CALCULATION OF RELAXATION TIMESFOR COMPLEX MECHANISMS
2319
Chemical Relaxation Spectra: Calculation of Relaxation Times for Complex Mechanisms'
by Gordon G. Hammes and Paul R. SchimmeP Department of Chemistry, Cornell University, Ithaca, New York
(Received February 2,1966)
A theoretical treatment of chemical relaxation spectra is given; the only restrictive assumption employed is that for a given mechanistic scheme no more than two steps which equilibrate at comparable rates are coupled together via rapid reactions. A determinantal method for explicitly computing relaxation times is developed. For the general mechanism where the ith reaction is only coupled to the i - 1 and i 1 reaction and where the products of the ith reaction are the reactants for the i 1st reaction, the relaxation times characterizing the mechanism may be written by examination (wit,hout performing the usually complex mathematical operations) regardless of the number of steps in the mechanism or the molecularity (or order) of each reaction. Under certain conditions, the relaxation times for more complex mechanisms also may be written by examination.
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Introduction Chemical relaxation techniques are now being applied to a wide variety of reaction^.^ Several theoretical treatments of chemical relaxation have been de~ e l o p e d . ~Except for the most simple mechanisms, the mathematical analysis required for calculating the relaxation spectrum is somewhat complex and laborious. A similar difficulty exists in the mathematical analysis of other complex reaction kinetic phenomena and methods for simplifying the mathematical labor are highly desirable. Special mention should be made of the schematic method of King and Altman' which has greatly reduced the computational labor in treating steady-state enzymatic mechanisms. We present here an analysis of the relaxation spectrum of a mechanism which permits individual relaxation times to be readily calculated. The general theoretical treatment is based on the formulation of Kirkwood and Crawfords and more recently, Castellan.6 Under certain conditions the relaxation spectrum can be written by inspection with the use of a schematic method. The mechanism we will consider in detail is ki
A+ B
+ ... =Xi= k-i
ka k -a
ki+i
X t z k-ci+l)
ki
...
k-i
kn
... Z
k -n
Q
+ P + ...
(1)
+
where the number of intermediates, reactants, and products is arbitrary. Actually, as we will discuss later, the critical feature of this mechanism is that the ith step is only coupled to the (i - 1) and (i 1) steps (i f 1, n) and the products of the ith reaction are the reactants of the i 1st reaction; each step can be of arbitrary molecularity or order. An exact computation of the n relaxation times when n is greater than 2 is virtually impossible; however, approximate calculations can be made which will be adequate to describe most of the situations encountered in actual practice. The simplifying assumption which is made is that no more than two steps which equilibrate at comparable rates are coupled together via rapid reactions. The method which is developed eliminates the necessity of solving a large number of simultaneous linear differential equations and under certain conditions may be ex-
+
+
(1) This work was supported by a grant from the National Institutes of Health (GM13292). (2) National Institutes of Health Predoctoral Fellow. (3) See, for example, the reviews by M. Eigen and L. de Maeyer in "Techniques of Organic Chemistry," Vol. V I I I , Part 11,S. L. Friess, E. S. Lewis, and A. Weissberger. Ed., Interscience Publishers, Ino., New York, N. Y., 1963, p 895, and M. Eigen and G. G. Hammes, Advan. Enzymol., 2 5 , 1 (1963). (4) E. L. King and C. Altman, J. Phya. Chem., 60, 1375 (1956). (5) J. G. Kirkwood and B. Crawford, ibid., 56, 1048 (1952). (6) G.W. Castellan, Ber. Bumengea. Phyaik. Chem., 67, 898 (1963).
