RUTHERFORD A R E
This paper conducts a tour
through a maze that links the divergent fields
of kinetics and mathematics
Mathematical
Aspects of
CHEMICAL REACTION
t might be useful to introduce the term “formal chemkinetics” to denote that segment of natural
philosophy that concerns itself with those aspects of
chemical kinetics which are independent of the specific
nature of the substances taking part in a reaction. T o do
so is not an attempt to canonize the notorious A+B or
the ubiquitous Z a Z j A , = 0, but is a recognition that
the fundamental principles will become much clearer
and important distinctions can be sharpened, when the
subject is treated more abstractly. Such an abstraction
has proved its worth repeatedly in diverse branches of
science, as, for example, when the distinction among
kinematical notions, dynamical laws, and constitutive
relations allows the structure of fluid mechanics to be
perceived.
In an introduction to a recent ACS symposium of the
above title ( I ) , I used Kuhn’s term “paradigm” ( 2 ) .
This has been questioned (3) on the grounds that the
subject has not yet assumed that framework of accepted
ideas which a t once commands the attention of a sufficient number of workers and is sufficiently openended to leave them a number of problems to resolve.
That there are quite a few people working in this area
was clear from the interest in the symposium, and that
there are any number of interesting problems to be
solved has never been in doubt, but the relationship
among the various ideas is well worth inquiring into,
for what emerged at that meeting was an impression of
the intricate web of cross-connections among the ideas of
various scholars pursuing their studies quite independently.
This essay is not a comprehensive review of the kind
that this journal so usefully provides on various chemical engineering topics, though it will attempt to direct
attention to as many references as possible. Nor is it a
report on the symposium held a t the Illinois Institute
of Technology, though it will refer to the papers given
there, but it will be convenient to substantiate the claim
I ical
that there is a fabric of interconnections by attempting
to chart those papers (4-16) in the following diagram.
Such a diagram can only roughly indicate the relationships that we are trying to discern for these are
naturally more subtle and tenuous than a block diagram.
To take an illustration, the concept of reaction mechanism and the associated notion of the pseudo-steady
state is a pervasive one. Seller’s combinatorial approach
Figure 1. Some interconnectiom among the papers of the 35th ACS
December Symposium
VOL. 61
NO. 6 J U N E 1 9 6 9
17
(6, 78) provides a way of relating the mechanism to the
overall reaction and of describing the complete set of independent mechanisms. I t is algebraic and does not
take directly into account the analytical aspects that
come in when the stoichiometry or kinematics of a
reaction system is invested with kinetic laws or a dynamics. The motivation behind the analysis of lumping by Wei and Kuo (77, 79, 79a) is similar to that of
the kineticist in respect of the desirability of simplification of complex systems, and it clearly has a bearing on
the notion of mechanism. Their interest, however, was
first to discover the conditions under which a lumped
system would remain monomolecular, whereas a reaction mechanism is almost always an example of what
they would term improper lumping-z.e., it leads to a
totally different kinetic expression.
Zimmerman’s presentation of the kinetics of replicating
macromolecules (72,20,27) paid some attention to the
mechanism though it was primarily concerned with a
very beautiful analysis involving stochastic and combinatorial ideas and the solution of the differential equations, as was Gavalas’ example of an autocatalytic
reaction (9) though it principally featured the elegant
methods of nonlinear differential operators. Similarly,
Bartholomay’s valuable exposition of stochastic formulations of kinetics ( 4 ) )necessarily made passing reference
to mechanism though it has had his more direct attention
elsewhere (22). I t was in Silveston’s (75) and Higgins’
(76) works that the whole question of the pseudo-steadystate hypothesis was raised most directly. I n the first case
by direct calculations intended to check out some criteria
proposed for its validity; in the second, by showing the
degree to which assumption would break down under
certain frequencies of perturbation. Again, Higgins’
paper was related to the notion of lumping, and an
extremely complex biological pathway was broken down
into a smaller number of lumped steps.
The notion of a reaction mechanism could scarcely
fail to have a prominent role to play in any discussion
of chemical reaction, but to illustrate the role of a less
obvious concept we may turn to that of fading memory.
In the mechanics of materials, it is now commonplace
to recognize that the present state of stress can be a
functional of the whole past history of strain. The
mixtures considered by Bowen (5) [extending his earlier
work (23)]and by Coleman (8, 24) are of materials
with this memory. But Othmer and Scriven (73), in
discussing the interaction of diffusion and reaction in
surfaces, pointed out that at the limits of their analysis a
hereditary or memory effect could not be ignored.
Halsey (25) has used the phrase “short term memory”
to describe nonequilibrium adsorption in catalysis, and
it has been shown (26) that the same assumptions that
justify the pseudo-steady-state hypothesis will obliviate
the memory in a reaction rate functional. Here again
is a link back to the notion of mechanism.
With these indications of the intricate weave that we
must expect to see in the fabric of formal chemical
kinetics, let us attempt to sketch a more systematic
cartoon.
18
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
Axiornatics
Risible though it might appear to the experimental
chemist, a strictly deductive theory of chemical kinetics
is a worthy goal. Bunge has outlined this program for
various branches of physics (27), and Basri has actually
carried it out in detail for the fundamentally important
theory of space and time (28). ’IVe are still a good deal
farther away from this goal in chemical subjects than in
physical, for chemistry has, on the whole, been much
more resistant than physics to the incursions of the
philosophy of science. The only attempt at axiomatization of kinetics is that of Wei (29), who presents
the following simple and attractive set for a closed reaction system:
1. The total mass of the mixture is conserved
2. The masses of each species are never negative
3. T h e rates of change of each mass are smooth functions of the masses
4. The principle of microscopic reversibility obtains
5. There exists an appropriate Liapunov function
assuring the stability of a unique equilibrium point
I t may well be true that “Tot homines, quot propositiones per se notae, et de gustibus non est disputandum,”
but it is interesting to inquire how primitive and complete this set may be. Thus, if we accept the continuum
equations of continuity of distinguishable species we
can replace item 1 above by two propositions which
might be taken to be axiomatic :
l a . Every property of the mean motion of a mixture is
a mathematical consequence of the properties of
the motions of its constituents
l b . If all effects of diffusion are taken into account
properly, the equations for the mean motion are
the same as those governing the motion of a simple
medium
From these two postulates it follows as a theorem that
if the mean motion of a heterogeneous medium is to
satisfy the ordinary equation of continuity, then it is
necessary and sufficient that the sum of the rates of
change of the masses of the several components should
be zero a t each point [cf. Truesdell (30)]. Thus, two
propositions can be made to yield Wei’s first axiom as a
theorem, though whether they are more reasonably
“per se notae,” as the Doctor Communis would have
put it, is open to discussion. O n the other hand, if we
start with a proposition of structure, that the molecular
species may be uniquely defined in terms of constituent
atomic species, we may add a conservation law, that
AUTHOR Rutherford Aris is Professor of Chemical Engineer-
ing at the University of Minnesota, Minneapolis, Minn. 55455.
