Stochastic Approach to Reaction
E. A. Boucher University of Sussex Brighton BNI 9QJ, England
and P~YS~CO-Chemical Kinetics
The familiar introductory kind of treatment of chemical kinetics, often starting with simple reactions and involving the concepts of reaction order and molecularity, can be called the deterministic approach, to distinguish it from the stochastic approach which is based on the probability of prescribed events occurring. Stochastic processes (the mathematical abstraction of physical processes) are by no means limited in applicability to chemistry (reactions or physico-chemical changes such as nucleation), but can be used in the wide field of applied probability ( I , 2), e.g., for the description of genetics, for the time-dependence of epidemics, and for Brownian knotion and sedimentation. In chemistry even, the applicability is not limited to kinetics. Finding the probability that a family name (surname) will survive, where, to simplify the problem, a generation is taken as a fixed (discrete) interval, is a classical problem which has many features in common with branching processes in nuclear react~onsor with gel formation. One of the reasons for studying the kinetics of a chemical reaction is to be able to confirm a reaction mechanism. It is usual to predict the time-dependence of extent and rate of reaction by expressing in mathematical terms the consequences of a supposed mechanism. The main concern is therefore ultimately with the mechanism, and not with the time scale (e.g., not with whether 50% reaction occurs in 1 s or 1 hr). With physico-chemical phenomena the kinetics will not necessarily reflect the behavior of chemical reactions, but may describe situations where intermolecular forces are physical rather than chemical in nature. It sometimes becomes clearer to see the basic kinetic method when the process is physico-chemical (e.g., kinetics of critical nucleus formation) rather than a chemical reaction. The purpose of this article is not to give a comprehensive account, but only to highlight the stochastic approach-to show some of its scope compared to that of the deterministic approach, and to point out problems for which the stochastic a ~ ~ r o a cishmost suited. and indeed for which other mefhods are often not applicable. This article and the references cited should therefore provide a source of teaching material to complement that in more orthodox courses centered around elementary reactions, and transition state and collision theories. Elementary Reactions Deterministic Approach Suppose a reaction is represented by the irreversible change A+B A student with very little knowledge of kinetics, when asked to write down some kinetic equations for this reac. tion, would write something like d[Al/dt
=
-h[A]
d[B]/dt
=
h[A]
("
might even go on to give some initial conditions, solve these equations for the time dependence of [A] and [B], and even say that the reaction is first-order. The symbolism of differentiation, of [ 1, and the use of signs would 580
/ Journal of Chemical Education
readily be taken as understood. One can well imagine that few would note, let alone question, the nature of A or the B, i.e., whether it is cheminature of the conversion A cal or physical. By "the nature of A one could mean, whether i t is a liter of gas, or a crystal, or whether-in some state-it is only ten molecules. The approach to the kinetics of the conversion as envisaged here (with detailed specifications of species and nomenclature) is called the deterministic approach. From i t [B] [A]& - e-"1 (3) [A] The change in time of the reaction given by these equations would normally be expected to be shown by a real system obeying the same broadly supposed mechanism (mode of conversion) within the limits of observation: they give an auerage behavior. The deterministic approach would not be applicable to a small number of molecules where fluctuations will be important. It is possible to look on the same conversion in a different way, and so learn more about the basic meaning of the ensuing kinetic equations. (The novice must accept the new jargon and symbolism-here kept to a minimum with sacrifice of rigor-and work through the simple but unfamiliar equations, because once he or she has the idea, much more complicated cases can be handled.) +
-
-
Stochastic Approach Let the A-type molecules in reaction (1) undergo some change with time to B, i.e., a t any time t there will be some A-type molecules and some B-type molecules; whereas, for convenience, a t t = 0 there were only A-type present.
