present to the wide audience of chemists, biologists,
physicists and material scientists the present status of the theory and the practice of
dispersive kinetics which is nowadays endemic in condensed media. The book id
written at the level of graduate students or specialized upper-year undergraduates.
Reactions by their very nature have to disturb reactivity distributions of the reactants in
condensed media, as the more reactive species are the first ones to disappear from the
system. The extent of this disturbance depends on the ratio of the rates of reactions to
the rate of system internal rearrangements (mixing) restoring the initial distribution in
reactivity of reactants. If the rate of internal rearrangements exceed markedly the rates
of chemical reactions, then the extent of disturbance is negligible and classical kinetics,
with a constant specific reaction rate, termed the reaction rate constant, may be valid. If,
however, the rates of chemical reaction exceed the rates of internal rearrangements, then
the initial distributions in reactant reactivity are not preserved during the course of
reactions. Many time scales coexist; specific reaction rate change in time, kinetics is
called dispersive.
The concept of energy profile along areaction path helps one to visualize the main
feature of the chemical reaction, including its mechanism. Along the reaction path, the
reactant are separated from the products by a potential energy barrier. Specific reaction
rates are related to thermally activated over-barrier transitions or to quantum mechanical
tunneling through the barrier. In classical kinetics, for a constant specific reaction rate a
single potential energy barrier is envisaged for the whole reaction course. In dispersive
kinetics, for time-dependent specific reaction rate, the potential energy barrier
separating the reactants from products has to evolve during the reaction course. The
evolution of energy barrier during the reaction course is described in terms of
distribution ftmctions for activation energy for the case of over-barrier transitions or in
terms of tunneling distances for the quantum-mechanical tunneling through the barrier.
These distribution ftmctions are related directly to the distribution ftmction of
logarithms of lifetimes calculable from the kinetic equations with time-dependent
specific reaction rate.
The above given phenomenological approach is compared with that in which the kinetic
equations with time-dependent specific reaction rates are interpreted in terms of
superposition of classical reaction patterns. Special attention is paid to renormalization
of rate coefficients following from the stochastic theory of renewals (structural
relaxation) in the reaction system. The phenomenological approach to kinetics is also
taken as a convenient ground to present a number of comprehensive models of
dispersive kinetics and to show what one gets directly from the experimental data. All
quoted examples are far to complete the broad field of dispersive kinetics and the author
fills it necessary to apologize to all whose contributions to the development of
dispersive kinetics were omitted or inadequately dealt with.
AP
Spring 2001
Chapter 1. lntroduction
To illustrate the thesis that the distribution in reactivity of reactants is only seen when
not preserved during the reaction course it seems highly instructive to consider the
simple example of two states of the reactants, Aland Ab giving the product P according
to the simple kinetic scheme
For this reaction scheme, the solution of a pair of simultaneous differential equations
(1.1)
- dA2 =(k +aA)A -M (1.2)
dt
2 2 1
is
where
(1.5)
and Yl and Y2 are negative roots ofthe algebraic equation
(1.6)
equal to
A. Plonka, Dispersive Kinetics
© Springer Science+Business Media Dordrecht 2001
2 Chapter 1
For
A(t) = AI (t) + Az (t) (1.9)
one gets
or
A(t) = A [17 exp( -rlt) + (1 -17) exp( -r zf)] (1.11)
where
(1.12)
For equal rates oftransition between states 1 and 2, i.e. for a = 1 , YI and rz reduce to rl'
and rz' given by
(1.13)
(1.14)
and 17 yields 17' equal to
(1.15)
Solution for this particular case was presented and discussed recently by Goldanskii et
al.l,2 It was shown that for A »kl - k2, in our notation,
(l.I6)
which follows from 17'~ 0 and r2'= 21 (kl + k2) under this condition. It was also shown
that for A ~O
lntroduction 3
A(t) = (A 1 2)[exp( -k,t) + exp( -k2t)] (1.17)
which results from 1]'= 1/2 and y,' = k, and y 2' = k2 in this case. For the more general
case, considered presently, of unequal rates of transitions between the states 1 and 2, the
solution was not shown. The authors restricted their attention to discussing the most
interesting special limiting cases corresponding to equations (1.16) and (1.17). Here
these special cases are shown to follow from the general solution, equation (1.11).
O~------------------------------------------------.
'<: ....
.... ,
.... ,
,
,
,
,
,
,
,
,
,
,
,
,
-2 ,
,
,
,
,
,
,
,
0.4,..---------------------" , , I.. = 1.0
,
,
,
,
,
,
,
0.3 ,
-4 ,
,
,
,
,
,
,
0.2 ,
,
,
,
,
,
,
0.1 '-o'- ---------::'-2:5: ,-----------5'0 .... .... .... ,
1..1 k2
-60~------------~--------~1----------------------~2
k2t
Fig. 1.1. Two-state kinetics. The solid curves depict the decay in time 01 A(t) according
to equation (1.11) lor k,/k2 = 10, a = 2, and;t equal to 0.1, I, and 10. The asymptotes,
equations (1.18) and (1.19) are depicted by dashed fines. The insert shows the changes
01 the mean lifetime lor A with ;t
The first one is when the relaxation rate is high as compared to the reaction rate, i.e.
;t » k" k2• Instant relaxation of the matrix to the stationary distribution of states
occurs. Under this condition 1] ~ 0 and Y2 ~ ak, 1(1 + a) + k2 1(1 + a), which yields,
cf. equation (1.11),
)t]
A(t)=Aexp[-( a k,+_I_k2 (1.18)
1+a 1+a
4 Chapter 1
i.e. the monoexponential decay with the specific reaction rate equal to the weighted
mean of specific reaction rates for state 1 and 2.
The second case is when relaxation is much slower than chemical reactions. No
relaxation of the matrix occurs during the time of conversion of the reagent. In our
notation A ~ 0 and from equations (1.7) and (1.8), respectively, YI = kl and Y2 = k2;
furthermore, from equation (1.12) 1] = a /(1 + a) wh ich yields, cf. equation (1.11),
A(t) = A [ ~ exp( -klt) + _1 exp( -ki )] (l.l9)
I+a I+a
i.e. the biexponential decay. For illustrative example see Figure 1.1.
0.8
I-
l0J..... E-
:2
X
0.4
log (barrier f1uctuation rate)
Fig. 1.2. Mean first passage time, MFPT in dimensionless units, versus the
dimensionless barrier jluetuation rate, on log seale, for E+ = 8T and E. = 0, where T
denotes the temperature. Solid fine ealeulated from the model, diserete data depiet
Monte Carlo simulations (20" error bars). Inset illustrates the setup for the problem: the
center height of the triangular barrier jluetuates between E+ and K at rate y. Adapted
from Doering and Gadoua6
The idea that partieles can escape from potential weHs is so familiar3,4 that the
complexity of attempts to broaden the problem is a surprise.5 Doering and Gadoua6 and
Zureher and Doering7 considered the problem of crossing a thermally activated potential
barrier in the presence of fluctuations of the barrier itself. For a piecewise barrier,
switching between two values as a Markov process, exact and Monte Carlo results