The author is indebted to Octave Levenspiel and Dan Luss for
raising several points of interest in correspondence. In particular, the former points out that the term formal kinetics (la cinitique formelle) was used by Jungers to denote the area of the
dzferential equations of kinetics and determination of rate constants and mechanisms. Since writing this paper the author has
also seen a manuscript by F. J . Krambeck on ‘‘ The Structure of
Chemical Kinetics” which develops an axiomatic treatment of
kinetics paying particular attention to the thermodynamic basis,
the masses of the latter are all constant and obtain the
equivalent of Wei’s first two axioms.
Then Bowen has shown that in the absence of diffusion, or in a closed, uniform system, the changes
in composition can be uniquely expressed in terms of a
fixed number of independent reactions (31, Theorem 6).
We can then impose Wei’s third axiom on the reaction
rate expressions and, because the reactions are independent, the principle of microscopic reversibility emerges as
a theorem. Wei shows that the existence of an equilibrium point is a consequence of his first three axioms
and the fixed point theorem of Brouwer, but this does
not ensure its uniqueness or stability. Likewise the
principle of microscopic reversibility only ensures an
ultimately monotonic approach to equilibrium by
guaranteeing the symmetrizability of the local linearization (32). Wei’s fifth axiom is therefore necessary if
the commonly observed uniqueness of equilibrium is to
be reproduced in the mathematical system. This uniqueness is a feature of statically thermodynamic conditions (33) and so can be claimed for all mass-action
kinetic expressions which are consistent with equilibrium (34, 35).
This brief discussion does not exhaust the question
of axiometic foundations but is intended to show that
further work is needed along these lines. We shall
allude to that question again in connection with Bartholomay’s stochastic formulation and very briefly in
the next section. As indicated above, it needs to be
related to the whole business of constructing deductive
theories (cf. 27, 36-40) in the context of the physical
and chemical sciences, but this is a large matter.
matrix of the products of the reaction (in fact, the steps of
such a reaction can be written down as a sequence of
such matrices), but this does not appear to be fruitful.
An approach more directly related to truly topological
notions is that of Lederberg (45-47). This has involved the classification of regular trivalent graphs (since
the diagram of the organic molecule rarely has more
than three branches at any one node) and the association of these with certain polyhedra. Nothing has been
done to connect this theory with a representation of
reaction. Topological notions have been introduced to
elucidate complex molecular structures by Wasserman
(48). Sellers’ treatment of chemical complexes is
capable of sustaining a representation of structure (18))
but little of the detail has been worked out.
Combinatorial Problems
The work of Sellers (6, 17, 78) represents a remarkable
synthesis of combinatorial topology and chemistry,
and its articulation should give lead to great advances
in the art of discovering and proving the mechanism of a
reaction. Though there is a need to soften the forbidding
aspect that his principal account (78) may have for the
Structure
The mathematical representation of molecular structure is not our main concern here; but since it has a
bearing on reaction we can scarcely avoid some mention
of it, even though this connection has been little developed. J. J. Mulckhuyse has an axiomatic treatment of
structure in organic chemistry (41)) but this does not
seem to have been followed up. Spialter has devised a
computer-oriented chemical nomenclature in which the
names of the atoms are literal entries A , on the diagonal
of a matrix, and the numerical off-diagonal elements b,,
denote the number of bonds between A i and A , (4244). Since such a representation should be invariant
with respect to permutations of rows and columns,
Spialter suggest that the characteristic polynomial is a
good representation. The difficulty here is that if there
are two atoms of the same chemical element in the
molecule and if they are not literally distinguished,
then the matrix often cannot be uniquely reconstructed
from its polynomial. O n the other hand, if every
atom is distinguished only the three highest powers of
the polynomial are needed. Moreover this representation does not seem to lend itself to the discussion of a
reaction. It is true that a mixture of molecules can be
represented by a partitioned matrix. If a matrix of
the reactants is written down, a bond-breaking and
-making matrix can be added to it, which will give the
Figure 2. Model of enzyme-catalyzed synthesis
nonmathematician, a brief summary of the representation of a reaction and its mechanism may be useful.
Consider an enzyme-catalyzed synthesis of two substances, R and S, to form a product, P. If this is a
balanced reaction the empirical formula for P must in
fact be the sum of those for R and S and we might write
P
SOIR, when 01 represents a bond of some sort
between the two parts of the molecule. Let the sides
of the triangle, 2, in Figure 2 represent the species R, S,
and P as shown.
The triangle itself may represent the overall reaction
which, since it is a combination of S and R , we write as
S g 1 R where 8 1 is an abstract “product” of the two
entities R and S. I n fact, we can write the chemical
equation of the reaction in various ways. The stanS = P is easily seen to be equivalent to
dard form R
R
S - P = 0, where all the species are written on
+
+
VOL. 6 1
NO. 6 J U N E 1 9 6 9
19
one side of the equation and the product species is given a
negative stoichiometric coefficient. Writing R
S P = 0 in its equivalent form
+
R - SOlR
+S = 0
suggests a simple relation to the abstract product
S@ 1R
I n fact, if we put an orientation on the face, 8,as shown
by the anticlockwise arrow, we see that R - SOIR
S
is just the boundary of the face, the signs being given
according to whether the orientation of the side is in
the same (+) or the opposite (-) direction to that of
the face. This may be written
+
d(SC31R) = R - SOlR
+S
where the rule for the boundary operator, b, is to delete
the first term of the abstract product, subtract the species
obtained by replacing @ I by the bond 0 1 , and finally
add the species obtained by deleting the last factor
of the product. This rule of deletion and changing of
@ to 0 can be extended to any number of terms and
corresponds to the geometric notion of taking the boundary. Because the face S@IR represents a properly
balanced chemical reaction, its boundary is zero, for
S O I R or P is just the sum of R and S.
Xow suppose the reaction mechanism involves a
catalyst, E , which first binds to R to form a complex, C,
which then allows S and R to combine to P in a double
complex, D, and finally releases itself from the catalyst.
