J. Phys. Chem. 1991, 95, 4166-4171
4166 40 30
-
20 h
10 -
-10 -
-20 -
-30 -40 -
1-50 -
,
50
I
40 -
30 20 -
10 -
5 >
0-
-10 -20 -
-30 -
-40 -
I
"
, ,
/
,
,
,
-50-40 -30 -20 -10 0
,
,
,
,
,
10 20 30 40 50
x (A, Figure 19. Instantaneous views of a replicated 12-A-thicksegment of the system down the hexagonal axis, after 45 and 105 ps.
Discussion In the previous sections we discussed the average structure of the hexagonal mesophase E and, where possible, made comparisons with results for the L,phase; the respective compositions being 90938 in the former case and 15718 in the latter. In many respects, it is surprising how closely the results for the cylindrical micelle agree with those for the quasi-spherical micelle, even though the surface area available to each head group differs by roughly a factor of 2. In particular, the degree of water penetration into the core region and the mean conformational structure of the hydrocarbon tails are surprisingly similar. However, the degree
of counterion condensation undergoes a small, but distinct, change and a shift of preferred 'location" in the concentrated solution. The calculation commenced with one hydrocarbon tail actually pointing out of the core region into the solution. It was gratifying to see the hydrophobic effect at work and the subsequent retreat of the tail toward the core. In summary, we have presented results of a simulation on the structure of an hexagonal liquid-crystalline phase of sodium octanoate and water. The length of the simulation and the system size precluded the study of self-assembly or even monomer diffusion between aggregates. Nevertheless, we have obtained some indication of the nature of the solvation of the polar head groups, the degree of counterion condensation, the conformational structure of the hydrocarbon tails, and the extent of water penetration into the micelle core. For the future, serious consideration needs to be given to improved modeling of both amphiphile and the solvent. Recent work on polarizable water models has revealed important effects on the solvation characteristics of simple anions.44 It is to be expected that the use of a polarizable water model should also yield interesting effects for micellar systems. Effects of polarization for the micelle core may well also be important. Recent molecular dynamics studies of dense monolayers of long-chain molecules45 has revealed significant deficiencies in the usual pseudoatom potential parameters for the >CH,group. On the other hand, all-atom potentials that explicitly include H atoms46are appreciably better in this It will therefore be important to reinvestigate the present problem with an all-atom potential model for the hydrocarbon chains. Another possibility is to develop a more consistent set of pseudoatom parameters for long-chain molecules. Finally, it would be of great interest to explore in more detail the nature of the shape fluctuations in micellar aggregates that are hinted at in Figures 18 and 19.
Acknowledgment. We thank the National Institutes of Health for their support under GM-40712 and RR04882. (44) Sprik, M.; Klein, M. L.J . Chem. Phys. 1988,89. 7556. Sprik. M.; Watanabe, K.; Klein, M. L.J . Phys. Chem. 1990,94. Sprik, M. J. Phys. C (Condensed Mutter) 1990, 2, SA 16 1. (45) Bareman, J. P.; Klein, M. L. J . Phys. Chem. 1990, 94, 5202. (46) Williams, D. E. J. Chem. Phys. 1967, 85, 1613. Ryckaert, J.-P.; McDonald, 1. R.; Klein, M. L. Mol. Phys. 1989, 67, 957.
Conditions for Reactlon Mechanisms P.L.Corio* and &MY
G.Johnson
Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506-0055 (Received: August 31, 1990)
Several conditions are developed for mechanistically complete reactions and illustrated by examples, including the commercial carbonylation of methyl acetate to acetic anhydride, and a mechanism proposed for the control of nitrogen oxides in engine exhausts. The conditions, which are based on conservation principles and the theory of linear equations, apply to reactions in closed systems, but are otherwise independent of the energy, the kinetics, and the physical states of the components. In addition, a method is given for constructing all possible complete reaction mechanisms consistent with a given set of reaction components.
Introduction An arbitrary reaction mechanism may be written
k o , ~=,o
a = I , 2,
...
