Thermodynamic Study: Steady-State Approximation & Transition State Theory, Lecture notes of Chemistry

Download Thermodynamic Study: Steady-State Approximation & Transition State Theory and more Lecture notes Chemistry in PDF only on Docsity! 1 Manuscript ID: ed-2016-00957b.R2 (Revised) Some Considerations on the Fundamentals of Chemical Kinetics: Steady State, Quasi-Equilibrium, and Transition State Theory Joaquin F. Perez-Benito* Departamento de Ciencia de Materiales y Quimica Fisica, Seccion de Quimica Fisica, Facultad de Quimica, Universidad de Barcelona, Marti i Franques 1, 08028 Barcelona, Spain 2 ___________________________________________________________________________ ABSTRACT: The elementary reaction sequence A I Products  is the simplest mechanism for which the steady-state and quasi-equilibrium kinetic approximations can be applied. The exact integrated solutions for this chemical system allow inferring the conditions that must fulfil the rate constants for the different approximations to hold. A graphical approach showing the behavior of the exact and approximate intermediate concentrations might help to clarify the use of these methods in the teaching of chemical kinetics. Finally, the previously acquired ideas on the approximate kinetic methods lead to the proposal that activated complexes in steady state rather than in quasi-equilibrium with the reactants might be a closer to reality alternative in the mathematical development of Transition State Theory (TST), leading to an expression for the rate constant of an elementary irreversible reaction that differs only in the factor 1  ( being the transmission coefficient) with respect to that given by conventional TST, and to an expression for the equilibrium constant of an elementary reversible reaction more compatible with that predicted by chemical thermodynamics. ___________________________________________________________________________ KEYWORDS: Audience: Graduate Education/Research; Domain: Physical Chemistry; Pedagogy: Misconceptions; Topics: Kinetics, Reactive Intermediates 5 where parameters 1 and 2 are algebraic combinations of the three rate constants involved: 1 2 2 1 1 -1 2 1 -1 2 1 2 1 = { + + + ( + + ) 4 } 2 k k k k k k k k      (6) 1 2 2 2 1 -1 2 1 -1 2 1 2 1 = { + + ( + + ) 4 } 2 k k k k k k k k       (7) The integrated laws (eqs 4 and 5) have been obtained by integration of the corresponding differential laws (eqs 2 and 3) using a matrix-based method. 10 The validity of these exact solutions can be verified by consulting different sources, 11,12 and they have been experimentally confirmed for the oxidation of thiols by Cr(VI). 13-16 Otherwise, the integrated laws can be checked straightforwardly by differentiation, leading to the differential laws derived from the mechanism considered (eq 1). An interesting parameter to discuss later the application of the steady-state and quasi- equilibrium approximations is the time elapsed when the intermediate reaches its maximum concentration. By obtaining d[I]/dt from eq 5 and equating the result to zero, it can be inferred that: 1 max 1 2 2 1 ln t       (8) First Approximation According to eqs 6 and 7, both parameters 1 and 2 are positive, but with 1 > 2 . This means that the exponential function whose exponent is 1 t approaches the zero value much 6 faster than the function whose exponent is 2 t (Figure 1). Thus, there exists an instant that we may call 2t so that for 2 t t the only exponential function still different enough from zero will be the second, and then eqs 4 and 5 can be approximated to: 2o1 1 1 2 - ( ) [A] [A] tk e       (9) 2o1 1 2 - [A] [I] tk e      (10) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 100 200 300 400 500 E x p o n en ti al f u n ct io n s Time / min Figure 1. Dependence of the exponential functions 1 te  (green plot) and 2 te  (purple plot) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k1 = k-1 = k2 = 1.00 10 -4 s-1. Since the approximations that will be developed in the following sections (steady state and quasi-equilibrium) will assume the fulfilment of eqs 9 and 10, it is important to notice that these new approximations can be applied only after a certain instant of the reaction course ( 2 t t ) and never from the beginning itself. As we will see later, this happens after the instant at which the intermediate attains its maximum concentration ( max2 t t ). 