These steps might be written
S+C=D
R+E=C
D=P+E
+
and the sum of them is the overall reaction R
S = P.
But if we represent the binding to the catalyst by a second symbol 0 2 , then C is represented by ROzE and D
by S 0 1 R 0 2 E . Thus, the overall reaction
2: R
- SOiR
+S = 0
has three steps
+E = 0
+ ROzE = 0
II: E - SOlROzE + SOiR = 0
A:
I‘: R - ROzE
S - SOlROzE
and clearly
~ : = r - - n + ~
KOWjust as 2 may be represented by the abstract product &‘@ 1R so that the boundary b(S@1R) is the chemical
equation R - SOlR
S = 0, so may r, A, and 11 be
represented by R @ 2E, S@1R02E, and SOiR @ sE, re-
+
spectively. For example, in taking the boundary
b ( S O I R @ p E )we
, delete the first factor, SO& giving
E, subtract the result of replacing 8 2 by 0 2 , namely,
SOIR02E, and add SOlR obtained by deleting the
second factor, E; thus, d(SO1R@?E),= E - SOiROzE
SOiR.
Now the four triangles 8, r, A, and II can be fitted
together as shown in Figure 3, where the orientations of
faces and edges are shown by arrows.
+
20
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Figure 3. “Cutal3.zution model’’
Consider now the double “product” S@ 1R@SE, and
apply the extension of the rule for the boundary
d(S@lR@pE)= R@zE - SOiRC3zE
S@IR
=
+ SBiROzE -
r -n+A
- 2
Thus, just as the faces, which are two-dimensional triangles, have boundaries which are one-dimensional edges,
so the three-dimensional tetrahedron has a boundary
consisting of the faces. Moreover, just as the chemical
equation for the reaction associated with a face is obtained
by setting its boundary equal to zero, so the mechanism
is obtained by setting the boundary of the tetrahedron
equal to zero. For ~ ( S @ D R @ , =
E )r - II
A -Z
= 0 can be written Z = F - II
A , just as above.
Sellers has called the three-dimensional complex, here
a tetrahedron, a “catalyzation.” The boundary of a
catalyzation is a signed collection of reactions or a
mechanism. The boundary of a reaction is a signed
collection of chemical species or a reaction equation.
This is a bald description of Sellers’ concepts, but we
hope it shows that the notions of chemical reaction and
mechanism can be tied in with those of combinatorial
topology. The power of these ideas arises from the
fact that the resources of combinatorial mathematics are
now harnessed to the tasks of the kineticist, such as that
of computing the full independent set of possible mechanisms of a reaction.
Another place where combinatorial considerations
must be entertained has been shown by Zimmerman
(72, 20, 21) in his work on the kinetics of polymerization. If growth starts at two centers on a long molecule, the possibility of interference between independently propagating chains must be considercd. In fact,
all stochastic formulations have a combinatorial element
since the basic probabilities are usually calculated in
this fashion.
+
+
Stochastic Formulations of Chemical Kinetics
That the course of a chemical reaction can be regarded as a stochastic process has been the observation
of several authors (22, 49-67). This formulation has
the advantage over the deterministic approach of giving
a measure of the intrinsic variability of the process.
For example, if the irreversible first-order reaction A --c
B is considered deterministically, the number of molecules of A that time, t, would satisfy are expressed by
dn
_ -- -kn
dt
or
n(t) =
Algebraic Structure of Systems of Kinetics Equations
nee-"
O n the other hand, suppose that (no - n) molecules
have decomposed in the time interval (0,t) and that the
probability of a single decomposition in (t, t 4- st) is
knst
o ( 6 t ) and that the probability of more than one
decomposition is o ( 6 t ) . Then, the probability of going
from a state with i molecules to one withj(<i) in time
interval, t, is
+
Thus, starting with i = no at time t = 0, the expected
number of molecules at time, t, is
nn
which agrees with the deterministic process. However,
we can also find the variance of the number of molecules
about the expected value, namely
2 ( t ) = noe-"(l
- e-")
if the total number of molecules is large. It is always
valid if the processes are all first order, but only approximate for higher-order processes when the numbers
are small. Fredrickson (59) has given a careful treatment of the triangular reaction, which could easily be
extended to a n arbitrary linear system. Very similar
methods are used and similar equations obtained in
discussions of the dynamics of microbial cell populations
(62, 63).
6
no/4
T h e standard error of the process is proportional to
no-'I2.
When a number of experimental runs are performed
and observations taken at times t , = sr, s = 0,1, . . . S =
T / T ,N , , sbeing the observation of n(t,) in the rth run, then
the maximum likelihood estimate of k is
The least-squares estimate based on the deterministic
model differs of course from this, though the estimates
are usually close. The difference lies in the fact that
the deterministic model regards the random error as
lying solely in the experimental procedure whereas the
stochastic model supposes that there is an intrinsic
randomness in the reaction process itself.
For more complicated reactions, the equations for
the transition probability are not so readily solved and
approximations have to be made. The set theoretic
approach takes over the deterministic mean and takes
exp --kt and (1 - exp -kt) to be the probabilities that a
given molecule is A or B, respectively. This gives the
expression for Pno7(t) directly, by calculating the probability that j of the total number of molecules are A,
rather than by setting u p and solving equations for the
transition probabilities. This assumption of equivalence of deterministic and stochastic means is reasonable
First-order or monomolecular systems. A general
system of first-order kinetic equations for the concentrations ci(t) for n species will take the form
5
dt
=
5 k,F,, i
= 1, 2,
. .n
j=1
-c kji.
n
where k,, > 0 if i # j and k,, =
The structure
j=l,i#z
of this first-order system has been well known in chemical
engineering circles since Wei and Prater published a
study notable both for its elegance of theory and usefulness of practical application (64, 65). This systematized a subject that had been touched on rather lightly
in various ways by various workers (66-69) and which,
in fact, bore close resemblance to the so-called compartmental analysis of biologists (see references in 70).
If an equilibrium composition ci* exists with c,* > 0,
all i, the system is fully reversible.
These equations can be written in matrix form
dc
_ -- KC
dt
and the equilibrium composition is such that
Kc* = 0
By the principle of microscopic reversibility ki,cj* =
kf2ct*, and this implies that the matrix is symmetrizable
since kt,(cf*/cZ*)1/2 = k3i(ct*/c3*)1/2. I t follows that
the eigenvalues are all real and further, that they are
all nonpositive. I n the concentration space, there are
straight paths which are solutions of the equations.