(1)
i= I
where diu is the coefficient of component X iin step a,taken with a plus or minus sign according as X, is a product or reactant, and 0022-3654/91/2095-4166$02.50/0
K is the total number of chemical components. Aside from the reversibility or irreversibility of each step, which must be established by experiment or assumption, the reaction matrix u = (u,J determines the chemistry. In particular n = rank u (2)
is the number of independent mechanistic steps. 0 1991 American Chemical Society
7'he Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4167
Conditions for Reaction Mechanisms In a closed system, the elements of u satisfy the orthogonality relations I
Ca,juj, = 0
a = 1, 2,
...
(3)
j=I
where the av are integers determined by the conservation conditions I
in which [X,] is the molar concentration of Xi at any time, and c, is a constant determined by the initial conditions. If N denotes the number of distinct atoms comprising the components, and the conservation conditions are realized as atom and charge conservations, then the index i ranges from 1 to N 1 or N,according as there is or is not a charge conservation condition. The number of independent conservation conditions is equal to the rank of the matrix A = (aij): v = rank A (5) For mechanisms that are complete in the sense that they conform to the principles of mass and charge conservation, and for which catalysts are completely regenerated and intermediates entirely eliminated by the chemistry'*2 ~ = p + r + i = n + v (6) where p is the number of reactants, A the number of products, and L the number of intermediates.' For example, a suggested mechanism' for the coupling of hexachlorccyclopentadiene to bis(pentachlorocyclopentadienyl) is c5c16 cucl a csc6'cuc~ C~Cl~*CUCI CSCI5 CUClZ
+
+
--
+
2C5CI5 ClOCllO (7) Equations 2 and 5 yield n = v = 3, and since K = 6 and p = A = L = 2, eqs 6 are satisfied. Note that a reversible reaction is counted as a single mechanistic step. On the other hand, the following mechanism, proposed as a method for controlling nitric oxide in engine exhaust^,^ does not conform to eqs 6: HNCO NH CO NH + N O - H N2O H + H N C O NH2 + CO NH2 N O N2H + O H NH2 + N O N2 H2O NZH N2 H OH C O + H + CO2 (8)
-
+ +
+
+
+
-
4
+
+
+
The first equality of (6) is satisfied ( K = 12, p = 2, A = i = 5 ) , but the second is not ( n = 7, v = 4). It is necessary to append an eighth independent step, using the given components, to satisfy eq 6. One possibility, as shown in the Appendix, is NH2 O H N H H20 (9)
+
-
+
The values of p and A can also be obtained from the stoichiometric relations 2c5cl6 2 c u c I ClOCllO 2cucI2 (10) 4HNCO + 6 N 0 2CO + 2C02 3N2 2N2O 2H20 (11) but in the case of catalyzed reactions each catalyst must be counted once in computing the sum p + A. For example
+
+
-
+
+
+
+
(1) Corio, P. L. J . fhys. Chem. 1984, 88, 1825. (2) Corio, P. L. Topics in Current Chemistry; Springer-Verlag: Berlin, 1989; Vol. 150, p 249. (3) The rank of a matrix is zero if and only if every one of its elements is zero, so that, excepting the case when there is no reaction whatever, nand Y must be 21; hence rand p + T must be 22, while0 5 L 5 K - 2. (4) Roberts,C. W.; Haigh. D. H.; Lloyd, W. G. J. fhys. Chem. 1960,64, 1887. ( 5 ) Miller, J . A,; Fisk, G. A. Chem. Eng. News 1987, 65 (Aug 31), 38.
