Research: Science and Education
The Application of the Concept of Extent of Reaction Adela E. Croce Departamento de Química, INIFTA, Universidad Nacional de La Plata, La Plata 1900, Argentina;
[email protected] The concepts of reaction rate equation and elementary reaction step are introduced in basic chemistry courses. However, much confusion is provoked by the plethora of mathematical manipulations that compete in handling the various possible kinetic model behaviors. The definitions of the “extent of reaction” and the related “degree of advancement” variables have been used to express the reaction rate independently of the rate of concentration change of a given reactant or product. But in the case of multistep reactions, cryptic equations, difficult to deduce for the student, often arise. Moreover, the misuse of these concepts may even lead to wrong results. In the following it is shown how this tool can be used effectively without adding to the general confusion.
Even though it is introduced in classical textbooks on chemical kinetics, the degree of advancement variable is not explained for the general case of multistep reactions (2). Moreover, the misuse of such a variable may lead to incorrect results (1). In this paper a systematic procedure to define the degree of advancement variables related to the extent of reaction is discussed in exemplary cases of multistep reactions. Definitions Let us consider a chemical reaction (5) ν1A1 + ν2A2 + … → νn A n +νn+1A n+1 + …
(1)
This can be written more briefly as Introduction The course of a chemical reaction can be followed by determining the concentration of a given reactant consumed (or product formed) after a certain time interval (1–5). However, the assignment of the temporal change of the concentration of a given species to the rate of the reaction requires a careful consideration of the stoichiometry of the reaction. The International Union of Pure and Applied Chemistry (IUPAC) has recommended the use of the extent of reaction (6 ); this concept and the related definitions of rate of conversion and stoichiometric coefficients have been extensively treated in this Journal (7). The rate of reaction defined in terms of the extent of reaction has the clear advantage of being independent of whichever concentration is used to monitor the rate (8). On the other hand, a principal aim of kinetic studies is the determination of reaction rate constants. The traditional strategy of chemical kinetics points to establishing the reaction rate equation: an ordinary first-order differential equation that gives the dependence of the rate on the concentrations and may contain empirical coefficients. In particularly simple cases, the integration of the reaction rate equation allows the determination of reaction rate constants (1–5). For instance, such integration is straightforward in reactions that take place in a single step. In fact, a single advancement variable defined from the extent of reaction plays a decisive role to simplify the mathematical procedure (6, 7). However, most experimental observations in chemical kinetics cannot be modeled in terms of a single elementary step but require a series of elementary reactions. In these cases, one degree of advancement variable has to be defined for each elementary step. Exceptionally, the rate equation may be simplified by using a single variable—in certain cases by application of the elegant methods of linear algebra (9). Otherwise, a system of differential equations has to be solved. However, it may still be possible that no analytical integration method is available, rendering always useful the application of numerical methods (4 ). 506
Σi νi A i = 0
(2)
where A i denotes reactants and products and νi stands for the stoichiometric coefficients, defined as positive for products and negative for reactants. A progress variable, the extent of reaction, ξ, is defined as ni = n0, i + νi ξ
(3)
where ni stands for the number of moles of A i present in the system and n0, i the number present initially. Accordingly, the rate of reaction takes the form (2) v ≡ dξ /dt = (1/νi) dni /dt = (V/νi) dci /dt
(4)
where ci denotes the molar concentration of the products or of the reactants at the time t, the volume V being independent of t. If the chemical change is measured at constant volume V, the rate of a homogeneous reaction (2) per unit volume is given by v/V = (1/νi) dci /dt
(5)
The practice of referring to v/V loosely as the “rate” is widespread and well established, and most textbooks on physical chemistry and chemical kinetics define the rate of reaction by means of eq 5, as reviewed in ref 7. The dependence of the rate on the concentrations of the substances taking part in the reaction leads to the so-called rate equation. As pointed out above, the rate of a homogeneous reaction, measured at constant volume, is given by the intensive quantity defined by eq 5; and in particular cases the rate equation takes the form (1/νi) dci /dt = k Πj cj αj
(6)
where k is the reaction rate constant at a given temperature and the exponents αj are constants defined as the order of the reaction with respect to the species j. This is the case of elementary unimolecular, bimolecular, and termolecular reactions (1–5). However, complex reactions, which take place through a sequence of elementary steps, may also present such a rate equation (10).