Recently Wei, Prater, and Silvestri (77-73) have investigated the case when there are irreversible reactions
and some of the ci* may actually be zero.
I n practice, even when knowledge or optimistic
ignorance allows a system to be regarded as of the first
order, there are many more components than one would
wish to deal with and Wei and KUO'Srecent attack
on the problem of "lumping" is of the first importance
(79, 7 9 ~ ) . By "lumping" we undertsand that m(<n)
linear combinations of the concentrations c, are to be
taken in place of the full set. Thus, if M is a matrix,
the vector of lumped species ^G is given by
I = Mc
and two compositions c1 and c2 are M-equivalent if
they map into the same lumped composition-i.e.,
Mci = Mc2
VOL. 6 1
NO. 6 J U N E 1 9 6 9
21
But if there exists a matrix
then
K
such that MK = KM,
de
- =
dt
and the system is said to be exactly lumpable. The
compositions in any two lumped subsystems of a n exactly lumpable system remain M-equivalent if the
initial compositions are M-equivalent. This notion of
equivalence has far-reaching consequences, and Wei
and Kuo have gone on to discuss semiproper, improper,
and approximate lumpability. The normal concept
of a mechanism for a reaction usually involves improper
lumping, since even if the original equations are first
order, the kinetic expression obtained by using steadystate hypotheses is rarely that simple. When infinitely
many components are involved, a first-order system may
be lumped into virtually any kinetic law (35), and probably a n approximate form of such lumping accounts for
the success of some of the less enlightening empirical
rate expressions.
Higher order systems. The first-order system is very
fully understood because the resources of linear algebra
are available. The equivalence classes of reactions are
those of the matrix K, and lumping represents a projection into a lower dimensional space. With higher
order reactions the same degree of insight has not yet
been obtained, though a canonical form has been proposed (74). A beginning study of second-order reactions was made (75) using the relationship between quadratic equations and nonassociative algebras that Marcus
had discovered (76). This is not an easy approach,
though it gives a certain insight, and Wei has shown that
the simplifying features of the first-order system, such
as straight line paths, will rarely be present (77). Some
of the advances in polynomial differential equations (e.g.,
78, 79) have yet to be interpreted in the context of
chemical kinetics. One difficulty has been that the
nonassociative algebras that have attracted mathematicians' attention have been rather special ones such
as Lie (80),Jordan (81), or genetic algebras (82, 83);
Schafer's is the only general survey of the field (84),
though there are passing references to some of the basic
problems in many other papers and books.
Systems with a denumerably or continuously infinite number of components. Large systems of
equations which contain certain regularities, such as
the hydride exchange scheme of Wei and Mikovsky
(85) or some multiplace enzyme systems (86),can often
be solved by the introduction of generating functions.
This can be extended to denumerably infinite sets of
equations and in simple cases gives solutions in closed
form (87). An extensive review of polymerization
equations is given by Liu and Amundson (88),and in
later work Amundson and Zeman set up a continuous
model in which the polymer P, of integral length, j ,
was replaced by P, and j regarded as a continuous
real variable (89). This method goes back to the work
of Bamford and has been used extensively. We shall
22
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
not attempt to review all these methods in detail as this
has been done elsewhere (90). The notion of the continuous mixture and the connection with monomolecular systems is worth indicating, however; a more
complete treatment has been given elsewhere (97).
I n place of a discrete index i to denote the ith component and its concentration c d ( t ) , we use a continuous
parameter x, a 6 x 6 6 , and let c(x,t)dx denote the
concentration of material in the parametric interval
(x, x
dx). Balances can be done for this interval just
as for a discrete species and the analog of Wei's monomolecular system
+
n
CAt) =
is
t ( ~ , t )=
sa
c
(kijcj
1-1
- h,cJ
( k ( x , u ) c(u,t) - k(u,x) c ( x , t ) ) du
where the dot denotes d/dt. This is an integro-differential equation, with many of the simplicities of
linearity, but distinctly more complex than the discrete
equations. However, it can be shown that the principle
of microscopic reversibility ensures a unique and stable
equilibrium point, though it does not guarantee the
reality of the spectrum. Nonlinear kinetic equations can
be postulated, even if they cannot be readily solved and
the partial differential equations of polymerization can
be obtained from the same formalism. For example
the sequential first-order reactions A I -+ A2 -+ A a -+ . . ,
generalize to d A ( x ) / d x = 0 and the concentration density
c(x,t) satisfies the equation
ac
-
at
b
+(kc) = 0
ax
where k(x) is the rate constant for substance of parameter, x.
The continuous mixture can also be used to simulate
independent parallel reactions, and the remark made
about lumping comes up again. Thus, if c(x,t)dx is
the concentration of material which decomposes (e.g.,
desulfurizes) with first-order constant x, then a t time, t,
c ( x , t ) = c(x,o)exp - x t
where c(x,o) is the original concentration of material
of index, x. If x can take any positive value and we
lump the whole mixture into one (ix.,consider only,
total sulfur), then
Prn
In particular if c ( x , o )
=
Cok-' exp -x/k, then
C(t) = Co(1
+ kt)-'
which is the way in which a substance decomposing
irreversibly by a second-order reaction would behave.
Almost any kinetic law could be simulated by taking the
appropriate initial distribution
c(x,o) = .e-'[C(t)]
Determination of kinetic constants. One of the
beauties of Wei and Prater's work on monomolecular
systems was to show how the straight line paths could be
used to determine the rate constants. Constants in
any set of differential equations may be estimated by
choosing them to give the best fit between the experimental data and values of the same quantities calculated
from equations (92). The usual criterion of goodness
of fit is a form of weighted least squares and a search
procedure, though Bayesian methods of estimation also
have been proposed (93) and a large number of papers
have appeared (94-700).
More recently, it has been questioned whether much
is to be learned by fitting overall reaction rate expressions
without supplementing kinetic data with theoretical insight (707). I n the so-called Langmuir-Hinshelwood or
Hougen-Watson kinetic expression, commonly used in
heterogeneous catalysis, the constant in the denominator
need not be the equilibrium adsorption constants if, in
fact, the rates of adsorption and desorption are comparable with that of reaction (cf. Sec. 6.2 of 87). The technique of slight displacement from equilibrium, used by
Eigen on elementary reactions with great success (702) is
now being applied to biological systems by Czerlinski
(703, 703u) and to heterogeneously catalyzed reactions by
Parravano and others (704). There may be problems of
mathematical interest in this area. Some of the fundamental mathematical questions involved in various
schemes of measuring rate constants were examined by
Cannon (705, 706). Power series solutions have also
been proposed for determining kinetic constants (707).