V3t
+ Fe3+
-
V4+ + FeZt
+
(12)
+
is catalyzed by cupric ion, so that p A = 4 1 = 5. Although eqs 1 are useful in constructing reaction mechanisms, they do not constitute a complete set, that is, incomplete mechanisms can also satisfy these equations; for instance A+B+C+D 2D E
-
C+E-F+G
(13)
The difficulty in this mechanism is easily discerned by inspection, but intuition may not be so trustworthy in more complex cases. In this paper, several additional relations are developed that impose further constraints on the mechanistic parameters and disclose the difficulties with mechanisms such as (13) by straightforward tests. Mechanisms and Stoichiometries A proposed reaction mechanism must generate the observed stoichiometry, and internal consistency requires that it be possible to obtain the stoichiometry by algebraic elimination of the intermediates. For this purpose, we assume the reactions are arranged so that the first n steps of the mechanism are independent. Any dependent steps can be ignored insofar as the elimination of intermediates is concerned, since everyone of them can be expressed in terms of the independent steps. Furthermore, all of the i intermediates must occur as components in the independent reactions selected. For suppose an intermediate occurred in the pth reaction, p > n, but not in any of the chosen independent reactions. If we write the reactions as K-dimensional vectors R, = (uia, uZu,..., uK,), the dependent reaction Rp can be expressed in the form Rp = cd,R, u=I
for suitable constants dp,. Now suppose that the coefficient of the intermediate occurring in Rp, but in no R,, cy = 1, 2 , ..., n, is the qth entry, so that uqp# 0. But the qth entries in R I , R2, ...,R, are zero by assumption, so the right side of the preceding equation requires uw = 0. We may therefore confine our attention to the first n reactions. For convenience, we assume the reaction components are labeled so that the last i components are the intermediates. To eliminate the intermediates, we multiply eq 1 by a, and sum over a,obtaining
The intermediates will be eliminated from this expression if and only if the coefficient of each intermediate vanishes. Therefore n
~ U , ~ U , = O i=p+*+l,p+r+2
,..., K
(15)
,=I
The nonvanishing terms of (14), namely P+* n
C ( C QiaaJXi = 0
i=l ,=I
(16)
with the au's obtained from the solutions of ( 1 9 , provide the stoichiometric relations defined by the mechanism. Equations 15 constitute a system of i equations for the n unknowns a,, a2,..., a,. The coefficient matrix of this system, which will be denoted u', consists of i rows of the matrix u, so that rank U' I rank u = n. We can disregard the case where rank u' = n, since the necessary and sufficient condition for solutions other than the trivial one, when all ai are zero, is rank u' < n (17) This leads to the following conclusions: (I) If there are fewer intermediates than independent reactions, eqs 15 always have solutions other than the trivial solution, since (17) is necessarily
4168 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 true when i < n. In other words, the intermediates can always be eliminated when i < n. (11) If the number of intermediates equals the number of independent reactions (i = n), the necessary and sufficient condition for solutions of (15) other than the trivial one is det u' = 0. (111) If the number of intermediates exceeds the number of independent reactions (i > n), condition ( 1 7 ) requires that the determinant of every n-dimensional matrix contained in u' vanish. The number of such determinants is
- n)!n!
i!/(i
(18)
which reduces to case (11) when i = n. In either case, the vanishing of these determinants imposes stringent conditions on the elements of u. In the case of a mechanism containing m independent steps without intermediates, these m steps evidently constitute m independent stoichiometric relations, so the preceding discussion need only be applied to n - m independent reactions of the mechanism containing intermediates. It is now easy to detect the difficulty within mechanism 13, namely, n = i = 3, but 1 0 & t o = 1 -2 10 1
-1 0 #O -11
so that the intermediates cannot be eliminated. Indeed, it is evident that C and D are generated in the same ratio in step one, but are not consumed in the same ratio in subsequent steps. Some of component C will remain a t completion, so that it functions ambiguously as a product and an intermediate. Each independent solution of eqs 15 when substituted into (16) yields an independent stoichiometric relation, the total number of stoichiometries being n - rank u'. Now the number of such relations as determined by the observed stoichiometry',2 is p + A - us, where us is the number of independent conservation conditions determined from the observed stoichiometry. The integer p A - us. called the multiplicity, satisfies
+
p+x-u,ll
(19)
since there must be at least one (nontrivial) stoichiometry. For internal consistency the multiplicity determined by the mechanism must equal the multiplicity determined by the observed stoichiometry; hence rank u' = n - p - A
+ us
(20)
It should be remarked that in the case of a catalyzed reaction, each independent solution of eqs 15 should produce a (net) zero coefficient for each catalyst when substituted into (16). As an illustration of these results, consider the mechanism proposed6+' for the commercial carbonylation of methyl acetate to acetic anhydride CH,C(O)OCH, CH31
+ --
+ HI
CH,C(O)OH
+ Rh(C02)12-
Rh(CO)(COCH3)13-
Rh(CO)(COCH,)I,-
CO
Rh(C0)2(COCH,)I3LiI
+ CH3C(0)OLi
CH3C(O)OH
+ CHjC(0)I
Rh(CO)l(COCH3)13-
is equal to the seventh minus the fifth plus the first, so that n < 7. In fact, rank u = n = 6. It is also clear that conservation of rhodium is equivalent to conservation of negative charge, so that u < 7. In fact, rank A = u = 6. It follows that the mechanism conforms to eqs 6. Since n = i,the necessary and sufficient condition for elimination of the intermediates is that det u' = 0, and an easy calculation shows that this condition is satisfied. On omitting the dependent sixth step and eliminating the intermediates from the remaining independent steps, we obtain CH3C(O)OCH* CO .-* CH,C(O)O(O)CCH3 (22)
+
the catalysts cancelling, as they should. The application of (20) to this case is straightforward after noting that the value of us must include a contribution from each catalyst.2 Since there are 3 catalysts, and (22) yields only two independent atom conservations, us = 3 + 2 = 5. It follows from (20) that rank u' = 6 - 6 5 = 5, which may be easily confirmed by computation.