A B Already there is more feeling for the Pact that individual molecules of type A are converted to those of type B, and that it is not a magical process without molecular foundation. The nature of the actual molecular changes which might be involved can also he subjected to an analysis using the stochastic approach, as indicated in the concluding section below. The problem can now be set out with three basic assumptions (the analysis is based on the treatment by Bartholomay (3) and by McQuarrie (4)). (1) The probability that one of the A-type molecules is converted to a B-type molecule in the time interval between t and t + At is kA t + o(A ti; i.e., it is proportional to the small time interval since o(A t)/At- 0 as At- 0. (2) At is small enough so that only one molecule is converted in
that interval: i.e.. "exactlv simultaneous" conversion of more than one mol&le is not considered. (3) BY using- rather than t,we imply that the ~ n b a b i l i of t ~a B~typemolecule becoming an A-type molecule is zero. The number of A-type molecules present at time t isXA. Let Px(t) be the probability there are exactly x molecules of type-A present a t time t; i.e., P,(t) is the probability that X, = x. An equation can be written for Pz(t + At), the probability that there are x A-type molecules present a t t + A t
P,(t
+ At) = k(x + 1)AtP,+,(t) + (1 - kxAt)P,(t)
(4)
The first term on the right-hand side signifies that the probability that one of (x + 1) A-type molecules becomes a B-type molecule in At is proportional to (x + I), to At, and to the prohability that a t t there are (x + 1) A-type molecules present. The prohability that if there are x molecules present a t t, probability P,(t), then none are converted in At is (1- kxA t). Now P,(t At) - P,(t) + dP, (5) I n At dt A,-"
+
and so by rearranging eqn. (4) and dividing by At, in the limit Initially, t = 0 and let x = xo, and P,(x > xo) = 0 always. The first equation (boundary condition) is dP". P,,,(O) = 1 (7) dt = -k,.P,, Equation (6) represents a set of linear first-order differential-difference equations for integer x (= XO,xo - 1, xo - 2, . . . 2, 1, 0). Sequential solution starting by substituting eqn. (8) into (6)does not usually lead to a general form of the solution, and generating functions (see Appendix)are often used to get the solution. The required solution is -
(8)
PJt)
= eCx'*f
For the whole range of x values the P,(t) values constitute a probabiiity distribution. The mean value m of the number of A-type molecules present a t time t is often called the expected value E[XA] -see Appendix E[XJ
= .zoechz
(9)
and just as in any other random process (random variable XA), there will be scatter about the mean which could be given by the standard deviation a, hut which is often expressed as the variance (Appendix)
Whereas the deterministic approach gives only a continuous function of t, the stochastic approach uses a discrete random variable XA for the number of A-type molecules. We do not get the variance by the deterministic approach, but we can say that in this example the stochastic and the deterministic results agree in the mean. The stochastic approach is particularly useful for prohlems related to polymers (synthetic and biological, including enzymatic reactions), and to colloids and surfaces. A problem is solved here to which the deterministic method is not applicable. Marvel (5) and Flory (6)drew attention to the reaction between zinc dust and poly(viny1 chloride) in boiling dioxan solution (101.5"C). which. was supposed to eliminate pairs of chlorine atoms from 1,3-positions (provided the polymer exists in the head-to-tail arrangement) giving cyclopropane units, and, if the reaction were "random," a proportion e-2 ( ~ 1 3 . 5 3 % )of the chlorine atoms would eventually become isolated. While there is in fact about this amount of chlorine isolated, there is now considerable evidence (7) that the reaction is not one of simple abstraction of chlorine neighbors (and should no longer be cited as a simple pairing reaction showing a "statistical effect"): the approximate agreement between the limit of reaction (8) and the theoretical value seems to be fortuitous. Nonetheless, pairing between neighboring substituents (pendant groups) may well be applicable to some reactions. either in the solid state or in solution (8-
10) e.g., to anhydride formation on heating poly(acrylic acids), and is here analyzed in model form. With circles representing suhstituents, a portion of a reacted molecule might be 3 .