From the mathematical point of view the situation is
perplexing. A great variety of methods exists and finds
useful application in all branches of chemistry and biochemistry, but no major organizing principle emerges.
The state of the art would seem to be comparable to that
of non-Newtonian mechanics a few years ago when exponents were being fitted to equations for power law
fluids and no really searching analysis of the concept of a
constitutive relation had been made.
Oscillating Reactions
A closed chemical system obeying the principle of
microscopic reversibility cannot approach equilibrium
in an oscillatory manner ( 3 2 ) . One can, of course, construct rather artificial systems such as A -+ B -+ C-c A
in which each step is autocatalytic and, for example, ci =
kzcu - k&.
This has solutions which are closed paths
kJog(u/uo)
kzlog(b/bo)
kslog(c/co) = 0 in a triangular diagram. A very ingenious example constructed
by Wei (708) obeys the principle of microscopic reversibility and supposes that A -+ B is catalyzed by three
'
, Y,Z,according to the scheme
forms of a catalyst, A
A +X=B
Y,A + Y e B + Z ,A Z - - , B + X .
When the concentration ratio ( x
y
z)/(u
6) is
small, the concentrations of X,Y,and 2 oscillate about
their equilibrium values a finite number of times before
settling down to a monotonic approach. Moreover this
number may be made arbitrarily large by taking the
concentration ratio sufficiently small.
+
+
+
+
+ +
+
I n open systems where certain reactants are supplied
and products removed, the situation can of course be
quite different. The simplest example is the stirred
tank reactor, where even A 4 B can show limit cycles
under nonisothermal condition ( 709), but many other
examples have been given, usually of biological significance (67, 770, 717). A valuable and extensive review
is given by Higgins ( 7 72) and it would be impertinent
to cover the same ground here; we shall only attempt
to point out topics that may be of mathematical interest.
For first-order systems, or systems that can be made
pseudo-first-order by assuming certain concentrations to
be constant, the eigenvalues are always always real,
provided that the principle of microscopic reversibility is
imposed ( 7 70). If the system possesses closed loops and
the product of the equilibrium constants around the loop
is not unity (this is the so-called Wegscheider condition,
which can only be violated for pseudo-first-order reactions), then damped oscillatory behavior may be
observed ( 7 72).
For higher-order reactions, Higgins has developed the
notions of inhibitition and activation corresponding to
negative and positive signs of the partial derivative of a
rate of reaction with respect to a concentration; thus
self-activation would correspond to autocatalytic behavior, for example. For two components Higgins
suggests that some of the self-coupling terms must be of
opposite character, as also must be some of the crosscoupling terms, if oscillatory behavior is to be observed.
This conclusion is obtained partly from the use of
theorems in nonlinear oscillation theory and partly from
a very considerable experience with computer solutions
and biological experiment. I t would be most valuable
to see them established rigorously and extended to larger
systems for they are backed by a great deal of insight into
physical systems. Similar oscillations in inorganic systems have been discussed by Bray (773) and more recently by Mason and Bischoff ( 7 74).
The important interaction of physical and chemical
process has interesting mathematical overtones. We
shall note a few of them in connection with diffusion
problems but phase changes may also be of significance
(775).
Pseudo-Steady-State Hypothesis
There are few more pervasive or useful assumptions
than that of the steady state introduced into chemical
kinetics a little over 60 years ago. Indeed its introduction, acceptance, and elucidation would be an interesting
study in the history of science. Quite evidently, it has
been and is still being used in many quarters with great
practical success and complete indifference to its conceptual and theoretical foundations, for its cruder
expositions are often mathematically ludicrous, and the
literature abounds with loose statements. Yet it should
provide a case study of the function of mathematics in the
refinement of concepts-the role which makes her the
queen, as well as the handmaiden, of the sciences. By
this hypothesis we understand the doctrine that in many
circumstances certain changes are so slow as to be negligiVOL. 6 1
NO. 6 J U N E 1 9 6 9
23
ble, and that important simplifications can thereby be
introduced. This has been stated rather vaguely, and in
practice it takes one of three forms:
1. Steady state of intermediate concentrations
2. Rate-determining step
3. Long chain approximation
I n the first case, by setting the derivative of such concentrations equal to zero, we obtain algebraic equations
which can be used to eliminate these concentrations.
I n the second, equilibrium relationships are used for the
same purpose. In the third, only the propagation
reactions are used to determine relative concentration
distributions, and the initiation and termination reactions are brought in to give the overall reaction rate.
Hypotheses of this kind were introduced in the context
of enzyme kinetics by Henri in 1903 (776) and again by
Michaelis and Menten in 1913 (777) in the form of a
rate-determining step. Thus, if in the scheme
1
2
S+E+C-+P+E
-1
the first reaction is much faster than the second, we might
assert the equilibrium relation klc = k-Ise, which
together with the stoichiometric relation e
c = eo gives
c = eos/(K’
s), K’ = k - l / k ~ . The overall rate of
reaction would then be
+
+
This was criticized by Briggs and Haldane in 1925 ( 7 78)
who asserted rather that the derivative of the intermediate concentration
dc
_
dt -- kles
-
(k-1
+ kz)c
+
+
+ s) and eokns/(K + s)
with
K’
= k-l/kl
and K =
(k-1
+ k*)/kl
only the second can be rigorously justified (724). I n
fact, the hypothesis that 1.1 = eo/so 4 0 does lead to the
limiting form in the second case whereas the other
hypothesis kz/kl and kz/k-1
0 leads to a cumbersome
rate expression quite different from the first. It is ironic
that the name Michaelis-Menten kinetics has become
firmly attached to the Briggs-Haldane kinetic equation.
-
24
S = - k leos
6 = kleos
-
+ (kls +
(kls
+
k-1
k-1)~
+ k2)c
would first lead to solving the second equation in the
form
c = (hens - t ) / ( k l s
k-1
kz)
+
s) and a
was negligible. This leads to c = eos/(K
rate expression of the same form but with K = ( k z
k - l ) / k I . We see this to be a hypothesis of the first
rather than of the second kind, and such a hypothesis
had in fact been suggested carlier by Bodenstein (779)
in 1913. [The history of this basic enzyme kinetics
has been reviewed by Reiner (86, 720) and by Walter
(727), and Benson gives some references on the chemical
side (722). See also Christiansen’s paper (723).] The
interesting feature of this case is that though the two
hypotheses lead to similar reaction rate expressions,
namely
eokzs/(K’
The specious argument, that when p is small, the concentrations e and c are small and so dcldt is negligible, was
accepted in support of the Briggs-Haldane kinetics
long before mathematical rigor was invoked. Fragmentary though the mathematical argument was, its
consequences had a tremendous practical importance
and such mechanisms were established by experiment,
as in Chance’s ingenious work with peroxidase (725).