+
Useful Inequalities When the stoichiometry is known, the value of us provides a lower limit to the value of u. Furthermore, if N is the number of distinct atoms in the stoichiometric relation, the value of u cannot exceed N + 1 or N , according as there is or is not a charge conservation condition, so that us I u I N (or N + 1 ) (23)
In particular cases, the upper limit may be smaller since certain groups of atoms (methyl, phenyl, hydroxyl, etc.) often participate in chemical reactions as individual entities. Although the value of u is in principle determinable by experiment, in practice the matter may be quite difficult. It would be necessary, for instance, to determine all components in the reaction system, some of which might not be present at the moments chosen to make the observations, or might be present in amounts beyond the limit of detection. In such cases, (23) can be useful, and might even settle the matter. For example, the pyrolysis of acetaldehyde to methane, carbon monoxide, ethane, and hydrogen yields us = 3, N = 3, so that, by (23), u is 3 or 4. If it is assumed that no ions are generated during the reaction, then u = 3, and any mechanism must conform to the conditions K = 5 + i = n + 3. It is often true that us = u, but, in general, us I u. The nitration of the aromatic ArH illustrates the point: ArH
-
nx
ArN02 + H 2 0
+ -
H2N03++ X-
HZN03" z HZO
+ LiI CH3C(O)O(O)CCH3 + HI
CH3C(0)O(O)CCH3
(21)
This mechanism has five reactants, three of which are catalysts (HI, Rh(CO),I;, LiI), one is a product, and six are intermediates, so that K = 12. It can be seen by inspection that the sixth step (7) Zpeller, J. R.; Cloyd, J. D.; Lafferty, N . L.; Nicely, V. A,; Polichnowski, W.; Cook, S. L. Adu. Chem., to be published.
(24)
+
H N 0 3 + HX
+ CH3C(O)I CH3C(0)OLi + CH3J
( 6 ) Haggin, J. Chem. Eng. News 1990,68 (May 21), 38.
+ HNO,
which is catalyzed by any strong acid HX. The rank of the matrix obtained from the conservation conditions for Ar, H, N, and 0 is 3, so that us = 3 1 = 4. Assuming that the aryl group participates in the mechanism as a unit, then N = 5, namely, H, N, 0, Ar, X. Strong acids in the presence of water suggest a charge conservation condition, so that 4 5 u 5 6 , as required by (23), and p + A = 5 . A suggested mechanism* for the nitration is
Rh(C0)ZIz-
+ CH3C(0)OCH3
CH3C(O)I
+ CH3I
Corio and Johnson
NO2+ + ArH
ArN02H+ + X-
-
NOZ+
ArN02H+
ArNOz + HX
(25)
From the mechanism, we find, using ( 5 ) , that u = 5. Furthermore, n = 4, so that with K = 9, i = 4, and p + A = 5, eqs 6 are satisfied. Since n = L = 4, the intermediates can be eliminated if and only if det u' = 0, and it is easily verified that the mechanism satisfies this condition. In fact, the rank of u' is 3, as required by the general formula, eq 20, and elimination of the intermediates leads to (24). ~
~
(8) Gillespie, E. D.; Ingold, C. K.; Reed, R. I . Nature 1949, 163, 599.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4169
Conditions for Reaction Mechanisms From (17) and (20). we find the general relation
(26)
nlp+u-v,
the sign of equality holding when u' = 0, that is for a reaction without intermediates. Thus, the number of independent reactions in a mechanism cannot be less than the multiplicity. For the pyrolysis of acetaldehyde mentioned above, n 1 2. Combining (26) with (6), we find3 Y
- Us 5 L 5 K - 2
(27)
1
(28)
while (19) and (6) yield n+u-u,-tl
Note that in the particular case u = us, which occurs frequently in practice,'v2 the number of intermediates cannot exceed n - 1.
