Research: Science and Education
The Relationship between Stoichiometry and Kinetics Revisited Jim Y. Lee Department of Chemical and Environmental Engineering, National University of Singapore, Singapore 117576;
[email protected] In the February 2000 issue of this Journal, Toby illustrated the relationship between stoichiometry and kinetics through four examples of elementary reactions (1). However, the interdependence of concentrations in Toby’s derivation could be obtained more generally from material balance considerations alone, or in more complex cases, with the added assumption of negligible reactive intermediate concentrations. In other words, the concentration relationships could be derived without detailed knowledge of the kinetics of the elementary steps in the reaction mechanism. It is the purpose of this note to decouple the suggested link between stoichiometry and kinetics. The decoupling between stoichiometry and kinetics is particularly important for dealing with nonelementary reactions. The following derivations are based on material balance of reacting species, summarized most conveniently in tabular form as practiced in most chemical engineering texts on reaction engineering (2). The first example involves three species in the following isothermal reactions; only A is present initially. Reaction A → 2B A+B→C
Extent of Reaction ξ1 ξ2
The extent of reaction, or the number of moles extent, for the j th reaction, is defined as ξ j = ∆nij /νi j, where νij is the stoichiometric coefficient of species i in the j th reaction, a signed quantity (νi j < 0 for reactants and νi j > 0 for products) and ∆nij is the moles of species i reacted in the j th reaction. A stoichiometric table can be set up as follows: Species
Moles Present Initially
Moles Reacted
Moles Present after Time t
A B
nA(0)
᎑ξ 1 – ξ 2
nA(0) – ξ 1 – ξ 2 (= nA)
—
2ξ 1 – ξ 2 ξ2
C
—
2ξ 1 – ξ 2 (= nB) ξ 2 (= nC)
Elimination of the ξ’s between the entries in the last column of the table yields 2nA + nB + 3nC =2nA(0). For a constantdensity system where the reaction volume is invariant (reactions in solution or dilute gas phase reactions), this leads to 2[A]0 – 2[A] = [B] + 3[C]. In Toby’s article, the concentration relationship was obtained by assuming the reactions are elementary, which enables the determination of reaction orders from the stoichiometric coefficients. Any attempt to linearly combine the two reactions to produce the “overall” reaction 2A → B + C is improper. For this case a similar stoichiometric table analysis will produce [A]0 – [A] = [B] + [C] instead. The two concentration relationships are congruent only when ξ1 = ξ 2, which implies that the two reactions must occur to the same extent. In practice the equality may occur only at some point during the course of reaction and not necessarily at the end of reaction as implied
by the word “overall”. When the equality is not satisfied, one sees an apparent change in “reaction stoichiometry” with time, as described by Laidler (3) and cited by Toby. It is an anomaly due to the under-representation of the stoichiometries of a system of reactions by a single, “overall” stoichiometry. If the stoichiometries of all the underlying reactions of a complex reacting system are determined in full, there will not be any time-dependent stoichiometry to speak of. The other more complex examples can be worked out likewise (see Appendix), with the added assumption of zero concentrations for the reactive intermediates ([I] ≈ 0). This simplifying assumption is based on the negligible contribution the reactive intermediates made to the overall concentration. The assumption is a mathematical contrivance in the same genre as the pseudo-stationary-state approximation d[I]/dt = 0 or prior equilibrium assumption. It is therefore acceptable only if the experimental results bear out its prediction. In the context of this discussion, this means the validity of the concentration relationships of Toby in actual experiments. Again the material balance approach is able to arrive at these results without the demanding assumption of elementary reactions to provide the rate law for each step of the reaction mechanism. Conclusion It is generally accepted that reaction rate laws cannot be deduced from reaction stoichiometry unless the reactions are known to be elementary. The concentration relationships in Toby’s article (1) are the consequence of species material balance and could be satisfied by any reaction kinetics and reaction mechanism that are consistent with a known overall reaction stoichiometry (i.e., no time-dependent stoichiometry). An additional issue should perhaps be raised here. The derivations are based on a constant-density reacting system, where the volume of the reaction mixture is held constant either by an incompressible reaction medium or by the conservation of the total number of moles in the reaction. In the event that neither of these conditions is satisfied, concentration changes will also be caused by the expansion and contraction of the reaction mixture and hence are not directly proportional to the reaction extent. In this case the concentration relationships must be replaced by molar relationships, and an equation of state must be used for the calculation of concentrations. The detailed procedures are covered in most chemical reaction engineering texts such as ref 2. Literature Cited 1. Toby, S. J. Chem. Educ. 2000, 77, 188–190. 2. Fogler, H. S. Elements of Chemical Reaction Engineering, 3rd ed.; Prentice Hall: Englewood Cliffs, NJ, 1999. 3. Laidler, K. J. Chemical Kinetics, 3rd ed.; Prentice Hall: New York, 1987; p 8.
