Research: Science & Education
Validity of the Quasi-Stationary-State Approximation in the Case of Two Successive Reversible First-Order Reactions V. Viossat and R. I. Ben-Aim Laboratoire de Chimie des Surfaces, Université Pierre et Marie Curie, 4 Place Jussieu, Paris, France
In chemical kinetics, the quasi-stationary-state approximation (QSSA) is often used. It states that for two successive reactions the rates of formation and consumption of the intermediate are equal. This approximation leads to some erroneous results, but it may be considered satisfactory if, under certain conditions, these results are not too far from the exact ones. Also, it is important to determine the domains of applicability of this assumption. In textbooks, the simple mechanism 1
3
A → B→ C
(I)
is completely solved, because analytical calculations are easy. It is used to introduce QSSA for more complex mechanisms. The conditions of validity of QSSA for the mechanism A
1 2
3
B→C
(II)
are discussed in many papers. If we note respectively the rate constants k1, k2, k3, the reaction rates v1, v2, v3, the initial conditions A0, B0, C0, the exact and approximate concentrations A, B, C, Ass, Bss, Css, four different methods are described: 1. An intuitive approach with the representation of free energy profiles (1); 2. The study of factor DSS (deviation from the quasistationary state) representing the extent of deviation from steady state (2). DSS is the ratio of the reaction rates (v2 + v3)/v1. 3. A comparison between exact and approximate solutions. Pyun (3) states that Ass + Css = A0 and gives a concentration for B 0 different from the exact one. Gellene (4) states that Ass + Bss + Css = A0 + B0 + C0 , which is the mass balance condition. The approximate results depend on A 0, B0, C0, and CSS0. 4. In a previous paper (5), we suppose that QSSA holds at the very beginning of the reaction, which implies adoption for A and/or B of initial conditions different from the exact ones. We named them fictitious initial conditions. The approximate solutions, with different initial conditions satisfying QSSA, are calculated and the mass balance is used as a validity criterion.
The last three papers (3–5) insist that mass balance is an important criterion for the validity of the approximation. All the authors arrive at the well-known condition k1 Q > 0 ⇒ λ2 > λ1 It is possible to determine two particular values of DSS: 1. At the initial time, DSS(0) = 0. 2. At a very long time, t = ∞, the system reaches a thermodynamic equilibrium; that is, v1 = v2; v3 = v4; so DSS(∞ ) = 1
time tm only if DSSmax is close to 1. Table 1 and Figure 1 show these results. If k1 ≤ k4, B and DSS are always increasing and tend respectively to the asymptotic values Beq and 1 (Table 2 and Fig. 2). DSS ≈ 1, once fulfilled, is a necessary and sufficient condition of validity, so QSSA can be applied after a certain time and there is no risk of error. DSS and B versus Time for Mechanisms I and II The mechanisms I and II lead to a complete reaction. They evidently correspond to the case k1 > k4 = 0; so B always presents a maximum occurring at the same time tm as DSS = 1. The evolution of the function DSS can be studied directly: DSS = (k2 + k3)(T5 e ᎑λ 2t + T6 e ᎑λ 1t )/k1(T2 e ᎑λ 2t + T3 e ᎑λ 1t ) d(DSS)/dt = (k2 + k3)e ᎑(λ 1–λ 2)t/[T3e ᎑(λ 1–λ2)t + T2 ] 2 The derivative is always positive, so DSS is an increasing function of time. When t increases infinitely, the expression DSS(∞) presents the indeterminate form 0/0. It is possible to circumvent this by the following: λ2 > λ1 ⇒ e ᎑λ 2t 0, tm has a positive value if ( λ2 – k4)/ ( λ1 – k4) > 1. Let us consider three cases:
If k1 ≥ k3 then λ2 = k1 and DSS(∞) = ∞. If k 1 < k 3 then λ2 = k3 and DSS(∞) = k3/(k3 – k1) > 1. The more k3 surpasses k1, the closer DSS(∞) is to 1. The finite value of DSS(∞) has previously been indicated by Volk et al. (2). However, these authors did not indicate that DSS(∞) could tend to very large values because they considered only the case k1 < k3.