Additional Conditions on the Reaction Matrix The discussion of eqs 15 led to certain conditions that the elements of the reaction matrix u must satisfy. We now call attention to some conditions imposed on the a,, by eqs 16 on the assumption that eqs 15 admit solutions in which the a, are integers not all zero. The stoichiometric coefficients will then be integers, determined to within a multiplicative factor. Aside from this factor, the coefficient of X,in (16) is n
Zu,.a,=b,
u= I
i = 1 , 2,..., p + u
(29)
+
there being one set of b's for each of the p u - Y , independent stoichiometries. By virtue of (26),there are more equations than unknowns, so we may delete any dependent equations, leaving, say, m equations in n unknowns, whose consistency requiresgJO that the greatest common divisor of all m-rowdeterminants formed from the matrix of coefficients be equal to the greatest common divisor of all m-row determinants in the augmented matrix. If the stoichiometry is known, this theorem imposes additional conditions upon the uh, which may be used in the construction of a mechanism. In the following section, we shall describe a procedure for constructing mechanisms that takes these conditions into account automatically. The ula also satisfy certain inequalities determined by the function of the reaction components. For example, if the first subscript in u, refers to a reactant, then uSa< 0 for at least one mechanistic step a. It may be true that component s has a negative coefficient in other steps, but the inequality is only guaranteed for one step. There will, however, be one such inequality for every reactant. Similarly, if s refers to a product, um > 1 for at least one step a. If s refers to an intermediate, then us, > 1 for some a,but u , < ~ 1 for some 0 # a. In this way, the uia are subject to p + T + 21 inequalities.
Construction of Reaction Mechanisms The n independent steps of a mechanism with K components are determined by Kn elements of the matrix u. These elements are subject to the orthogonality conditions, eqs 3, which imply that the Kn elements can be expressed in terms of n2 parameters. For if all components of the reaction are known or assumed, we can solve eqs 3 for the uiarfor some fixed a. Since the rank of the system is Y , there are n = K - u independent solutions which form a basis, and every solution may be expressed as a linear combination of this basis. The customary procedure for solving linear systems is not applicable, since we require not merely solutions in which the ,a are integers, but all integral solutions. It is not sufficient to take integral linear combinations of any set of n independent solutions. To see this, consider the gas-phase reaction of nitrogen dioxide and chlorine monoxide'' 2N02 + C120 N03C1 + NO2Cl (30) +
(9) Smith, H. J. S. Philos. Trans. R. Soc. London 1861, 151, 312. (IO) Hcger, 1. Denkschr. Akad. Wiss. Wien. Malh.-natunv. KI.Bd. 1 8 9 , I 4 (II), 1 1 1 .