Of course the structure of the solution of the complete
equation is such that the concentration of c rises very
rapidly during a brief “induction period” and decays
slowly enough thereafter that during the main part of the
reaction its derivative is indeed negative. But how
should this feature, which is also exhibited in much more
complex systems, be made precise?
The validation of the pseudo-steady-state hypothesis
has been approached from a number of different angles.
Walter (727, 726, 727) has made detailed maps of the
following kind. Given the rate constants, k,, and initial
concentrations, one can find the per cent reaction a t
which some arbitrary, but small, value (say, 0.01, 0.02,
or 0.05) of (e/#) is still encountered or, conversely,
values of (t/b)at various small values of conversion may
be determined. Such maps are interesting, for the
biochemist commonly takes “initial” rate data a t some
low conversion, but they do little to clarify the concept
involved. Hirschfelder (728) suggested an iterative
scheme, which when applied to the enzyme reaction
equations gives
INDUSTRIAL A N D ENGINEERING CHEMISTRY
+
+
+
k-1
kz) and
Setting t = 0 gives: c1 = kleos/(kls
SI = kzeosl/(K
S I ) as an equation to be solved. The
next iterate is
+
cz
=
( k m
- Il)/(kls
+
k-1
+ kz)
where t 1 = (dcl/ds)S1, and substituting c1 into the first
equation gives the equation for s1. I n general,
e< =
c,+1
~ , + 1=
= (kleos,+l
((klsi+l
(dc,/ds)S,
- dJ/(k1+1
+ k-1 + k z )
+ k - - d t t + knkleost+l}/
{klJ,+l
+ k-1 + k z )
Thus, the first approximation is that of the pseudosteady-state, and successive iterates can be calculated.
Giddings and others have looked a t the second term
(729, 730) and suggested using its magnitude as a test
of the validity of the hypothesis, but in the absence of any
real understanding of the conrrergence of the iteration,
this also is inconclusive. Hirschfelder’s method is used
by Higgins (76) to compute the solution with an impressed oscillation in the feed. If the steady-state
hypothesis were perfectly valid, the path of the solution
would always lie in the surface given by the hypothesis
( i , e , , the surface 6 = 0 in the above example). This it
does at low frequencies of perturbation, but as the impressed changes become more rapid the path oscillates
out of the steady-state surface. This approach reflects
the biological importance of oscillating systems in
Higgins’ work.
O n the basis of some numerical calculations, Benson
(731) suggested certain criteria for validity based on the
degree of conversion at the end of the induction
period and this type of criterion Silveston (75) has subjected to much greater scrutiny. Similar considerations
have been raised by Frank-Kamenetski (732). All these
methods, however, suffer from an ineradicable element of
arbitrariness, entering either in the definition of the
induction period or in the fixing of a certain percentage
deviation. Credit for discovering the true mathematical
significance of the steady-state hypothesis as an instance
of singular perturbation is due to Acrivos and coworkers
(733). Later work by others (724) is really only an application of more recent techniques of asymptotic expansion (134)to their idea.
By throwing the kinetic equations into dimensionless
form using the variables
y = s/so, z = c/co,
T
= kleot
Pi =
and the parameters
+ kd/krso, X kdklso,
we have the equations
y
-y + ( y + - X)z
K
= (k-1
=
=
p = edso
H
pk = y
- 0, + x)z
withy = 1, z = 0 at T = 0. I n the degenerate case
p = 0, the second equation is algebraic rather than differential and gives the dimensionless form, z = y/(y
N), of the steady-state relation c = eokls/(kls
k-1
kz). Substitution in the first differential equation then
H) which is just the dimensionless
gives i, = -Xy/(y
form of the Michaelis-Menten kinetic expression ds/dt =
-kzeos/(s
K ) . The value of the dimensionless form of
the equations is that it shows clearly that the parameter
p is indeed the one leading in the limit p + 0 to the degenerate case corresponding to the pseudo-steady-state
hypothesis. What is even more important is that the
theory of the singular perturbation of differential equations tells us what to expect in the solution. There
will be an “inner solution” which lasts for a few multiples
of the dimensionless time interval p / ( l
H) [;.e., a real
time interval of (klso
k-1
k ~ ) - during
~]
which z ,
or c, will rise to its maximum value and after which the
pseudo-steady-state assumption is a good approximation
and the so-called “outer solution” obtains. But all this
can be made precise in a systematic way and a solution
in powers of p, say y = Yn(7),z = Zn(7),can be developed which is related to the true solution y = Y ( T ,
p ) , z = Z ( T , p ) in the sense that as p + 0, the bounds
+
+
+
+
+
+
lYn(7) - y(7,
+
~ l ) l < % and
~ ~ ~I z ~n ( 7 )
The least upper bound of ao thus provides an unambiguous measure of the validity of the steady-state
hypothesis which is free from any arbitrariness.
The notions of the singular perturbation approach
have been sketched for a particularly simple reaction,
but they can be readily applied to more complicated
ones revealing what the consequences of the limiting
hypothesis are and how many parameters must be considered in the system. If the accuracy of the actual
method of estimating the rate constants (say by a Lineweaver-Burke plot) is to be discussed, then further calculations may be necessary to be faithful to the method of
estimation. From the conceptual point of view, the important point is that the singular perturbation approach
reveals the analytical nature of the projection implicit in
the notion of a chemical mechanism, just as the work of
Sellers (77) or Aris (35) shows the algebraic aspect of this
project. The justification of the long chain approximation also involves the ideas of perturbation theory as
Gavalas has shown (735).
The notion of memory introduced by Halsey (25) may
be justified by observing that if y ( ~ is
) an arbitrary function of time, then the equation
+
- ~ ( 7p)l<b,p’+’
,
hold in the interval 0 < T 6 T. The constants a,
and b n depend only on T , and in many cases are bounded.
y
- (x + r)Z
z(0) = 0
can be solved for z as a function of the whole past history of y. Substituted in the rate equation it therefore
makes the rate of formation of the product a functional of
the past history of the concentration Y ( T - u), 0 6 u 6
T.