Journal of Chemical Education • Vol. 79 No. 4 April 2002 • JChemEd.chem.wisc.edu
Research: Science and Education
Actually, the variable ξ cannot be used to integrate the rate equation because it is precisely the dependence of the rate on the concentrations that is obtained experimentally. Therefore, a further suitable transformation ci = c0, i + νi x
(7)
has to be made using the variable x = ξ/V in eq 3 and assuming that ξ vanishes at initial time (7). The variable x is also referred to as “degree of advancement” (9) and dx/dt = (1/V )dξ/dt = (1/νi) dci /dt
(8)
The case in which the concentration of only one species intervenes in eq 6, for example for A1 → products, 2A1 → products, and 3A1 → products, is discussed in ref 3. By applying eqs 7 and 8, the rate equation takes the general form dx/dt = kc1α 1 = k(c0, 1 + ν1x)α 1
(9)
than one elementary reaction, eq 7 should be rewritten as ci = c0,i + Σj νi, j xj
(7′)
where i identifies the reactant or product and j, the reaction step.
Case I. Multistep Systems Leading to a Single Differential Equation as a Function of a Single Degree of Advancement Variable (1–5, 9) As an example, the procedure to obtain the rate equation is shown for the system of parallel first- and second-order reactions interpreting the hydrolysis of an organic halide (A1) in the presence of hydroxide ion (A2) (2). This hydrolysis may take place as an SN1 or first-order reaction or by an SN2 or second-order reaction. The first-order mechanism includes the reaction k1
A1 → A4 + A5
(11)
This is a first-order linear differential equation in the single variable x. For a second-order reaction of the form A1 + A2 → products, eq 9 has to be slightly modified:
to form a halide ion (A4) and an organic cation (A5), followed by the fast reaction
dx/dt = k Πj cjαj = k Πj (c0, j + νj x)α j
A2 + A5 → A3
(10)
where x is independent of j. This equation is also valid for a third-order reaction, for which the reaction stoichiometry may be A1 + A2 + A3 → products or 2A1 + A2 → products, etc. All these examples lead to a single differential equation in the single degree of advancement x (9), as long as one deals with an elementary reaction step.
Example The reaction rate equation for the elementary reaction 2A1 + A2 → products may be written as or equivalently as
(᎑ 1Ú2)dc1/dt = kc12c 2 ᎑dc2/dt = kc1 c 2 2
Even though these nonlinear first-order ordinary differential equations are coupled, they lead to a single equation with the condition d(c1 – 2c2)/dt = 0; that is c1 – 2c2 = constant. On the other hand, the application of eq 7 leads to c1= c0,1 – 2x and c2 = c0,2 – x. By replacing the expressions for c1 and c2 in the rate equation, a single equation with a single degree of advancement variable dx/dt = k(c0, i – 2x)2(c0,2 – x) is directly obtained. This equation yields k upon application of standard integration methods as shown in ref 3. Application of the Degree of Advancement to Multistep Reactions Many reactions that occur through a sequence of elementary steps do not present a reaction rate equation of the form of eq 6. In such cases, other methods must be applied to obtain reaction rate constants; for instance, the integration of a system of differential equations. However, in certain cases a single differential equation as a function of a single degree of advancement variable may be obtained. Basically, the same set of eqs 3–5 may be applied, one for each reaction step. However, as a given reactant or product may intervene in more
k2
(12)
in which an alcohol (A3) is formed. In the second-order mechanism reaction 11 is replaced by k3
A1 + A2 → A3 + A4
(13)
However, cases are known in which both mechanisms occur side by side (3); that is, reactions 11–13. For this case, with the conditions k2 >> k1 and k2 >> k3, the rate of reaction is given by (2, 3) dy/dt = k1(c0,1 – y) + k3(c0,1 – y)(c0,2 – y)
(14)