-
0
0
0
- 0
0
The reaction is conveniently described in 'terms of the number N, of sequences containing x unreacted members (8). Initially there are M chains each containing m members (suhstituents), i.e., at t = 0, Nm = . M . After a time t has elapsed, the mixture of chains will have reacted in various ways giving a range of N, for the system, which describes the average behavior. It is possible to describe the entire system in terms of the probability that potential (unreacted nearest neighbors) pairs will react. The initial M chains can undergo reaction at any one of the m - 1 pairing positions, and so such chains are destroyed and cannot be created. Without going into lengthy justification, hut by assuming that only one pairing event occurs in dt, and by assuming that kdt is the probability that reaction occurs between a hitherto unreacted pair in the interval between t and t d t dN, = -k(m - l)N,df (11)
+
giving
,'
m
=
Me*m-"*'
(12)
A sequence of x = m - 2 unreacted members is formed from one of m members when the first pairing occurs a t one of the ends. A sequence of m - 2 will he lost by pairing a t any one of the m - 3 potential pairing sites; so the net rate of formation of the Nm - 2 sequences is
and more generally
Obviously N,,, - = 0 always, and by substituting eqn. (12) into eqn. (131, the solution for Nm - 2 is found, and so by eqn. (14) sequential solutions can be found for all N,. This is very laborious and i t is more convenient to simplify the problem and proceed to a solution as follows (8). The time variable can be written as a = exp(-kt) (15) since it occurs most often in this form. If we write N, and use a = m
=
(16)
a"-"Q,,M
- x, the set of eqns. (14) reduces to
and
The initial conditions can now he written as or = 1, Qo = 1, Q, = 0 for z > 1. A generating function Q(y,a) for the Q, can he defined by
which, from the difference dQr/da eqn. (1'0, must satisfy
-
adQ,-
I/&
by
subject to the initial condition Q(y,l) = 0. The solution of eqn. (20) is
Volume 51, Number 9, September 1974
/
581
This does not seem straightforward to expand, but by picking out the first few terms (coefficients of y), i t can he shown that Q(y,a) factorizes to the form
and by summing to m in eqn. (34)
I and so separate generating functions f(y) and gfy) can he defined by
~
)
~
e
=
- all
(39)
The rate (i--dfldt) of reaction is given by
ilk
(24)
-
" 2aexp[-2(1
- a)]
(1 - EXln(1
- 5) +
(25) ~
-
lim(N,/mM) m-
(40)
21
(41)
-- (1 - a)'
exp[-XI - a)]
(42)
and so a t the end of the reaction (ol = 0)
(26)
y
Another interesting aspect of generating functions now arises. By expanding eqn. (21) to get the first few coefficients of y, it can be shown that the Q2 are polynomials of the form
Q,
exp[-2(l
The number of isolated (unreactable) substituents is given by Nl/mM, and for long molecules
whence by inspection of eqn. (21) Q(y,o) = g(ay)f(y) and f(y) = (1 - y)-Ze-2r g(y) = (1 -
= 1-
Cf,igd
(27)
i-0
The appropriate f f and g's are thus paired off in different ways for each value of z; i.e., they are numhers which can be obtained separately. If the generating functions ffy) and gfy) given by eqns. (25) and (26) are expanded as powers of y, the coefficients of y are sequences (the f, and the gt), and the members of such sequences may generally satisfy a recurrence (difference) formula. In the present examples, expansions of f(y) and gfy) show that fo = 1, fi =Oandforz>2
as found by Flory. The build-up of isolated suhstituents as the reaction proceeds is shown in Figure 1 together with the build-up and decay in the numhers of sequences of two unreacted members. Note that by eqn. (38), N,+ IIN, = u . The dependence of extent of reaction on reduced time k t is shown in Figure 2, and the rate Elk of reaction as a function of f is compared with overall first-order behavior in Figure 3. While some structural information about the product is given by the N,, the composition of individual chains remains unknown since N,/mM only gives the av-
and with go = 1, g~ = 0
For example Qz =
f2g0+ f l g l a
+
f&e2 (32)
and so form = 5, say, from eqn. (19) NJmM = aY1 - a )
Figure 1. Dependence of
N, and Nz on a = exp ( - k t ) for pairing (m
ml.
-
(33) In this way, for any m the appropriate Qz can be used to get all the N, for 1 5 x 5 m - 2, and with N m - 1 = 0 and N, by eqn. (12). the entire system is described. The fractional extent of reaction can he defined as the total proportion of the reacted suhstituents
-
When m is large, asymptotic relations accurate in the limit m m can be used for the N,. The reader might like to verify that for large z (35) f a D ( 2 + 3)ec2 that the number of suhstituents ultimately isolated is N , / M = Q,_,(O) = f,-, and that, also for larger Q, .- L(z 3x1 - a)% 2aZ(1- a)] exp[-2(l
+
Noting that z = m 582
/
+
- x , by eqn. (16)
Journal of Chemical Education
(36)
- a)] (37)
Figure 2. 6pendence of extent dom pairing.