and suppose. there is only one intermediate, namely, OC1. Labeling the components NOz, CI20, N02C1, OC1, and N03Cl 1-5, in that order, the conservation conditions for nitrogen, oxygen, and chlorine are Qla +%a +osa = 0
+cYh +20h
241,
-4,
+3050! =
0
(31)
242, -3Cr + ~ 4 a *sa = 0 Two independent solutions of these equations are (0,-1, 1,2, -1) and (-2, -1, 1,0, l), and linear combinations of these solutions with integral coefficients yield infinitely many solutions of ( 3 l ) , but infinitely many solutions are omitted, for example, for the solution (-1, -1, 1, 1,0), and all integral multiples of it. On the other hand, the solutions xI = (-2, -1, 1, 0, 1) and x2 = (1, 0, 0, 1, -1) are independent and generate all integral solutions of (31) in the form clxl c2x2,with cl and c2 integers; that is, xl and x2 constitute a fundamental system of solutions. This follows from a theorem of Smith,I2 which states that the independent solutions of the system (eqs 3 ) form a fundamental system when the nonvanishing determinants of highest order formed by compounding the solutions into a matrix are relatively prime, that is, their greatest common divisors are f l . A fundamental system of this type will be called a Smith basis. A systematic procedure for constructing a Smith basis is described in the Appendix. It should be noted, as illustrated in the Appendix, that although a Smith basis has certain essential mathematical properties, it is not unique; the various steps of a mechanism can be expressed in terms of any Smith basis, which need not be a set of independent elementary reactions. If we use the Smith basis x1 and x2, the two independent reactions of the mechanism may be written pxl qx2 and rxl sx2, where p, q, r, and s are integers satisfying ps - qr # 0 to ensure independence of the reactions. In chemical notation, the reactions are ( 2 p - q)N02 pCl20 .+ pN02CI + q O C l + (p - q)NO3Cl (32) (2r - s ) N 0 2 + rC120 rN02Cl + sOCl + ( r - s)N03CI (33) Thus, the ten elements utaare expressed in terms of four parameters. Since L < n, the intermediates can be eliminated, and when this is done the stoichiometry of (30) is obtained after cancellation of the nonzero factor ps - qr. To determine all possible mechanisms, we must consider the inequalities satisfied by p, q, r, and s. Suppose reaction 32 is the first step of the mechanism. In that case, we must have p ( q ) > 0, since the intermediate OC1 must be formed in that step. Furthermore, the first step must include NO2, as otherwise atomic chlorine would be generated which is not assumed to be a component; hence 2p - q > 0. As N03CI is a product, it may or may not appear in the first step, so that p 2 q. Therefore, for any integer p 1 1, we must have q = 1, 2, ...,p. Consequently, the molecularity of the first step is 3p - q, which is 2, for p = q = 1, but is 2 4 for values of p > 2. The first step of lowest molecularity is NO2 + Cl20 N02CI OCI (34)
+
+
+
+
-
+
-+
which is suggested by the kinetics." With p = q = 1 for (32),the product N03CI must be generated by the second step, so that r > s, with s < 0, since the intermediate must be consumed in that step. Thus for s = -& (& a positive integer) r = -k + 1, -k + 2, ... Fuithermore, with p = q = 1 and r > s, ps - qr # 0, as required. Only one bimolecular reaction is possible for the second step (s = -1, r = 0) NO1 OC1 NO3CI (35)
+
-
( 1 1) Martin, H.; Meise, W. Z . Elektrochem. 1959, 63. 162. (12) Smith, H. J. S.Philos. Trans. 1861, 151, 293.
Corio and Johnson
4110 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 so (34) and (35) constitute the most reasonable mechanism on
the basis of lowest molecularity, and the observed kinetics is consistent with it, provided (35) is much faster" than (34). There is, however, an interesting termolecular reaction: 20C1 NO2CI -.+ C120 + NO3CI
+
Although CI20 is regenerated in this step, it does not function as a catalyst, since it appears in the left member of the stoichiometric relation with a nonzero coefficient when OCI is eliminated between (34) and (36). Assuming the steady-state approximation is applicable, it is easily shown that (34) and (36) are also consistent with observed kinetics, but low concentrations of the intermediate OC1 would render the third-order reaction less probable. The preceding example illustrates the construction of reaction mechanisms from elementary principles but is scarcely representative of the numerous possible mechanisms that can arise when the number of independent mechanistic steps increases. The treatment of such problems is best carried out with computers using algorithms specifically designed for such purposes.I3
Concluding Remarks The conditions developed in this paper apply to all reactions occurring in closed systems, presumed to be describable in terms of a complete mechanism with a finite number of independent steps. In particular, the conditions are independent of the kinetics and whether the reaction occurs homogeneously or heterogeneously. They are also independent of the energy, and are therefore applicable to reactions in which the temperature does not have a uniform value throughout the system, such as reactions in flames. In addition, a method has been developed for the construction of all independent reactions consistent with a given set of components. Although the procedure cannot determine dependent steps of a mechanism, any dependent step in expressible in terms of the independent steps, so that, in principle, all steps of the mechanism are determined by this method. The implementation of this procedure by means of computers will be described in a subsequent publica t ion.I3 Appendix Consider the system of Y independent equations in K unknowns U,IXl
+ + a,& 0'.
