Mass Conservation Implications of a Reaction Mechanism

Research: Science and Education

Mass Conservation Implications of a Reaction Mechanism William R. Smith*† Department of Mathematics and Statistics, and School of Engineering, University of Guelph, Guelph, Ontario, Canada N1G 2W1; *[email protected] Ronald W. Missen Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3E5

In a general sense, a proposed reaction mechanism in chemical kinetics must be in accordance with the reaction stoichiometry. That is, one must be able to obtain from the mechanism the mass-conservation equations applicable to the system being modeled. Obtaining these equations involves responses to precisely worded questions that have been addressed by various authors over the past quarter century. In this paper, we describe, with examples, the underlying basis for algorithms answering such questions. Most recently in this Journal, the problem of determining mass-conservation equations implied by a reaction mechanism has been addressed by Toby (1), who used rate expressions arising from elementary steps of the mechanism to “give a general method for obtaining the time-dependent stoichiometry . . . of a many-step mechanism.” Among the examples used by Toby were mechanisms for the decomposition of azomethane and of acetaldehyde. From each decomposition reaction he obtained two independent mass-conservation equations, rather than the three required, as shown below. Lee (2) subsequently showed that the use of rate laws is irrelevant for the solution of this problem and pointed out that there is no “time-dependent stoichiometry”. Lee used a material-balance method based on the extent-ofreaction variable to obtain Toby’s two mass-conservation equations in each case. The third equation required is also contained in Lee’s analysis, implicitly in one case and explicitly in the other, but it is not pointed out by him. We agree with Lee that the rate-law approach is irrelevant, but we also show that use of the extent-of-reaction variable is an unnecessary complication in a more efficient approach that reveals the fundamental nature of the problem. In the method presented here, the only information required is the stoichiometric matrix, N = [ν′ij], where ν′ij is the stoichiometric coefficient for species i in reaction j of the mechanism (+ for a species on the right side and – for a species on the left side). The number of species is N and the number of reactions is R′, and hence N is (N × R′ ). One of our goals is to provide a universally applicable method for using N to determine the number and an actual set of independent mass-conservation equations implied by a given reaction mechanism. If the formula matrix A = [aki], where aki is the subscript to element k in the molecular formula of species i, is also known, as is usually the case, further information can be obtained about the nature of the conservation relations. This leads to a complementary goal: to show whether a set obtained from N must be used, or whether the element-conservation equations can be used. The number of elements is M, and hence A is (M × N ). † Current address: School of Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada

In pursuit of these goals, three important questions arise: 1. Is N consistent with a closed-system description of the system? If this is true (the usual case), N is called “conservative” (3). If it is not true, N can only represent an open system and is “nonconservative”. 2a. If N is conservative, how many independent mass-conservation equations (C ′ ) are implied by N? 2b. If N is conservative and A is known, what is the number of independent element-conservation equations, C (where C ≤ M )? 3. If N is conservative and A is known (a common situation), what does the value of C ′ − C ≡ r tell us about the number and origin of a set of mass-conservation equations implied by N? Here, r is the number of stoichiometric restrictions (i.e., the number of mass-conservation equations over and above the element-conservation equations themselves; ref 4 ). If, however, A is unknown, how do we generate an appropriate set of mass-conservation equations?