JChemEd.chem.wisc.edu • Vol. 78 No. 9 September 2001 • Journal of Chemical Education
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Research: Science and Education
Appendix
Chain Reactions Reaction
A more complex example was provided in the second example. For simplicity A is used to represent azomethane (CH 3N=NCH3), M is CH 3 , B is CH 2 N=NCH 3 , MED is methyl ethyl diimide (C2H5N=NCH3), C is (CH3)2N-NCH3, and TMH is tetramethylhydrazine (CH3)2N-N(CH3)2: Reaction
M+A→C
ξ5
M + C → TMH
ξ6
Species
Before Reaction
After Reaction
A
—
ξ1 – ξ3 + ξ4 – 2ξ6
B
—
ξ1 – ξ2
CO
—
ξ2+ ξ4
C
—
ξ2 – ξ5
CH4
—
ξ3
H2
—
ξ5
After Reaction
D
—
ξ3 – ξ4 + ξ5
C2H6
—
ξ6
A
nA(0)
nA(0) – ξ1 – ξ2 – ξ5
M
—
2ξ1 – ξ2 – 2ξ3 – ξ4 – ξ5 – ξ6
N2
—
ξ1
CH4
—
ξ2
B
—
ξ2 – ξ 4
C2H6
—
ξ3
MED
—
ξ4
C
—
ξ5 – ξ 6
TMH
—
ξ6
Hence ξ 1 = nN2 ; ξ 2 = nCH4 ; ξ 3 = nC2H6 ; ξ 4 = nMEH ; ξ 6 = nTMH If it may be assumed that all reactive intermediates have negligible existence, nC = 0 gives ξ 5 = ξ 6 = nTMH nB = 0 gives ξ 2 = ξ 4 nM = 0 gives 2ξ 1 = ξ 2 + 2ξ 3 + ξ 4 + ξ 5 + ξ 6
Hence ξ 3 = nCH4 ; ξ 5 = nH2 ; ξ 6 = nC2H6 From nC = 0 ξ 2 = ξ 5 = nH 2 nD = 0 ξ 3 = nCH4 = ξ 4 – ξ 5 = ξ 4 – nH2 ξ 4 = nCH4 + nH2 nCO = ξ 2 + ξ 4 = 2nH2 + nCH4 From nB = 0 ξ 1 = ξ 2 = nH 2 nA = 0 nH2 = nC2H6 From nCH3CHO = nCH3CHO(0) – ξ 1 – ξ 3 – ξ 5 = nCH3CHO(0) – 2nH2 – nCH4 2nCH3CHO = 2(nCH3CHO)0 – 2nH2 – nCH4 – nCO or 2[CH3CHO] = 2[CH3CHO]0 – 2[H2] – [CH4] – [CO] 3nCH3CHO = 3nCH3CHO(0) – nCO – 2nCH4 – nC2H6 – 3nH2 or 3[CH3CHO] = 3[CH3CHO]0 – [CO] – 2[CH4] – [C2H6] – 3[H2] Zero-Order Reactions Reaction
From the stoichiometric table, nA = nA(0) – ξ 1 – ξ 2 – ξ 5 = nA(0) – ξ 1 – ξ 2 – ξ 6 = nA(0) – nN2 – nCH4 – nTMH
or [A] = [A]0 – [N2] – [CH4] – [TMH] for a constant-density system (eq 6, Toby) and 2ξ 1 = ξ 2 + 2ξ 3 + ξ 4 + ξ 5 + ξ 6 becomes 2nN2 = nCH4 + 2nC2H6 + nMED + 2nTMH For a constant-volume system this becomes 2[N2] = [CH4] + 2[C2H6] + [MED] + 2[TMH]
(eq 7 of Toby)
The proofs for the remaining two examples are also presented here for the sake of completeness. Symbols are again used to abbreviate the chemical identities of the species.
1284
ξ6
Before Reaction
CH3CHO
where M, B and C are reactive intermediates. The stoichiometric table analysis for the system of reactions is given below. A constant-density reacting system is assumed. Again, ξ j is used to indicate the extent of the jth reaction (1 ≤ j ≤ 6). Species
ξ5
C + CH3CHO → H2 + D
ξ2 ξ4
ξ3 ξ4
D → A + CO 2A → C2H6
ξ3
M + B → MED
ξ2
A + CH3CHO → CH4 + D
ξ1
M + A → CH4 + B 2M → C2H6
ξ1
B → CO + C
mExtent of Reaction
A → 2M + N2
mExtent of Reaction
CH3CHO → A + B
mExtent of Reaction ξ1
E →G D+G→D+D
ξ2
D→C
ξ3
Species
Before Reaction
After Reaction
E
nE(0)
nE(0) – ξ1
G
nG(0)
nG(0) + ξ1 – ξ2
D
nD(0)
nD(0) – ξ2 + 2ξ2 – ξ3
C
—
ξ3
Hence nG + nD + nC = nG(0) + nD(0) + ξ1 or [G] + [D] + [C] = [G]0 + [D]0 + ξ1/V, where V is the reaction volume. The assumption of zero-order reaction will result in ξ 1 = Vkt and the concentration relationship transformed into eq 8 of Toby.
Journal of Chemical Education • Vol. 78 No. 9 September 2001 • JChemEd.chem.wisc.edu