where y denotes a degree of advancement variable. This equation can be deduced by applying the formalism described by eqs 4–10 and the steady-state approximation. In this case, the traditional formalism employing concentrations at constant volume to express reaction rates would lead to dc1/dt = ᎑k1c1 – k3c1c2
(15)
dc2 /dt = ᎑k2c2c5 – k3c1c2
(16)
dc3/dt = k2c2c5 + k3c1c2
(17)
dc4/dt = k1c1 + k3c1c2
(18)
dc5 /dt = k1c1 – k2c2c5
(19)
As can be readily recognized, these equations form a system of five nonlinear first-order ordinary differential equations in the six variables ci (i = 1, …, 5) and t. The integration of this system leads to the time dependence of the concentrations of the reactants and products, and reaction rate constants may be determined by numerical fitting of experimental results. However dc1/dt = ᎑dc4/dt and dc2 /dt = ᎑dc3/dt, so that the system may be reduced to three linearly independent differential equations. If the condition k2 >> k1 is fulfilled, the steady-state approach can be applied to eq 19 resulting in k 1c1 – k2c2c5 ≅ 0. Therefore dc1/dt = dc2 /dt and a single differential equation is left—for instance, eq 15. On the other hand, if the rate for each reaction step is expressed according to eq 4 with the substitution xj = ξ j /V,
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the following system is obtained: dx1/dt = k1c1 = k1(c0,1 – x1– x3)
(20)
dx2/dt = k2c2c5 = k2(c0,2 – x2 – x3)(c0,5 + x1 – x2)
(21)
dx3/dt = k3c1c2 = k3(c0,1 – x1– x3)(c0,2 – x2 – x3)
(22)
where the ci (i = 1, 2, 3) have been substituted employing eq 7′. For example, the time derivative of x1 in eq 20 expresses the rate of reaction 11, which is first order in A1. Then, c1 may be obtained by application of eq 7′: A1 is consumed via reactions 11 and 13, with a stoichiometric coefficient ν1,1 = ν1,3 = ᎑1, such that x1 and x3 have to be subtracted from c0,1. Fortunately, the problem can be further simplified. In this mechanism, A5 plays the role of an intermediate of reaction under the condition k2 >> k1. If the steady-state approximation is applied to eq 19, k1c1 – k2c2c5 ≅ 0 is obtained. This leads to dx2/dt = dx1/dt (i.e., x2 = x1) and therefore dx3/dt = k3c1c2 = k3 (c0,1 – x1 – x3)(c0,2 – x1 – x3) (23) By defining y = x1 + x3 and adding eqs 20 and 23, eq 14 is obtained. In practice, the numerical fitting of the experimental results to eq 14 has been performed to obtain the values of k1 and k3 (3). Other examples of multistep systems leading to a single differential equation as a function of a single degree of advancement variable (1–5, 9) are given in the following: First-order reversible reaction: A 1 → A 2, A 2 → A 1 Higher-order opposing or reversible reactions, such as
For this case, textbooks replace erroneously by a single degree of advancement and integrate the resulting equation. Thus the rate of reaction given in ref 1, page 32, dx/dt = k1(c0,1 – x)[(k1 + k2c0,2 + 2k3c0,1) – (k2 + 2k3)x] (29) is wrong, as will be shown in the following. Again in this case, if the rate for each reaction step is expressed according to eq 4 with the substitution xj = ξj/V, the following system is obtained: dx1/dt = k1c1 = k1(c0,1 – x1 – x2 – 2x3)
dx2/dt = k2c1c2 = k2(c0,1 – x1 – x2 – 2x3)(c0,2 – x2) (31) dx3/dt = k3c12 = k3(c0,1 – x1 – x2 – 2x3)2
dy1/dt = k1(c0,1 – y1 – y2) – 2k3(c0,1 – y1 – y2)2
(33)
dy2 /dt = k2(c0,1 – y1 – y2)(c0,2 – y2)
(34)
involving subsidiary variables y1 = x1 + 2x3 and y2 = x2, but by no means a single equation such as eq 29. Therefore, in this case it is not possible to express the reaction rate equation as a function of a single degree of advancement variable. Further examples of mechanisms that do not lead to a single differential equation as a function of a single degree of advancement variable are given in the following. The consecutive first order reaction involving two steps: A1 → A2, A2 → A3
The mechanism combining both first- and second-order reversible reactions, A1 → A 2 + A 3, A 2 + A 3 → A1