=0
(A. 1)
it being understood from now on that i = 1, 2, ..., Y and that the uyare integers. The problem of obtaining all solutions in integers was studied divisor Smith'* who showed by mathematical induction that the system A.1 has n = K - Y independent solutions x,, r = 1, 2, .,., n; s = 1, 2, ...,K , such that the determinants formed from the matrix (x,) have the greatest common divisor unity and that these solutions form a fundamental set of solutions in the sense that every solution in integers is expressible in the form
obtained from the latter by adding or subtracting a column of coefficients to or from another column of coefficients. Note also that, excepting the coefficients (I,h f ay, the coefficients in (A.2) are equal to the corresponding coefficients in (A.l), and that the coefficient ( I l h f all in the first of equations A.2 is absolutely smaller than alh. Although the systems A.l and A.2 are not equivalent, their solutions are related. To see this, suppose we have a fundamental set of solutions yrs,r = 1, 2, ..., n and s = 1, 2, ...,K , of equations A.2, so that y r l ,yr2,..,,yrKsatisfy (A.2) for each r, and that these r solutions satisfy the requirements of Smith's theorem. Regrouping terms in (A.2), we have a,Oh **. + a ~ ( y f i yh) + + ai,,)'* = 0 (A.3) Uilyl
+ +
'.'
+
It follows that if y,, r = 1, 2, ..., n and s = 1, 2, ..., K , is a fundamental set of solutions for (A.2) and, therefore the equivalent system, A.3, then
xrs = Yrs (all s # j )
xr/ = Yrj
Yrh
(A.4)
r = 1,2, ..., n and s = 1,2, ..., K , is a fundamental set of solutions for (A.l) with the properties required by Smith's theorem, for the determinants contained in a matrix are not altered by adding or subtracting one column from another; hence, the matrix of solutions (x,) has the properties required by Smith's theorem because it is formed from a matrix (y,) with those properties. Note that in the expression for xri in (A.4) the column operation used to go from (A.1) to (A.2) IS reversed. By repeated transformations of the type described, we can derive a system of equations in which the first has only one nonzero coefficient, say a l l # 0. The new variable associated with this equation is zero, so that we now have a system of Y - 1 equations in K - 1 unknowns. By repeated reductions of the sort described, we ultimately arrive at a system of zero equations in K - Y = n unknowns. Since there are no conditions on this ultimate system, we may assign to the n unknowns any values we please. The n solutions (1, 0, 0, ..., 0) (0, 1, 0, ..., 0), etc., obviously conform to Smith's theorem, and by retracting our steps we derive a fundamental system of solutions for (A.1) with the desired properties. As an example take eqs 31, the matrix of which is
(a 9 ; y ;) (a 4 H y y)
We now subtract column 1 from the third and fifth columns, obtaining
In the transformed equations corresponding to this matrix, the first variable is zero, so we replace the first column by zeros and omit the first row, for bevity:
where the tr are integers. SkolemI4 devised an ingeneous demonstration of Smith's theorem that includes an algorithm for constructing a fundamental set of solutions with the required properties. We shall outline Skolem's proof and illustrate it by an example. Suppose that in the first of equations A.1 the coefficients alh and al, are not both zero, and that lalhl 2 lal,& We then transform system A.l into a new system of equations, namely = 0 (A.2) Oily1 + '*' + ( ( l f h f ay)Yh *'. + aip,
Subtracting the fifth column from the second and fourth columns, we obtain
where the + or - is taken according as alhal/ < 0 or alhal, > 0; that is, according as alh and al have unlike or like signs. Note that system A.2 is nor equiva(ent to system A.1, having been
for which the new fifth variable is zero. Replacing the fifth column with zeros, and omitting the first row, we obtain
(13) Johnson, B. J.; Corio, P. L. To be published. (14) Skolem, Th. Diophclnfische Gleiehungen; Springer: Berlin, 1938; pp 5 , 6.