€ of reaction
on reduced time kt for ran-
and to see the application to a reaction which entails dissociation of a molecule acting as a harmonic oscillator of N levels, such that the dissociation energy is the energy required to reach the (N+ 11th level E,,,,, = ( N
+
1)hv
(48)
Colloids and Surfaces
Figure 3. Dependence of rate f / k of reaction on extent ( 01 reaction for pairing 11) and for overall first-order reaction IF).
eraee value per substituent present. Some polymer properties, e.g., s~ectroscopic,may depend on the distribution of sequences of reacted pairs, but this distribution cannot be fo&d from the present treatment. An extension of this model can be made to incorporate the so-called neighboring-group effect (10) by postulating that the probability that reaction of a particular pair [or any other number of adjacent members including ones (11)) will occur in d t will be kodt, k ~ d or t kzdt depending. on whether zero, one, or two of the neighboring pairs have already formed. As a quick exercise the reader might like to use the same technique for the reaction of singletons (without neighboring-group effect), and so get the distribution of unreacted substituents, the extent of reaction (as a check on the obvious answer), and to compare results for finite and infinite m. To contain the length of this article, while demonstratine the scope of a~olicabilitv of the stochastic approach. .. several other topics will now be briefly surveyed.
-
Miscellaneous Topics
-
The Molecular Basis of Reactions
B, treated stochastically above, In the conversion A there was no direct interest in what changes to a molecule of type A caused it to become one of type B. The stochastic approach is capable, however, of dealing with changes in energy levels associated with molecules and consequently with the actual molecular transformation, as perhaps illustrated most easily with reference to oscillators. Considerable interest in this kind of problem was generated by Montroll and Shuler (12, 13), although the method of approach used here is mainly based on that of Widom .114.. 15). . For convenience.. onlv " transitions between nearest-neighbor energy states are permissible. Molecular states (ex.. corres~ondineto oscillator levels) are indexed i and so that krj-is the probability per unit time that the state i of a molecule will change to J, the rate of change in the number of i-molecules being dn,/dt = Zh,,n, - k,,n,)
The stochastic approach is readily applicable to several aspects of a pbysico-chemical nature associated with adsorption a t surfaces (interfaces), and to the behavior of colloidal particles [i.e., dispersed particles in the size range 10-10,000A ( l ~ m ) ] . Nucleation is the formation of a small amount of new phase in the parent phase, as in the formation of ice from liquid water, or of liquid droplets from vapor; i.e., the new phase does not suddenly appear spontaneously and completely, but rather i t grows from locations of nuclei which for many systems can be represented as a process of cluster (embryo) formation by addition of one molecule A a t a time according to the scheme
A,_, A,
+A
# A,
+ A * A,,,
It suffices for the present purposes to point out that although the free energy change associated with the conversion of old bulk phase to new bulk phase is favorahle, the formation of the small clusters, often referred to as nuclei or embryos, is accompanied by the production of an interfacial tension or surface free energy (16). Since the surface to volume (of new phase) ratio is large, the production of interface outweighs the production of new bulk phase and a maximum occurs in a plot of free energy change AG against new-phase particle size (signified by particle radius r ) as shown in Figure 4 [based on eqn. (1) of ref. (1711. The rate of nucleation is largely dependent on the height AG* of the maximum, i t being the rate a t which the clus-
(44)
If some states are product states denoted by P and others R are reactant states, then the total number of molecules will be P + R where each depends on t. Except for an initial transient period where k i and kb are phenomenological rate constants for the forward and the reverse reactions; kr/kb is the equilibrium constant K K = (P/R),-.. (46) and h,
+
k b = -A
(47)
is the fundamental relaxation rate. The reader might like
to examine progress that can be made by this approach, including the interpretation of features which arise U5),
Figure 4. Dependence of free energy AG of embryo formation on embryo radius in nucleation.