Question 1 has been considered by various authors, for example, Oliver (5) and Schuster et al. and references therein (6, 7). These authors have provided a test criterion and algorithms to determine whether N is conservative, in the absence of knowledge of A. However when A is known, the procedure to determine whether the criterion is satisfied is much simpler (5), as described below. Questions 2 and 3 have been treated by Smith and Missen (4), and examples of stoichiometric restrictions have been given by Björnbom (8) and by Missen and Smith (9). The concept has also been illustrated by Alberty (10, 11). More recently, Smith et al. (12) have implemented the algorithm of Smith and Missen to determine C and a set of independent chemical equations by means of the interactive Java applet JSTOICH on a Web site, which was noted briefly by Judd (13). Here, we show how to use JSTOICH to determine C ′, and hence r, together with a set of C ′ mass-conservation equations implied by N. In the following, we first give examples of N and A. We then consider question 1, developing a simple criterion for the case when A is known, and citing examples of both nonconservative and conservative systems. Next, we consider questions 2a, 2b, and 3, developing procedures for answering the questions raised and illustrating them with examples. Finally, we summarize the important conclusions. Stoichiometric Matrix To illustrate a stoichiometric matrix, N, we use the decomposition of azomethane, (A; CH3N⫽NCH3), to the five products N 2 , CH 4, C 2 H 6 , methylethyldiimide (MED; C 2 H 5 N⫽NCH 3 ), and tetramethylhydrazine (TMH;

JChemEd.chem.wisc.edu • Vol. 80 No. 7 July 2003 • Journal of Chemical Education

833


(CH3)2N⫺N(CH3)2). A six-step mechanism, R′ = 6, given by Toby (1) involves N = 9 species, including, as reactive intermediates, the three free radicals, CH2N•⫽NCH3 (B), CH3• (M), (CH3)2N⫺NCH3• (C):1

A

2M + N2

(1)

A + M

CH4 + B

(2)

2M

C2H6

(3)

M + B

MED

(4)

M + A

C

(5)

M + C

TMH

(6)

N for this mechanism is obtained by arranging the coefficients of the species, ν′ij, in the reactions in a matrix in which the rows represent the species and the columns represent the reactions, with the (arbitrary) order of species being A, M, N2, CH4, B, C2H6, MED, C, TMH, and the order of reactions as above:

N ≡ νij′

=

0 1

0 0

0 0

0 0

0 0

0 0

1 0

0 1

−1 0

0 0

0 0

0 0

0 0

0 0

1 0

0 1

0 −1

0

0

0

0

0

1

N is conservative if and only if there exists at least one positive solution (all xi > 0) to the equation N

∑ ν ′ij xi

(7)

If we interpret xi in eq 9 as the total number of atoms per molecule of species i, the criterion states that there must exist a formula matrix, [aki] such that aki ≥ 0 and the total number of atoms per molecule is positive.2 In addition the total number of atoms is conserved in each reaction j.3 Algorithms for implementing the criterion, without knowledge of A, are given by Oliver (5) and by Schuster et al. (6, 7). If A is known, the test of the criterion given in eq 9 can be carried out in a relatively simple manner (5) and provides a test that the stoichiometric matrix is conservative in terms of the given A. The total number of atoms in molecule i is given by the sum of the entries in column i of A

N

∑ aki ν′ij

∑ ν′ij xi

i =1

To illustrate a formula matrix, A = [aki], we again use the decomposition of azomethane with the N = 9 species involving the M = 3 elements, C, H, and N. The species now form the columns and the elements form the rows. The system may be represented by, {(A, M, N2, CH4, B, C2H6, MED, C, TMH), (C, H, N)} showing the (arbitrary) order of both species and elements. From this choice of ordering,

2 1 0 1 2 2 3 3 4 6 3 0 4 5 6 8 9 12

(8)

2

The (M × N ) matrix A in this case is (3 × 9). Criterion and Algorithm for a Conservative System Implied by N (Question 1) If a proposed mechanism represents a closed system, N obtained from the mechanism must be conservative. A crite-

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i = 1, 2,..., N

(10)

k =1

= 0 ; j = 1, 2,..., R ′; k = 1, 2,..., M (11)

Multiplying eq 10 by ν′ij, summing over i, and using eq 11, the following equation is obtained,

Formula Matrix

2 0 2 0 2 0 2 2

M

∑ aki ;

When A is known, the values of xi from eq 10 are all positive and the test for conservativity of N is that eq 9 must be satisfied. To see the latter, we first note that the formula and stoichiometric matrices are related (4) via

N

=

(9)

i =1

The (N × R′ ) matrix N in this case is (9 × 6).