Two parallel reactions of first order to give the same product: A1 → A2, A3 → A2
The system of parallel reactions of first-order decay to different products: A1 → A 2, A 1 → A 3, …, A1 → A n
The system of parallel reactions of first-order decay to a common product: A 1 → A 2, A 3 → A 2, …, A n → A 2
Parallel higher order reactions, all of the same order:
Parallel second-order reactions: A 1 + A 2 → A 4, A 1 + A 3 → A 5 Two-step consecutive reactions, one of first order in a reactant and another of second order in a different reactant: A 1 → A 2, 2A 2 → A 3
Consecutive two-step reactions, one step first order and another second order: A 1 → A 2, A 1 + A 2 → A 3
Case II. Systems Leading to More Than One Degree of Advancement Variables As an example, the process to obtain the reaction rate equations will be explained here in detail for concurrent first and second order reactions, namely, k1
A1 → products k2
A1 + A2 → products k3
A1 + A1 → products
(24) (25) (26)
The traditional formalism employing concentrations at constant volume to express reaction rates leads to the system dc1/dt = ᎑k1c1 – k 2 c1c2 – k3c12
(27)
dc2/dt = ᎑ k 2 c1c2
(28)
which has not been integrated analytically to date. In fact, the analytical resolution of this system is difficult enough to let us prefer the numerical integration. 508
(32)
where ci have been replaced using eq 7′. From these three, it is possible to write two equations, such that eqs 30–32 can be replaced by
A 1 + A 2 → A 3 + A 4, A 3 + A 4 → A 1 + A 2
n1A 1 + n2A 2 → A 3, …, n1A 1 + n 2A 2 → A n
(30)
The important two-step first-order consecutive reaction with reversible first stage A 1 → A 2 , A 2 → A 1, A 2 → A 3 that has been subject of several papers in this Journal (11) and leads to a system of linear differential equations with known exact solution (1–5) The first-order cyclic reversible reaction: A1 → A2 , A2 → A1, A2 → A3 , A3 → A2, A3 → A1, A1 → A3
Conclusions The concepts of extent of reaction and the related variable degree of advancement of reaction are useful in expressing reaction rate equations and may be introduced even in general chemistry courses. At a higher level, in physical chemistry courses, these concepts turn out to be helpful not only in thermodynamics (8), but also in chemical kinetics. In this last case, they allow simplification of systems of reaction rate equations as presented in classical textbooks. However, extreme care has to be taken in applying the degree of advancement of reaction to multistep reactions. A systematic procedure as outlined in the examples discussed here guarantees the correctness of the result.
Journal of Chemical Education • Vol. 79 No. 4 April 2002 • JChemEd.chem.wisc.edu
Research: Science and Education
Acknowledgments This research project was supported by the Universidad Nacional de La Plata, the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), the Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CICBA) and the Agencia Nacional de Promoción Científica y Tecnológica (PICT 6786). Literature Cited 1. Benson, S. W. The Foundations of Chemical Kinetics; McGrawHill: New York, 1960. 2. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1981. 3. Capellos, C.; Bielski, B. H. J. Kinetics Systems; Wiley-
Interscience: New York, 1972. 4. Steinfeld, J. I.; Francisco, J. S.; Hase, W. L. Chemical Kinetics and Dynamics; Prentice Hall: Englewood Cliffs, NJ, 1989. 5. Berry, R. S.; Rice, S. A;. Ross, J. Physical Chemistry; Wiley: New York, 1980. 6. Laidler, K. J. Pure Appl. Chem. 1981, 53, 753. 7. Cvitaˇs, T. J. Chem. Educ. 1999, 76, 1574–1577. 8. Baird, J. J. Chem. Educ. 1999, 76, 1146–1150. 9. Mauser, H.; Gauglitz, G. Comprehensive Chemical Kinetics, Vol. 36; Compton, R. G.; Hancock, G., Eds.; Elsevier: Amsterdam, 1998. 10. Croce, A. E.; Castellano, E. Int. J. Chem. Kinet. 1982, 14, 647–667. 11. Bluestone, S.; Yan, K. Y. J. Chem. Educ. 1995, 72, 884–886, and references therein.
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