1. λ1 – k4 > 0. As (λ2 – k4) > (λ1 – k4); the previous ratio is greater than 1 and t m exists. 2. λ1 – k4 < 0 < λ2 – k4. The ratio is negative and tm does not exist. 3. λ2 – k4 < 0; λ1 – k4 < λ2 – k4 < 0; |(λ1 – k4 )| > |(λ2 – k4)|. The ratio is less than 1 and tm does not exist.
Therefore B reaches a maximum and tm exists only if λ1 > k4. The condition λ1 – k4 > 0 gives (P – Q)/2 > k4 P – 2k4 > Q > 0 (P – 2k 4 ) 2 > Q 2 If this expression is developed, we obtain k1 > k4. In conclusion, if k1 > k4, there exists a time tm for which B presents a maximum and for which DSS reaches the value 1. Afterwards, B decreases to the asymptotic value Beq while DSS increases, reaches a maximum, and then tends to the asymptotic value 1 (Table 1). The maximum of DSS is noted DSSmax and is reached at the time tM. QSSA can be applied from
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λ2 = (P + Q)/2 < P = k 1 + k2 + k3 ⇒ λ2 – k1 < k2 + k3
Mechanism I: k2 = k 4 = 0 In this case,
Mechanism II: k4 = 0 with k2 ≠ 0 In this case, λ2 = {k1 + k2 + k3 + [(k1 + k 2 + k3)2 – 4k1k3]0.5}/2 >
{k1 + k3 + [(k1 + k3)2 – 4k1k3] 0.5}/2 According to the previous result if k1 ≥ k3, then λ2 > k1 and λ2 – k1 > 0; if k1 < k3, then λ2 > k 3 > k1 and λ2 – k 1 > 0. It appears that the term λ2 – k1 cannot be null or negative. This means that DSS(∞) always has a finite value greater than 1. The evolution of the functions B and DSS for mechanisms I and II are displayed in the Table 3 and illustrated by Figures 3 and 4. The results for the three mechanisms are collected in Table 4. The condition k1 > k4 fixes the existence of a maximum for the concentration of the intermediate product with
Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
the caracteristic value DSS = 1. For mechanisms I and II, the maximum value of DSS corresponds to DSS(∞). For mechanism III, it is always greater than DSS (∞) = 1. The condition k1 ≤ k4 corresponds only to mechanism III and the maximum value of DSS corresponds to DSS(∞) = 1. Methodology to Apply QSSA From the previous results, it is possible to propose a methodology to determine the conditions of validity of QSSA for the mechanisms I, II, and III. It is based on the value of DSS, which must be close to 1. However, this condition is not sufficient because this value can be attained at a given moment.
First Case: k 1 > k4 B presents a maximum and DSS reaches the value 1 at instant tm. QSSA is applicable selectively at tm, but it is necessary to determine the value of DSSmax to know if QSSA can be applied for t > tm. Mechanism I Two cases are to be considered: (i) k1 ≥ k3, DSS(∞) → ∞, consequently QSSA cannot be applied; (ii) k1 < k3, DSS(∞) = k3/(k3 – k1) > 1. The smaller k1 is before k3, the more DSS(∞) approaches 1 and the more QSSA is valid for long times. It is possible to determine the time from which QSSA can be applied according to the degree of precision desired. DSS(t) is easily calculated: DSS(t) = DSS(∞) [1 – e(k1–k3)t ]. If it is decided, for example, to apply QSSA when DSS is in the interval [0.99; 1.01], the first condition is DSS(∞) = k3/(k3 – k1) < 1.01 ⇒ k3 > 101k1 The second condition is DSS (t) ≅ 1[1 – e᎑100k1t ] > 0.99 Hence e᎑100k1t < 0.01 ⇒ t > 4.605 × 10᎑2/k1. This value was