The functional obtained satisfies the principles of fading memory laid down by Coleman and Mizel (736,
737) and the same hypothesis p + 0 that leads to the
steady-state assumption also obliviates the memory and
leads to the Michaelis-Menten kinetics (26). This notion
can also be generalized, though it remains to be seen what
its power may be.
Diffusion ahd Reaction
When diffusion as well as reaction is brought into the
picture we rise immediately from systems of first-order,
ordinary differential equations to systems of parabolic,
second-order, partial differential equations. While this
ascent brings a fascinating new landscape of mathematical problems into view, it also reveals that the exploration of this new terrain will involve much rougher
going, particularly in the nonlinear direction. We suggest three general areas in which problems of mathematical interest may arise. The first is largely the
province of the physical chemist and includes such topics
as the diffusion limitation of the reaction of highly reactive
species (138) and chemical reaction regarded as diffusion
through phase space (739). Some results of considerable
generality for complex geometries have been obtained by
Prager (749) using variational principles : We shall take
note of these methods a little later in another context.
The second area is that of diffusion and reaction in
catalyst pellets which has engaged the attention of chemical engineers for the past 30 years. Some reflections on
the history of this subject have been given by one of its
pioneers, E. W. Thiele (747). I t has already accumuVOL. 6 1
NO. 6
JUNE 1969
25
lated a literature of almost overwhelming proportions.
I n this area there is a large amount of straightforward
computational work giving the effectiveness factors for
various shapes of particle, kinetics of reaction, and
thermal conditions. I t would be a work of merit to bring
this all together in one place, but it is the theoretical
problems that come to light in the process which are of
real mathematical interest. These concern the uniqueness and stability of the steady states and the existence
periodic solutions of the nonlinear partial differential
equations.
An excellent review is to be found in Gavalas’ book
(f42), and an elementary survey of some of the recent
ideas on stability has been presented elsewhere (143),so
that we shall not attempt to cover this ground again.
Three points are worth noticing, however. The first is
the power of the maximum principle as Luss has applied
it to show that, when the Lewis number of the particle is
one, the unique steady state is globally stable and the
central unsteady state of three lies in the separatrix of
function space (144). This is a beautiful use of a general, and to the engineer a still somewhat recondite result
of differential equation theory. The second is the
existence of periodic solutions when the Lewis number is
not one, which Hlav6Eek and Marek (145) have reported
and Luss has investigated more fully (10, 146). The third
is range of multiplicities of solution that are coming to
light. For a single exothermic reaction taking place nonisothermally in a catalyst particle it has long been known
that three steady states exist for certain sizes of the
particle. Now it appears that five can be obtained with
a zeroth-order reaction in a sphere, though not in a slab
(747), or when external transfer limitations are also
important (148).
The variational methods used by Prager (749) and
others deserve more attention. He has used them with
Strieder (150) to discuss Knudsen flow and it is this work
that Strieder is taking further in the consideration of
reaction (14). Luss and Aniundson also used variational arguments in discussing the effect of the shape of a
particle (157). Finlayson and Scriven have shown how
to exploit upper and lower bounds (152). An interesting
question, within the context of these methods, is the relationship of the commonly accepted model of a catalyst
particle to reality. I n the usual modcl, the catalyst
particle is treated as a homogeneous phase with a uniform distribution of catalytic area. If c is the concentration of a species disappearing by reaction at a rate, kc, per
unit catalytic area, u is the area per unit volume and D
the effective diffusion coefficient, then for the homogeneous model of Figure 4a, we have the equations
DV2c = kac in V
c = cs (constant) on
S
(A)
I n reality, the species is diffusion through a free space of
immensely complex geometry and reacting on an inner
surface 2 distributed through V. If 2 is so uniformly
distributed that for every element of volume dV, the reaction area is d 2 = a&’, we would expect the two models to
26
INDUSTRIAL A N D ENGINEERING CHEMISTRY
b
a
Figure 4a,b.
Catalyst particle models
give the same result. But the equations for the second
model, illustrated by Figure 4b, are clearly
D‘v~c‘ = 0 in
V’
bc ’
bn
D ’- = kc’on 2
c ’ = co
(constant) on S
(B)
where a prime is used to denote quantities that may be
different in this model and V’ denotes the free space
within V .
Now it can be shown that of all the functions which
have a constant value co on S, the solution of Equations A
minimizes the functional
Moreover the effectiveness factor, T ~ as, it is usually defined, is given by this minimum value, for by Green’s
theorem,
Again, of all the functions constant on S, the solution of
Equations B minimizes the functional
and
A complete investigation of these relationships is in
progress, together with other general problems of this
character. Tiot only does it have some mathematical
interest, but its practical importance is likely to be enhanced by the use of molecular seive catalysts in which
the surface 2 is indeed less uniformly distributed than i s
the case with conventional catalysts.
The third area of promise in diffusion and reaction
problems is where some geometrical structure is found in
the medium. O n the one hand, it may be deliberately
laid down as a fixed structure as in some discussions of
active transport (753, 154) or bifunctional catalysis (755).
Or, it may arise naturally as the result of structureforming diffusion and reaction (73, 156). I t is this
second possibility, opened up by the pioneer work of 1
Turing (757) but more fully developed by Scriven and
his coworkers (758, 759), that is so interesting and holds
promise of a greater insight into the problems of morphogenesis. Another crosslink that has been suggested by
Scriven’s work is that, a t the limit of high wave numbers
and frequencies, he sees the need to consider memory
effects in transport and even action at a distance. The
former has received little attention outside Scriven’s work
(cf. 760) though it links with the formulation of constitutive relations using functionals, the latter is suggestive of Knudsen flow and Strieder’s work (739, 750).
We have by no means exhausted the mathematical
problems of diffusion and reaction. Some of the questions of multicomponent diffusion are exceedingly important and not a little delicate (767-763). But it is
hoped that enough has been said to indicate the range
and difficulty of the problems that may be usefully pursued in this milieu.