A ≡ aki

= 0 ; j = 1, 2,..., R ′

i =1

xi =

−1 −1 0 0 −1 0 2 −1 −2 −1 −1 −1 1 0

rion for this, as a response to question 1 above, has been given by Oliver (5) and by Schuster et al. (6, 7):

=

N M

∑ ∑ ν ′ij aki

i =1 k = 1

=

M

N

∑ ∑ ν′ij aki

= 0

(12)

k =1 i = 1

An example of a nonconservative system is given by the hypothetical kinetics scheme (5)

A1

A2 + A4

(13)

A3

A1 + A4

(14)

A3

A4

(15)

where each Ai represents a species whose molecular formula is not known. This system is nonconservative, as shown by Oliver (5), since it does not satisfy the criterion of eq 9. An example of a conservative system is the decomposition of azomethane, described above. N, given by eq 7, is conservative, since, from eq 10, x = (10, 4, 2, 5, 9, 8, 13, 14, 18)T, which satisfies the criterion of eq 9. Criterion and Algorithm for Determining Mass-Conservation Equations Implied by N (Questions 2a, 2b, and 3) Once a kinetics mechanism is shown to be conservative, it is possible to address question 2a about the number, C ′, and question 3 about a particular set of linearly independent

Journal of Chemical Education • Vol. 80 No. 7 July 2003 • JChemEd.chem.wisc.edu


mass-conservation equations implied by N. We emphasize at the outset that C ′ for a given N is unique, but the set is not unique. For the given N, the number of linearly independent reaction steps is, Fs = rank (N)

(16)

which is also the number of stoichiometric degrees of freedom (4), and

C ′ = N − Fs

1. “Number of Components, (C ) = 6” means rank (N) ≡ Fs = 6. 2. “Number of Chemical Equations (R) = 3” means C ′= 3 (= N − Fs). 3. The three equations listed represent mass-conservation quantities involving the indicated species, either as mole numbers or as molar concentrations in a constant-volume system. In the latter case, the corresponding mass-conservation equations are written as:

(17)

Both C ′ and a set of C ′ linearly independent mass-conservation equations can be obtained without knowledge of A, by the procedure described below; that is, it is not necessary to know the molecular formulas of the species involved, as in the Oliver example above. If A is known, as in the azomethane example above, a further characteristic of the system can be determined, as discussed below (question 3). The interactive Java applet JSTOICH (12) was originally created to calculate C, the number, R, and a particular set of chemical equations from a specified formula matrix A. In the following examples, we show how it may also be used to calculate Fs, C ′, and a particular set of mass-conservation equations from a specified stoichiometric matrix, N.

Decomposition of Azomethane Figure 1 shows the main input and the output for the azomethane decomposition example, based on the stoichiometric matrix of eq 7; the “Element Names” are replaced by the reaction designations. Prior to the input shown, the initial input is the “Number of Species”, N = 9, and “Number of Elements” (in this case, the number of reactions) R ′ = 6. In Figure 1, the input consists of: columns headed by the reaction numbers, R1 to R6 for reactions in 1–6; rows indicated by the species names; and the entries of N. The output is interpreted as follows:

[CH4 ]

− [MED] − [B] = c1

(18)

2 [C2 H6 ] + [M] − [C] − 2 [ A ] − 4 [N2 ] − [B] = c2 (19)

[B ]

+ 3[N2 ] + [ A ] − [TMH] − [M] − [CH4 ] − 2 [C2 H6 ] = c3

(20)

where c1, c2, c3 are constants, evaluated from the initial conditions.