Volume 70, Number 7 July 1966
2320
GORDON G. HAMMES AND PAUL R. SCHIMMEL
tended to include even more general mechanisms than described by eq 1. General Theory A direct approach for obtaining then relaxation times for the mechanism of eq 1 is to linearize the n differential equations characterizing the me~hanism;~ the resultant n differential equations are of the form dAci -dt
It
=
k-1
(i = 1 . . . n)
a$kAck
(2)
mechanisms are the eigenvalues of the matrix b which is defined by dAl --
(7)
dt = bA[
where At is the column vector of the The matrix b may be written as a product of two matrices, and g, which may be computed by use of rules given by Castellan.6 The matrix r is a diagonal matrix whose typical diagonal element, ri, is
or dAc -= aAc dt
(3)
where Act is the deviation of the ith species from its equilibrium value, ci, Ac is the column matrix of the Ac,‘s, and a is a square matrix of the atk’swhich are known functions of rate constants and equilibrium concentrations. The reciprocal relaxation times, 1/7, are the n eigenvalues of the secular equation
la
-
=
0
(4)
where I is the unit matrix and la - 1/dI is the determinant of the matrix a - 1/71. The solutions to the n differential equations are of the form n
Ac, = CAije-t/’f j=l
(5)
n
+ Cviufa(i = 1 . . . n) u=l
n
Ani = Xv,ASa u=l
The element gis arises from the “coupling” between the ith and j t h reaction. Thus the ith reaction gives rise to the ith row and column of g and premultiplication of g by a diagonal matrix r yields b whose elements in the ith row and column arise from the ith reaction and its coupling to other equilibria. Direct computation yields the matrix b which is triple diagonal with all elements equal to zero except kt
+
k-i
b il - -kt; j = i
- l,i #
bi5 = -k+;
+ 1, i # n
j =i
I
If the first step in the mechanism is unimolecular, b11 = kl Ll; for two reactants bll = k1 (A B) Ll;
+
(a+
+ +
+
+
for three reactants bll = kl AC E) k-1 where the bars designate equilibrium concentrations. Extension to higher order reactions and to b,, is obvious. The relaxation times are given as solutions of the secular equation
(6) (i = 1 . . . n)
where via and 6, are the stoichiometric coefficientsof the ith species in the ath reaction and degree of advancement of the ath reaction, respectively; n,O is the reference value of the mole number of the ith species, and Ata is the deviation of the degree of advancement, Fa, from its equilibrium value, E,. Castellan6has shown that the relaxation times for any chemical reaction The Journal of Physdcal Chemistry
(9)
bit =
In general, when n is large an exact analytical solution to eq 4 is not possible. However, approximate solutions are easily obtained for many special cases. For this purpose, the form of the matrix a is inconvenient. Alternatively, we may introduce a change of variables which will yield a coefficient matrix of a more tractable form. Adopting the transformation employed by Kirkwood and Crawfords and Castellan,6 we write ni = n?
where the product is taken over only the reactants of the ith reaction. (Obviously, rr can also be written in terms of the reverse rate constants and concentrations of products if desired.) The typical element, gi5, of g is
lb
-
=
0
(10)
The analysis given above shows that the jth equilibrium is represented by the jth row and column of b. This fact will be of importance when it is desirable to reduce the order of the matrix for various special cases considered below. Case I: Calculation of Relaxation Times for Epuilibria Coupled to Only Relatively Fast Reactions. If the
CALCULATION OF RELAXATION TIMESFOR COMPLEX MECHANISMS
k1
+ k-1 -kz
0
-k-l
IC:!
+ 0
2321
k-2
0
-k-2
-kj
kj
+
IC-5
-1
/ ~ -k-j
0
0
-k,
(12) k,
+ k-,
I
=o Application of the theorem “If each element in one column is expressed as the sum of two terms, then the determinant is equal to the sum of two determinants, in each of which one of the two terms is deleted in each element of that column,”’ to t h e j t h column of b enables eq 10 to be written as
or
(13) where (bj5(is the cofactor of b,, and is formally obtained by deleting the j t h row and column of Ibl. The matrix b,, is a block diagonal form; the eigenvalues of a block diagonal form are the eigenvalues of the individual blocks. Thus, after the longest relaxation time is computed, corresponding say to the jth equilibrium, two sub-matrices are constructed, one consisting of all of the matrix elements above and to the left of b,,, the other consisting of all elements below and to the right of 74,. The longest relaxation time of each sub-matrix is then computed by the same procedure used for the original matrix, b. This matrix “factorization” procedure is continued until the shortest relaxation time of each sub-matrix is obtained; this is the diagonal element associated with the fastest reaction in each submatrix. (For a 1 X 1 matrix, [b15[is defined as unity.) This procedure may be used to obtain explicit expressions for each of the n relaxation times as long as steps which equilibrate at comparable rates are not coupled together via rapid reactions. Case 11: Calculation of Coupled Relaxation Times. If two of the equilibria, the kth and mth, equilibrate at a comparable rate and are coupled via rapid reac-
(14) where (b,,,,,l is formally obtained by deleting the kth and mth row and column of Ibl. (For a 2 X 2 matrix lblck,mml is defined as being equal to unity.) Therefore, the relaxation spectrum can be explicitly evaluated if no more than two steps, which are coupled via rapid reactions, equilibrate at comparable rates. In a similar fashion, solutions can be constructed for cases where more than two equilibria equilibrate at comparable rates and are coupled via rapid reactions. However, cubic or higher order equations must be solved so that such a procedure is of little practical utility. Evaluation of the Determinants. The determinant Ibl is triple diagonal in form with each diagonal element being a linear combination of the two adjacent offdiagonal elements, that is
Because of this relationship, the determinant can be easily evaluated. We perform the following operations : add column two to column one; multiply column one by k-l/kl and add to column two; multiply column one by kllk-1. These operations do not change the value of the determinant.’ We now add column three to column one; multiply column three by (1 k-l/kl) and add to column two; multiply column three by 1/(1 L1/ kl) ; multiply column two by k-z/kz and add to column three; and finally multiply column two by kz/k-:!. Repetition of this procedure n - 1 times yields eq 15
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(7) F. B. Hildebrand, “Methods of Applied Mathematics,” PrenticeHall, Ino., Englewood Cliffs, N. J., 1952, Chapter I.