Subtracting the second column from the third, we obtain a system in which the second variable is zero. At this point we have zero equations in K - Y = 5 - 3 = 2 unknowns, namely the third and
+
+ +
0 1 0 1 1 0 2 1 1 1
)
(
0 0 0 0 1 0 1 1 0 1
)
(
(0
1
1 0 0)
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4171
Conditions for Reaction Mechanisms fourth variables, so we write out the following solution matrix with the Smith properties: 0 0 1 0 0 0 0 0 1 0
(
)
columns three and four being filled in by (1,O) and (0, 1). We now apply to this matrix in reversed order the operations by which it was derived. Thus, we begin by subtracting the third column from the second, obtaining
(:
0'
! :)
required n - m independent reactions. Since the unknown reactions are expressed in terms of a Smith basis, no independent reaction will be overlooked. As an illustration, take mechanism 8, which requires one additional reaction, and number the components 1 through 12 in the order HNCO, N H , CO, NO, H, N 2 0 , NH2, N2H, OH, N2, HzO, and COP The solution of the orthogonality conditions, using the aij obtained from the conservation conditions for carbon, hydrogen, oxygen, and nitrogen, yields the following eight vectors of a Smith basis: X I = (-1, 1, l , O , 0, 0, 0, 0, 0, 0, 0, 0) xz = (0, -1, 0, -1, 1, 1, x3
and continue by subtracting the second column from the fifth, then subtracting the fourth column from the new fifth column, etc., ending with the Smith basis
(
-2
-1 1 0 1 1 0 0 1 - 1
1
The rows of this matrix are the vectors xl and x2 cited in the text. The rows of
2
(
-1
1 -1
)
-1 0 -1 1 1 0
also form a Smith basis, as may be verified by computing all of its 2 X 2 determinants. A Smith basis is evidently not unique, and this fact may be used to advantage in constructing reaction mechanisms." As another application of a Smith basis, suppose we have m independent reactions and wish to construct n - m additional independent reactions, using components in the given reactions. Solving orthogonality conditions 3 by the above procedure, with the ail obtained from conservation conditions 4, we 0btain.a Smith basis for the system. We then construct an n X n matrix M whose first m rows are formed from the coefficients expressing the given reactions as linear combinations of the vectors in the Smith basis. The remaining n - m rows are constructed from the unknown coefficients expressing the desired reactions in terms of the Smith basis. The necessary and sufficient condition that the n reactions defined by M be independent is det M # 0. Any choice of the unknown coefficients consistent with this condition will yield the
o,o,
0, 0, 0, 0)
= (0, -1, 0, 0, -1, 0, l , O , 0,
o,o,
0)
x4 = (0, -2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0) xg = (0, 1, 0, -1, -2, 0, 0, 0, 1, 0, 0,O) x6 = (0, -2, 0, 0, 27 0, 0, 0, 0, 1, 0, 0) x7 = (0, 1, 0, -1, -3, 0, xg
= (-1,
2 9
o,o,
0, 0, 1, 0)
0, -1, -1, 0,0, 0, 0, 0, 0, 1)
Note that xI and x2 are the first and second steps of mechanism 8, so that the first and second rows of the matrix M are (1, 0, 0, 0, 0, 0, 0, 0) and (0, 1, 0, 0, 0, 0, 0, 0), respectively. The elements in the next five rows are easily worked out from the remaining reactions of mechanism 8. To obtain the elements in the last row, denote the desired eighth reaction R, and write it in the form
Substituting for the xi, we obtain the elements in the last row of M. Expanding the determinant of M, we find that the ci must satisfy the condition cz 2cq - cs 2c6 - 2c7 # 0 ('4.6)
+
+
The coefficients cI, cj, and c8 do not appear in this relation, so they may be given any values whatever. For reaction 9, cI = c2 = c4 = c6 = cg = 0, but c3 = c5 = -1, and c7 = 1, so that (A.6) is satisfied. Finally, any dependent reactions may also be expressed linearly in terms of a Smith basis for the system, again with the advantage that no possibility will be overlooked.