In the azomethane example (1), since only A is present initially, c1 = 0, c2 = ᎑2[A]0, c3 = [A]0. Furthermore, as is usual, we may set the concentrations of the intermediate species, [M], [B], [C], to zero to obtain:

[CH4 ]

[C2H6 ]

− [MED] = 0

(21)

− [ A ] − 2 [N2 ] = − [ A ]0

(22)

3[N2 ] + [ A ] − [ TMH] − [CH4 ] − 2 [C2 H6 ] = [A]0 (23)

Any set of three independent equations obtained by linearly combining eqs 18–20 (or 21–23) may be used to express mass conservation for this system. For example, a different set of equations arises when a different ordering of

Figure 1. (Left) Input data and (right) output calculations from JSTOICH for the azomethane reaction mechanism. See text for explanation and interpretation.


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species is used as input to JSTOICH. Toby (1) only obtained two mass-conservation equations; his eq 6 is 2(eq 22) + (eq 23), and his equation (7) is (eq 21) + 2(eq 22) + 2(eq 23). We emphasize that the derivation of eqs 18–20, and hence of 21–23, does not require knowledge of the system formula matrix A (designations of all the species are merely arbitrary labels). The above use of JSTOICH is the procedure required when N is conservative and A is unknown (last part of question 3). Since A is known for this conservative example (eq 8), the most important issue is the number of stoichiometric restrictions, r (first part of question 3). If r > 0, we must use the three equations obtained above from JSTOICH (or any set of three independent equations obtained by linear combination thereof ) to represent the mass-conservation equations for the system. If r = 0, these equations provide no information that cannot be obtained from linear combinations of the element-conservation equations; in this example, these are for C, H, N in turn (with the concentrations of the intermediates set to zero) 2 [ A ] + [CH4 ] + 2 [C2 H6 ] + 3[MED] + 4 [TMH] = 2 [ A ]0 6 [ A ] + 4 [CH4 ] + 6 [C2 H6 ] + 8[MED] + 12 [TMH] = 6 [A ]0

(24)

(25)

2 [ A ] + 2 [N2 ] + 2 [MED] + 2 [TMH] = 2 [ A ]0 (26)

To determine the value of r, we use A as input to JSTOICH in the usual manner, with the following results for this example: C = 3 (question 2b), R = 6. Since C = C ′, r = 0, and hence any set of three independent mass-conservation equations may be obtained from the element-conservation eqs 24–26 (first part of question 3). Thus, eqs 21–23 from JSTOICH are obtained from these eqs as, respectively, −3(eq 24) + (eq 25), 2(eq 24) − (1/2)(eq 25) − (eq 26), −(eq 24) + (3/2)(eq 26). Similarly, Toby’s equations (6) and (7) are −3(eq 24) + (eq 25) + (1/2)(eq 26), and −(eq 24) + (eq 26), respectively. This example illustrates the usual case, that both N and A are known. Furthermore, in this particular case, r = 0, and the element-conservation equations are the only equations that need be used to express mass-conservation. The set of massconservation equations resulting from Toby’s procedure (1), from Lee’s procedure (2), and from JSTOICH are no more than manipulations (linear combinations) of the basic element-conservation equations.

Decomposition of Acetaldehyde Toby (1) has also applied his method to a chain mechanism for the decomposition of acetaldehyde to obtain two mass-conservation equations. Use of JSTOICH for this (conservative) system, as in the previous example, also reveals that C = C ′ = 3. That is, there are three independent mass-conservation equations, which can be written as the three element-conservation equations.4 From another point of view, Missen et al. (14) have used the same system and mecha-

836

nism to investigate the number (R = N − C = 2) and form of independent (phenomenological) rate laws. They used the formula matrix A to determine the number,5 and the same mechanism, together with the usual stationary-state hypothesis, to determine the forms of two independent rate laws. The latter procedure is a matter of kinetics and is outside the scope of the topic of this paper.