Volume 70, Number 7 July 1966
2322
1;
lbl
GORDON G. HAMMES AND PAULR. SCHIMMEL
=
0 k2 0
0 0
. .
. .
0 0
. . . . k, .
If reactions of higher molecularity than one are involved, the rate constants are modified as previously discussed.
. . .
Summary of Results and Illustrations of the Method For the reaction mechanism being considered, the relaxation time of the kth equilibrium which is coupled via rapid reactions to the ith thru k - 1 and k 1 thru mth equilibria is
+
where n-1
K?-l = I I k - , / k , ;
Kn-ln-l
1
= 1
r=i+l
and kl and IC-, are understood to be functions of equilibrium concentrations as previously discussed. This determinant can be easily evaluated to give n
Ibl
=
IIkf i=l
n-1
n-1
is1
j=o
+ IIk,k-nCKjn-'
where Pm8is defined by eq 17 and is easily evaluated by examination. If the jth and mth reactions equilibrate at comparable rates and are coupled to each other and the ith thru j - 1st and m 1st thru rth equilibria via rapid reactions, the two relaxation times are solutions of the equation
=
+
'Pm+;) ( 1 / 7 )2 (P!-'P~+; Pfm-'Pm+{)(1/7)
(Pj-
It will be useful to define the symbol i=m j=p+l
p=m-1
Pj+lm-
+
+ P:
=
o
(22)
As a particular illustration of the methods described, consider the mechanism
where products running from a higher to a lower integer equal unity and m is an integer greater than zero. Each term can easily be written by examination. For
slow
8
example, form = 1 the first term in the sum is IIk-,;
A
j-1
+B
kz
ki
Xi k-1
very slow
slow
fast
I r----~ r----~
i
ka
X2
k-a
7Xa k-3
ki
fast i
r
Xq k-r
ki
i
C k-a
+D
8
the second term is kl
II kqi, etc.