Questions of Equilibrium and Stoichiometry
I n this somewhat circular tour of the domain of formal
kinetics let us return by way of a discussion of some of the
questions of equilibrium and stoichiometry. As an outgrowth of the interest in continuum mechanics and
continuum thermodynamics it is natural that there
should be an inquiry into the nature of thermodynamic
equilibrium. T h e literature of modern continuum
thermodynamics is already considerable and references
764-777 and 23 are just a sampling. Its objective is to
set down general constitutive relations for a class of
materials and to see what restrictions and relations are
imposed upon them by the equations of balance, by the
second law of thermodynamics (the Clausius-Duhem
inequality), and by any symmetry assumed for the
material. I t is, for example, shown that the specific free
energy, specific entropy, and stress tensor cannot be
functions of the temperature gradient. Coleman and
Gurtin’s internal state variables, cy (766), may be interpreted as extents of reaction, so that their equations of
change, d! = f(F, e, g, a), are the kinetic equations (F is
the deformation gradient, 0 the temperature,p = W).
The internal dissipation u is defined as -e-‘?),$
f, where
$ is the free energy function, $(F, 0, CY). If q = Q(F, e, g,
a) is the constitutive relation for the heat flux vector and
p the density, the Clausius-Duhem inequality implies that
pu
+
- $.g>O
but in general the two terms in the inequality cannot be
separated. If there is no reaction f = 0 and so 4 . g 6 0,
meaning that heat flows down the temperature gradient.
If there is no temperature gradient, g 0 and the internal
dissipation u must be positive. But it is not generally
possible to assert that u 2 0 for nonzero g, nor cj’g 6 0
for nonzero f. If, however, the rate equations, f, are
independent of g, as is commonly assumed, then u 2 0.
T h e quantity -?)+/bat is, of course, the affinity of the ith
reaction and, a t constant strain and temperature and
zero temperature gradient, u is the specific entropy production-i.e., u = -e-’$
= when J?’ = 0, d = 0, and g
= 0.
7
+
This approach has allowed Coleman and Gurtin to say
some very precise things about equilibrium states,
Their definition of an internal equilibrium state is a
triplet (F*, e*, a*) at which f(F*, 8*,0, a*) = 0 and thus
it would correspond to chemical equilibrium. If such an
equilibrium state is asymptotically stable, then 9 must
have a minimum there and the affinities vanish. A
material is strictly dissipative if u > 0 except a t an equilibrium state. I n a strictly dissipative material, asymptotic stability at constant strain and temperature is
equivalent to the statement that $(F*, e*, cy) > +*(F*’
e*’ cy*) in some neighborhood of the equilibrium state.
I t is such results that Bowen has refined and extended
(5, 772). He distinguishes between
(a) Weak equilibrium, where reaction rates vanish,
(b) Classical equilibrium, where affinities vanish,
and (c) Strong equilibrium, where both vanish.
Bowen shows that weak equilibrium implies the classical
if baf is nonsingular and classical equilibrium implies the
weak if ?),& is nonsingular. He has demonstrated some
interesting results for waves propagation into a region of
weak equilibrium. Coleman and Mizel (8, 773) have
addressed themselves to the question of relating the thermodynamic condition for stability (that some free energy
or equilibrium response function should be minimum) to
the dynamical conditions. They have solved this problem completely for systems whose state can be described
by a finite dimensional vector, which, however, may be
governed by functional-differential equations.
If, on the one hand, the continuum mechanics approach has enlarged the theoretical understanding of the
nature of equilibrium, on the other, the resources of linear
algebra and optimization techniques have been used to
prove the uniqueness of equilibrium compositions and to
calculate them. Shapiro’s work (7, 774-777) has led the
way here to show very clearly the conditions under which
the minimum of the free energy functional as constrained
by the mass balance equations is unique (cf. 34, 35).
These questions are important in view of the increasingly
complex systems that are being considered, particularly
in biological contexts, and the need to harness the techniques of optimization and machine computation to the
efficient calculation of equilibria (778-789).
These considerations lie very close to others of a purely
stoichiometric nature, for Sellers has pointed out that
stoichiometric equations are mathematical equations
(790), and stoichiometric restrictions on equilibria have
long been seen to be important (797, 792). Some discussion has centered on the form of the stoichiometric equation, which for a set of r reactions between s species, A,,
can be written
Various normalization schemes have been suggested
(793-796), and Petho has developed some algebraic
apparatus for achieving a simple form (797-203).
From an abstract point of view, it is important to retain a
sense of the arbitrariness of stoichiometry (3, 3 7 , 34,
VOL. 6 1
NO. 6
JUNE 1 9 6 9
27
195, 204), for any nonsingular transformation of an
independent set must be equivalent. The maximum
number of independent reactions is governed by the rank
of the structure matrix expressing the species A , in terms
of their elements (34, 123, 204, 205). And here we return to some questions related to mechanism where
linear combinations of the reactions are to be made,
[the coefficients in this case are the stoichiometric
numbers, as Horiuti (206) has defined them] and to count- I
ing the possible mechanisms of reaction (78, 34, 207).
Stoichiometric considerations have also been invoked
in the discussion of thermodynamic coupling, the possibility that some reactions of a system may be absorbing
the entropy generated by the others. Hooyman’s
normalization (794) claims to have a certain “intrinsic”
property alleged to be necessary in determining the
existence of thermodynamic coupling (208) after Koenig
and others had shown that system of reactions could
always be uncoupled (209). Manes showed that near
equilibrium reactions could be coupled or uncoupled by
composition perturbations (270, 27 7) and the whole
question is reminiscent of the arbitrariness surrounding
the reciprocity relations of irreversible thermodynamics
(212). It is very clear that algebraic artifice should not
be allowed to replace real physical insight (213).
Epilog
I n this circular tour of some of the mathematical
aspects of chemical reaction it should be evident that
there is no neat linear succession of ideas. Rather there
is a web of interconnections between the several distinguishable areas of kinetics and mathematical disciplines. There is plenty of scope for investigation of the
many avenues that can be opened up, and the contributions of workers with quite diverse backgrounds and
intentions have many points of convergence.
One of the earliest mathematicians to recognize the
affinity between his work on invariants and (‘the new
Atomic Theory, that sublime invention of KekulC” was
J. J. Sylvester. Even if his ideas on the bearing of invariant theory (274) were to prove abortive (215) until
transmuted into “something rich and strange” (cf.
Weyl’s remarks, 216), his approach was consonant with
the largeness of outlook proper to natural philosophy.
Coming from an ampler age than ours, the Augustan
measures of his prose may fall quaintly on our ears but it
may be permissible to transpose one of his remarks to our
subject. (‘The beautiful theory of atomicity”-or
for
that matter of formal chemical kinetics-“has
its home
in the attractive but somewhat misty border land lying
between fancy and reality and cannot, I think, suffer
from any not absolutely irrational guess which may assist
the chemical enquirer to rise to a higher level of contemplation of the possibilities of his subject.”
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28
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
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