Oxidation of Formaldehyde An example that is conservative and illustrates stoichiometric restrictions (r > 0) is the mechanism proposed by Bawn and White (15) for the aqueous-phase oxidation of formaldehyde by cobaltic ion: HCHO + Co3 +

(HCHO⭈Co3 +)

(27)

(HCHO⭈Co3 +)

+ Co2 + + H + HCO• (28)

HCO• + Co3 +

Co2 + + H + + CO

(29)

HCO• + H2O

HCHO + OH•

(30)

HCO• + H2O

HCOOH + H•

(31)

HCHO + OH•

H2O + HCO•

(32)

HCHO + OH•

HCOOH + H•

(33)

HCO• + H2

(34)

HCHO + H•

For the purpose of indicating ordering of species and of elements, the system may be represented by {(HCHO, Co3+, (HCHO⭈Co3+), Co2+, H+, HCO•, CO, H2O, OH•, HCOOH, H•, H2), (H, C, O, Co, p)} where p represents positive charge as an element. Here N = 12 and M = 5 and there are four reactive intermediates: (HCHO?Co3+), HCO•, OH•, and H•. For this example, we know both N and A:

−1

N =

0

1

0 −1 −1 −1

−1 0 −1 1 −1 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0 0

1 1

1 1

0 0

0 0

0 0

0 0

0 0

1 −1 −1 −1 0 1 0 0

1 0

0 0

1 0

0 0

0 0

0 −1 −1 1 0 0 1 0 −1 −1

0 0

0 0

0 0

0 0

0 0

1 1

0 0

1 0 1 −1

0

0

0

0

0

0

0


1

(35)


Using N as input to JSTOICH, we obtain the following results: rank (N ) = Fs = 6

(36)

(37) C ′ = N − Fs = 12 − 6 = 6 Thus, any appropriate set of 6 independent conservation equations involving the species may be used to represent the conservation of mass. One such set is the following, obtained from JSTOICH (prior to setting the concentrations of intermediates to zero),

[H+ ]

+ [ Co3

[Co2 + ] [CO]

+ [ HCHO • Co3 +] = c1

(38)

+ [ Co3 +] + [HCHO⭈Co3 + ] = c2

(39)

+

]

+ [HCO• ] + [HCHO⭈Co3 +] + [HCHO] (40) − [OH• ] − [H2 O] = c3

[HCOOH]

+ [HCHO] + [HCHO⭈Co3 +

[H • ]

[

HCO•

]

+

]

+ [CO] = c4

(41)

+ [HCHO] + [Co3 +] + 2 [HCHO⭈Co3 +]

(42) + 2 [HCO• ] + 3[CO] − [H2O] = c5 2 [CO] + [HCO• ] + [HCHO⭈Co3 + ] + [Co3 + ] − [H2 ] − [H2 O] = c6

(43)

where the quantities on the left are conserved, in the amounts indicated by the constants {c1, c2,…, c6}. Without writing the formula matrix A in full, we cite the results of input of A to JSTOICH as follows: rank A = C = 5.6 Comparing the results from the JSTOICH outputs for N and for A, we obtain r = C′− C = 6 − 5 = 1

(44)

That is, the mechanism implies one stoichiometric restriction. When r > 0, as in this case, we must use the output from JSTOICH, eqs 38–43, as the mass-conservation equations for the system implied by N, rather than the element-conservation equations by themselves. To test a kinetics mechanism when r > 0 (as in the mechanism of Bawn and White; ref 15) so as to be able to “explain the stoichiometric relationships observed” (16), the full set of mass-conservation equations implied by N (in this case six equations, eqs 38–43) must be tested against experiment; failure of these equations to provide consistent results would indicate that the mechanism is incorrect with respect to stoichiometric considerations.7 Bawn and White (16) also investigated the similar oxidation of formic acid. The mechanism proposed by them for this reaction contains no stoichiometric restriction. It is interesting to note that they expressed more confidence in their formic acid mechanism than in their formaldehyde mechanism. Whether this is related to the existence of a restriction in the latter, or to the simpler overall stoichiometry of the