Figure 1. A
1 2
B
3 4
C. k 1 = 1; k 2 = 0.01; k 3 = 0.1; k 4 = 0.01.
proposed by Volk et al. (2). If a more precise condition of application of QSSA is desired, we obtain, successively, 0.9999 < DSS < 1.0001. Then DSS(∞) = k3/(k3 – k1) < 1.0001 ⇒ k3 > 10001 × k1 DSS(t) ≅ 1; [1 – e᎑1000k1t ] > 0.9999 ⇒ t > 9.212 × 10᎑4/k1 The times calculated above can be compared to the time tm corresponding to the maximum of B. tm = 1/(k1 – k3) × ln(k1/k3) With the previous conditions k 3 > 101 × k1 or k3 > 10001 × k1, tm is equal respectively to 4.61 × 10᎑2 or 9.21 × 10᎑4. A remarkable concordance is observed. So it can said that QSSA is valid from the time tm. Mechanism II We obtain DSS(∞) = (k2 + k3)/(λ2 – k1) > 1 Replacing λ2 by its value and writing k1/(k2 + k3) = a, we get DSS(∞) = 2/(1 – a + {(1 + a) 2 – 4a[k3/(k2 + k 3)]} 0.5) This expression must be close to 1. This is the case if a k 4 with k 1 > k3. B always presents a maximum. DSSa Ba k1 = 1.0 k2 = 0.01 k3 = 0.1 k4 = 0 DSSb Bb k1 = 1.0 k2 = 0.01 k3 = 0.1 k4 = 0.01
Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
Conclusion Application of QSSA to all the mechanisms corresponding to two reversible (or not) successive first-order reactions is based on the analytical study of DSS. It has been shown that the condition DSS = 1 is always satisfied at a given time (finite or very long); it does not mean that QSSA can always be applied. When it is satisfied at a given finite time, it remains a necessary but not a sufficient condition, as afterwards it can eventually reach very large values. A method is given to determine the conditions of validity of QSSA and the time from which it can be used. An important result is that comparison of DSS with the theoretical value 1, for which QSSA applies rigorously, can be done quantitatively, whereas in textbooks qualitative considerations are generally proposed. Literature Cited 1. Raines, R. T.; Hansen, D. E. J. Chem. Educ. 1988, 65, 757. 2. Volk, L.; Richardson, W.; Lau, K. H.; Lin, S. H. J. Chem. Educ. 1977, 54, 95. 3. Pyun, C. W. J. Chem. Educ. 1971, 48, 194. 4. Gellene, G. I. J. Chem. Educ. 1995, 72, 196. 5. Viossat, V.; Ben-Aim, R. J. Chem. Educ. 1993, 70, 732. 6. Zeng, G.; Pilling, M. J.; Saunders, S. M. J. Chem. Soc. Faraday Trans. 1977, 93, 2937. 7. Farrow L. A.; Edelson, D. Int. J. Chem. Kinet. 1974, 6, 787. 8. Edelson, D. Int. J. Chem. Kinet. 1979, 11, 687. 9. Nicholson, A. J. C. Can. J. Chem. 1983, 61, 1831. 10. Savage, P. A. Chem. Eng. Sci. 1990, 45, 859. 11. Froment, G. F.; Sundaram, K. M. Int. J. Chem. Kinet. 1978, 10, 1189. 12. Porter, M. D.; Skinner G. B. J. Chem. Educ. 1976, 53, 366. 13. Graedel, T. E.; Farrow, L. A. J. Phys. Chem. 1977, 81, 2480. 14. Hesstvedt, E.; Hov, O.; Isaksen, I. S. A. Int. J. Chem. Kinet. 1978, 10, 971. 15. Turanyi, T.; Tomlin, A. S.; Pilling, M. J. J. Phys. Chem. 1993, 97, 163. 16. Come, G. M. J. Phys. Chem. 1977, 81, 2560. 17. Bond, R. A. B.; Martincigh, B. S.; Mika, J. R.; Simoyi, R. H. J. Chem. Educ. 1998, 75, 1158–1165. 18. Johnson, K. J. Numerical Methods in Chemistry. Dekker: NewYork, 1980. 19. Menzmer, M. D. J. Chem. Educ. 1993, 70, 620.
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