7 ■ STEADY-STATE APPROXIMATION Mathematical Approach to the Steady-State Method Let us consider firstly the particular case corresponding to the mechanism for which the first, reversible step is slow in the forward direction and fast in the backward direction, whereas the second, irreversible step is also fast, that is k-1 >> k1 and k2 >> k1. Under these conditions, we can approximate: 1 2 2 1 -1 2 1 2 -1 2( + + ) 4 + k k k k k k k     (11) Replacing this result into eqs 6 and 7: 1 -1 2 + k k  (12) 2 1 1 2 k  (13) and the new results into eqs 9 and 10 (remember that k1 is negligible against both k-1 and k2): o 1 1 - 2[A] [A] k t e (14) o 11 -1 2 1 - 2 [A] [I] + k tk k k e (15) From eqs 14 and 15: 10 Figure 2. Comparison between the kinetic plots showing the [Intermediate]/[Reactant]o ratio as obtained using either the steady-state approximation (green plots) or the exact rate law (purple plots) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k1 = 1.00 10 -4 s-1, and k-1 = k2 = 5.00 10-5 s-1 (A), 1.00 10-4 s-1 (B), 2.00 10-4 s-1 (C) and 4.00 10-4 s-1 (D). In order to reach conclusions about the relative values that must take the rate constants of the elementary reactions ( 1k , -1k , and 2k ) for the steady state approximation to hold, k1 has been kept constant, whereas the other two rate constants have been progressively increased keeping k-1 = k2. In Figure 2, the four bell-shaped curves showing a maximum (in purple) A B o [I ] / [A ] Time / minTime / min o [I ] / [A ] C Time / minTime / min [I ] / [A ] [I ] / [A ] oo D 0.00 0.05 0.10 0.15 0.20 0.25 0 600 1200 1800 0.0 0.1 0.2 0.3 0.4 0.5 0 600 1200 1800 0.0 0.2 0.4 0.6 0.8 1.0 0 600 1200 1800 0.000 0.025 0.050 0.075 0.100 0.125 0 600 1200 1800 11 correspond to the exact concentration ratio (eq 20), whereas the four continuously-decreasing curves (in green) correspond to the steady-state approximate concentration ratio (eqs 21 and 22). We can see that, for each set of rate constants, the exact and approximate curves cross exactly at the maximum of the intermediate concentration, given that eq 18 (the steady-state condition) is only exact at that maximum (the only point with a horizontal tangent). We can also see that each time rate constants -1k and 2k are multiplied by a factor of 2 (keeping 1k constant), the approximate curve gets closer and closer to the exact curve after its maximum. For the case k-1 = k2 = 4 k1 (Figure 2, D) the steady-state approximate curve is almost coincidental with the exact curve once the maximum is reached, meaning that the steady-state condition qualifies as an excellent approximation for this set of rate constants. Thus, we can consider that the time necessary for the steady-state condition to be fulfilled is coincident with the time elapsed when the intermediate concentration reaches its maximum value max2 )( t t The ratio between the exact and approximate intermediate concentrations increases with time, reaching a plateau when t  (Figure 3). The asymptotic value of the ratio is (from eqs 9, 10 and 19): -1 2 ss 1 1 + [I] lim = [I] k k k  t (23) and, from eq 12, it can be inferred that: -1 2 -1 2 1 1, + lim ( ) = 1 k k k k k    (24) 12 0.0 0.5 1.0 1.5 0 600 1200 1800 [I ] / [I ] Time / min ss Figure 3. Ratio between the exact intermediate concentration ([I]) and that obtained using the steady-state approximation ([I]ss) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k1 = 1.00 10 -4 s-1, and k-1 = k2 = 5.00 10 -5 s-1 (red), 1.00 10-4 s-1 (orange), 2.00 10-4 s-1 (green) and 4.00 10-4 s-1 (blue). The dashed line shows the limit corresponding to a perfect fulfilment of the steady-state approximation ([I] = [I]ss). In