The "very slow" step is coupled to all of the other reactions; hence the relaxation time is
i=2
The matrix b,, is obtained by deleting the jth row and column of b . Since this matrix is in block diagonal form, the determinant is the product of the determinants of the two blocks (eq 18)
_1 --- P15 r4
P13Pb5
X
0
*
-k,-1
k,-1
+ k4-1) 1 k,+l
+ k-c,+u
-k,+2
---k-($+l,
. *
0 kn-1
-k n
0 By use of the procedure previously described for evaluating Ib], we find that
(hjjl =
P1j-'P1+ln
(19)
From a similar analysis it may be shown that the determinant ( b j f , k k l is the product of three determinants The Journal of Physical Chemistry
+ k-(n-1)
+
0 0 -k-(n-l) kn IC-n
+
(18)
Pi5 = k-lk-&-3k4k4' kltk-'2k-3k-4k4' f kl'k2k4k-4k-5' kltk~k3k-4k--6' kl'k2kSk4k-5' kltk2k3k4k6
P13= k-lk-2k-3
+
+ kl'k-2k-3
+ + + kl'k2k-3 + kl'kzlc3
CALCULATION OF RELAXATION TIMESFOR COMPLEX MECHANISMS
2323
Discussion The validity of the schematic method rests on the fact that the ith reaction in eq 1 is coupled to the i - 1 and i 1 reactions only (i f 1, n) and the products of the ith reaction are the reactants of the i 1st reaction; if this were not the case, Ibl would no longer be in the proper triple diagonal form and could not be easily computed. The method is applicable to any reaction mechanism where the individual reactions bear this relationship to each ot,her regardless of the molecularity or order of the individual reactions. For higher order reactions the associated matrix elements are obtained exactly as previously discussed for bll. Under certain limiting conditions the method may be employed to compute relaxation times for more complex forms of the general mechanism given by eq 1. For example, if the vertical pathways equilibrate rapidly compared to the horizontal steps in mechanism 23
+
C
The step X4 tions, so
+ D is not coupled to any other reac-
(note P64= 1). The sequence (below eq 23) ki’
k2’
+ B’ + . . . - X i ‘
A‘
A
A
k-1‘
TZ
11 11 kl”
It
+ B” + . . . - X i ”
I
k-1”
A
+B
slow
-. k2”
J
k-2’’
slow
fest
Z X i Z X z G X 3
is not coupled to the remaining reactions. The relaxation time of the fast step is
-1 = p 2 72
k2
+ k-2
k,’
P’ k-,,’
It
11
It I t A“
. . . Xn-l’
+
. . Xn-1”
+ Q’ + . . .
I t 11 k,”
It I t Pf’
k-,”
+ Q“ + . . .
the relaxation times for the horizontal steps are easily evaluated by the schematic method. Equation 23 corresponds to eq 1 if we regard X, in eq 1 as the total concentration of the jth species, i.e., X , XI, X”,. The rate at which t h e j - 1 species is converted to the jth species is
+
+
while the relaxation times for the two slow steps are solutions of the quadratic
where
P22 = k2 Pza = k-zk-3
+ k-2
+ k2k-3 + k2k3
Pia = k-lk--pk~ 4- kl’k-2k-3 f kl’k2k--3 4-kl’k&g
Pi2 = klkz
+ klk-2 + k-lk-2
Thus we have explicitly evaluated the relaxation spectrum of our illustrative mechanism.
If a reaction is of higher order than one, the situation is somewhat more complex, but the same procedure given above may be used to calculate k,*. For example, in the case of a bimolecular reaction, the rate at which A and B are transformed into XI is given by the equation Volume 70, Number 7 July 1966
GORDON G. HAMMES AND PAUL R. SCHIMMEL
2324
ki(A)(B)
+ kit(Af)(B') + klff(A")(Bff)= ki*(A + A' + A")(B + B' + B f f )
where
Proceeding exactly as before (cf. eq 8 and 9), it can be shown that the n X n matrix b describing the relaxation of the horizontal steps is identical with that for eq 1 except kt is replaced by ki*. Any number of rapidly equilibrating vertical steps can be added to the general mechanism. Since each series of vertical steps is separated by the slowly equilibrating horizontal reactions, the relaxation spectrum of each vertical group may be obtained by the methods described in this paper. A procedure such as that used above is applicable only if the vertical steps equilibrate rapidly relative to the horizontal steps (or vice versa). For the case of a single reactant and product, eq 23 is characterized 2 relaxation times, although 5n 2 equilibby 3n ria exist. If the vertical and horizontal steps equili-
+
The J o u r d of Physicat Chemistry
+
brate at comparable rates, the fact that some of the reactions are thermodynamically dependent must be taken into account. For this general case, the order of the secular equation, eq 10, can be reduced in a straightforward manner by the procedure given by Castellan.6 The determinantal method and matrix factorization procedure may then be applied to the reduced equation. The results obtained here are useful in that the relaxation spectrum of the general mechanism in eq 1 and certain more complicated mechanisms may be obtained without explicitly doing the associated complex mathematics. The determinantal method and matrix factorization procedure which are outlined here are a reformulation and extension of previously described results and may be applied to an arbitrary reaction mechanism; only the actual evaluation of the determinants will differ. In general, the cofactors Ib,,l etc., will not be in block diagonal form so that factorization into lower order determinants may not be possible. Thus, for example, after the longest relaxation time is found, the entire determinant Ib,,l must be considered (rather than two separate determinants) in finding the next longest relaxation time.