Volume 51, Number 9, September 1974
/
583
ters of m memhers (i.e., the critical embryos of size r*) are converted to clusters of (m 1)memhers. In a sense the growing embryo can he called a colloidal particle although one normally thinks of colloidal particles a s being of, say, silver iodide dispersed in water, of polymer latex particles prepared by emulsion polymerization, or of a liquid droplet produced from its vapor. The pmhlem can thus he presented more generally a s the growth of a particle by some "reaction" occurring a t the interface between the particle and its surrounding source of material. Goodrich (18, 19) has used the stochastic approach to describe the irreversible growth of a particle (by what is known mathematically as a pure hirth process), and the reversible growth (by what, naturally, is known as a hirth and death process). Processes of diffusion, Brownian motion, and sedimentation (20-21) can also he classified under the heading of colloid-related phenomena.
+
things from a population xo, uiz., x a ! / ( q - x ) ! x ! . Incidentally, the($),or "C, are coefficients of y r in the expansion of (1 + y P : i.e., the binomial function is the generating function for the is customary in mathematical language to call the mean value of a random variable, here XA,the expected value, which by definition is
+
+ 3P:, +
ax,] = P, ZP, which it is easily verified to be given by
...
(53)
The variancedor D4XA]is by definition Dz[XJ = E[XA~]- [E(XJP
(55)
where
Appendix
Generating functions have several uses, and it is not anticipated that the reader will be familiar with any of them. Let us define a funetion F(y, t) by the series expansion F(y,t) = P,y
+ P,yZ + ... + P,yx + . . + P,,yr" +
It can be shown that
... Literature Cited
The various P,(t) are therefore the coefficients of y; in this equation. P. for x > xo is zero permitting the sum to be taken to m. The object is to find an analytic expression F(y,t) called the generating function of the P, which when expanded gives the P, as coefficients. The method is demonstrated using the P, obeying the differential-differenceeqn. (6). It is easily shown that
and that the solution of this equation, with the initial condition F(y,O) =Yo,is F(y,t) = [I
+ ( y - l)eci'1'.
(51)
The right-hand side can be expanded to give the binomial series, the coefficient ofy; in which isP,, which can be shown to be
where(:)
584
denotes the number of different ways of picking r
/ Journal of Chemical Education
(1) Bharuehs-Reid, A. T., '"ElemenM of the Thwry of Markov Rote-s end Their Applications," McGraw.Hill Baok Co.. N w Y a k , 1960. I21 Takaer. L.. "Stochsstie Prncessos: R0blems and Solutions." Methue" and Comoa" ~ ~ l m i t olandon, d, 1960, and Seiena Paperback.. 1966.' (31 Bartho1omay.A. F., Bull. Math. Biophys.. 20.175 (19581. (41 MeQuanie, D. A . J A w l . Prob.. 4.413 (19671:reprinted aa"Stochastic Approachto Chemical Kinetics: MethuenReviewSenesin Appl. Pmb. Vol. S (19671. (51 Msrvel. C. 8.. Sample, J. H.. and Roy. M. F., J Amer Chem. Soc., 61. 3241
\.""",. ,,wm,
161 Plorv. P. J.. J. Amrr. Chem. Soc.. 61.1518 119391 i ~~ep;iekiaR j M., n o , P. Q., and Guyot, A . , ' E U ~ ; .poivm. J.. 5,795 (19691. IS1 Banon. T . H . K.,andBoucher,E.A., Traw Faradoysor.. 65,3301 (1969l. (91 Banon, T. H. K., and Boucher. E. A,. 7hm Faradoy Soc., 66.2320 119701, (101 Boucher. E. A,.J.C.S. Farador Trans.. 168.2281 119721. (111 Bouchez, E.A..J.C.S. ~ w o d o ~ T m w168,2295119721. .. (12) Montroll. E. W.sndShuler. K..Aduon Cham. Phys., 1.361 (195S). (13) Monlmll. E. W. "Encqeties in Metallurgical Phenomens." (Editor: Gordon and Breach), Val.3. New Yark, 1961. p 121. 1141 Widom.B..Scienre. 148.1555119651. widom; B.:J c h & piy6.. 55; 44 (IWI). (161 Dunning, W. J., "Nucleation," (Editor: Zettlemoyer, A. C.I. Chap. 1, Maral Dek. her hc., NevYork, 1669.
iw