former, or to the postulated reaction steps in either case is an open question. Summary 1. The only information required to determine a complete set of mass-conservation equations implied by a reaction mechanism is the stoichiometric matrix of the mechanism, N. 2. The mechanism, N, should first be tested to determine whether it is conservative (represents a closed system). A simple criterion is given for this, when the formula matrix, A, is known. 3. For a conservative system, if A is unknown, a procedure is given to show how the interactive Java applet JSTOICH (12) can be used to determine, from N, a set of mass-conservation equations for the system. 4. For a conservative system, if A is known, JSTOICH can be used to determine r, the number of stoichiometric restrictions (4) in the mechanism. If r = 0, the element-conservation equations may be used as the mass-conservation equations; any other set of mass-conservation equations is a linear combination of the former. If r > 0, JSTOICH provides a set of (non-unique) mass-conservation equations of the correct number. In both cases, the appropriate set of mass-conservation equations implied by N must be compared with experiment to test the correctness of the mechanism with respect to stoichiometric considerations. 5. Examples are given, taken from the literature, to illustrate the various situations. Acknowledgment Financial assistance has been received from the Natural Sciences and Engineering Research Council of Canada. Notes 1. The abbreviated notation is that of Lee (2). 2. Each species i, whose molecular formula is represented by a column of [aki], has at least one positive entry. 3. Charged species may be accommodated by means of the device of using the number of electrons for a species instead of the charge; see ref 6 for an example. 4. Toby’s equations and those of Lee can be obtained as certain linear combinations of these equations. 5. The formula matrix, A, was used to determine the number, N, on the tacit assumptions of a conservative system and r = 0. 6. JSTOICH output from A also includes the value R = 7 and a permissible set of seven chemical equations, but this need not concern us here. 7. The same approach applies when r = 0, except that the element-conservation equations may be used as the mass-conservation equations.

Literature Cited 1. Toby, S. J. Chem. Educ. 2000, 77, 188–190. 2. Lee, J. Y. J. Chem. Educ. 2001, 78, 1283–1284. 3. Horn, F.; Jackson, R. Arch. Rational Mech. Anal. 1972, 47, 81–116.


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Research: Science and Education 4. Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis; Wiley-Interscience: New York, 1982; reprinted with corrections, Krieger: Malabar, FL, 1991; Chapter 2. 5. Oliver, P. Int. J. Chem. Kinet. 1980, 12, 509–517. 6. Schuster, S; Hofer, T. J. Chem. Soc., Faraday Trans. 1991, 87, 2561–2566. 7. Schuster, S; Hilgetag, C. J. Phys. Chem. 1995, 99, 8017–8023. 8. Björnbom, P. H. Ind. Eng. Chem. Fundam. 1975, 14, 102– 106. 9. Missen, R. W.; Smith, W. R. J. Chem. Educ. 1990, 67, 876– 877. 10. Alberty, R. A. J. Phys. Chem. 1989, 93, 3299–3304.

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11. Alberty, R. A. J. Phys. Chem. 1991, 95, 413–417. 12. Chemical Reaction Stoichiometry Web Site Home Page. Smith, W. R.; Sikaneta, I.; Missen, R. W. 1998; http:// www.chemical-stoichiometry.net (accessed Mar 2003). In addition to the Java applet JSTOICH, the site also contains a tutorial on Chemical Reaction Stoichiometry. 13. Judd, C. S. J. Chem. Educ. 1998, 75, 1073. 14. Missen, R. W.; Mims, C. A.; Saville, B. A. Introduction to Chemical Reaction Engineering and Kinetics; Wiley: New York, 1999; p 172. 15. Bawn, C. E. H.; White, A. G. J. Chem. Soc. 1951, 343–349. 16. Bawn, C. E. H.; White, A. G. J. Chem. Soc. 1951, 339–343.


Mass Conservation Implications of a Reaction Mechanism

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