other words, when k1 is kept constant and both k-1 and k2 are gradually increased, the limit of the asymptotic value of the ratio [I]/[I]ss decreases approaching unity (Figure 4, bottom), corresponding to a perfect fulfilment of the steady-state-approximation. Simultaneously, the time elapsed when the intermediate reaches its maximum concentration (coincident with the time interval required for the steady-state approximation to hold) also decreases approaching zero (Figure 4, top). We can, therefore, define an intermediate in steady state as a very reactive intermediary chemical species that has already reached its maximum concentration. 15 1 -1 [I] [A] k k  (30) Since the equilibrium constant (referred to concentrations instead of to activities) associated to the first, reversible step of the mechanism considered (eq 1) can be expressed as the ratio of the rate constants corresponding to the forward and backward elementary reactions: 1c -1 = k K k (31) eq 30 can also be written as: c [I] [A] K  (32) which is precisely the equation obtained when the quasi-equilibrium approximation is applied to the first, reversible step of the mechanism considered. We can thus conclude that this approximate method requires that the reversible step to which is applied be fast in both directions with respect to the rate-determining step of the mechanism (for the particular case considered here, 1k >> k2 and k-1 >> k2). In other words, this method is useful for intermediates formed in very fast, reversible reactions only. Those formed in acid-base reactions (both reversible and fast) would be an excellent example of quasi-equilibrium intermediates indeed. 16 Graphical Approach to the Quasi-Equilibrium Method We can represent the exact solution for [I] at different instants during the course of the reaction (eq 5) and compare it with the approximate solution provided by the quasi- equilibrium condition (eq 30): 1qe -1 [I] = [A] k k (33) Figure 5. Comparison between the kinetic plots showing the [Intermediate]/[Reactant]o ratio as obtained using either the quasi-equilibrium approximation (green plots) or the exact rate A B o [I ] / [A ] Time / minTime / min o [I ] / [A ] C Time / minTime / min [I ] / [A ] [I ] / [A ] oo D 0.0 0.2 0.4 0.6 0.8 1.0 0 600 1200 1800 0.0 0.2 0.4 0.6 0.8 1.0 0 600 1200 1800 0.0 0.2 0.4 0.6 0.8 1.0 0 600 1200 1800 0.0 0.2 0.4 0.6 0.8 1.0 0 600 1200 1800 17 law (purple plots) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible, with k1 = k-1 = 2.00 10 -4 s-1 (A), 4.00 10-4 s-1 (B), 8.00 10-4 s-1 (C) and 1.60 10-3 s-1 (D) , and k2 = 1.00 10 -4 s-1. Since both [I] (exact solution) and qe[I] (quasi-equilibrium approximate solution) are directly proportional to o[A] , it will be enough to compare the exact ratio (eq 20) with the quasi-equilibrium approximate ratio (from eqs 4 and 33): qe 1 o -1 1 2 ( ) [I] 1 = [A] f t k k   (34) where f(t) has the same meaning than in eq 22. In order to reach conclusions about the relative values that must have the rate constants of the elementary reactions ( 1k , -1k and 2k ) for the quasi-equilibrium approximation to hold, k2 has been kept constant, whereas the other two rate constants have been progressively increased keeping k1 = k-1. In Figure 5, the four bell-shaped curves showing a maximum (in purple) correspond to the exact concentration ratio (eq 20), whereas the four continuously- decreasing curves (in green) correspond to the quasi-equilibrium approximate concentration ratio (eq 34). We can see that each time rate constants k1 and -1k are multiplied by a factor of 2 (keeping k2 constant), the approximate curve gets closer and closer to the exact curve after its maximum. For the case k1 = k-1 = 16 k2 (Figure 5, D) the quasi-equilibrium approximate curve is almost coincidental with the exact curve once the maximum is reached, meaning that the quasi-equilibrium condition qualifies as an excellent approximation for this set of rate constants. Thus, we can consider again (as happened with the steady-state condition) that the 20 Figure 7. Dependencies of the time elapsed when the intermediate reaches its maximum concentration (top) and the limit at infinite time of the ratio between the exact intermediate concentration ([I]) and that obtained using the quasi-equilibrium approximation ([I]qe) (bottom) on the fast/slow ratio of rate constants (keeping k1 = k-1) for the mechanistic scheme formed by two unimolecular consecutive reactions, the first reversible and the second irreversible. The dashed line shows the limit corresponding to a perfect fulfilment of the quasi-equilibrium approximation ([I] = [I]qe). 65 0.75 0.85 60 100 0.80 0.90 40 0 0.95 80 130 195 260 1.00 20 t / m in m ax k / k = k / k 2-121 li m ( [I ] / [I ] ) q e 8 t 21 ■ TRANSITION STATE THEORY Activated Complexes: in Quasi-Equilibrium or in Steady State? The thermodynamic formulation of TST assumes as a basic hypothesis that the activated complex of an elementary reaction is in quasi-equilibrium with the reactants. However, we have seen that, in the simple mechanism considered (eq 1), the combination of rate constants k1 << k-1, k2 leads to the steady-state approximation, whereas the combination k1, k-1 >> k2 leads to the quasi-equilibrium approximation. This situation is illustrated in Scheme 1. ___________________________________________________________________________ Scheme 1. Combinations of Slow/Fast Steps Leading to Each Approximation slow fast fast fast slow fast X X in steady state X X in quasi-equilibrium       ___________________________________________________________________________ Since, in the case of TST, the energy barrier is associated with the forward direction of the reversible step leading to the formation of the activated complex, whereas decomposition of the latter either in the direction of the reactants or in that of the products does not imply any energetic requirement, we may wonder whether the activated complex is actually in quasi- equilibrium or in steady state instead. At first view, and taking into account the information given in Scheme 1, it seems that the steady-state alternative might be more reasonable: formation of the activated complex is expected to be the slow step, whereas its decompositions in both the reactant and product directions are expected to be rather fast. 22 The mathematical expression obtained from TST (thermodynamic formulation) for the rate constant of an elementary reaction is the well-known Eyring equation: o o o 1B ( ) =      S H n R RTc k T k e e h (37) where  is the transmission coefficient, n is the kinetic order of the elementary reaction (molecularity), c o is the standard-state concentration (arbitrarily taken as 1 mol dm -3 ), oH and oS are the standard activation enthalpy and entropy, whereas Bk , h and R are the Boltzmann, Planck and ideal gas universal constants, respectively. The factor (c o )1 n did not appear in the original Eyring equation, but more recently it has been incorporated in order to assure its dimensional homogeneity. 17 Let us now consider the case of an elementary reaction, for instance (although not necessarily) that of a bimolecular process involving two different reactants, written as: 1 2 1 * * * - A + B [A B] P + Q    k k k (38) where A and B are the reactants, [A B] ≠ the activated complex, and P and Q the reaction products. The symbols above and below the arrows carry an asterisk in order to emphasize that they are not true rate constants, although there units are the ones corresponding to either second-order ( * 1k , in M -1 s -1 ) or first-order ( *-1k and * 2k , in s -1 ) rate constants. We may designate these new parameters ( * 1k , * -1k , and * 2k ) as microscopic rate constants to differentiate them from the macroscopic rate constant for the elementary reaction ( k ).  25 Using eqs 45-48: of f o (1 ) = 1 K k          (49) Finally, TST allows us to write: o 1 o o B = ( ) n S H R RT k T K c e e h          (50) and replacing into eq 49: o 1 o o of f B o (1 ) = ( ) 1 n S H R RT k T k c e e h            (51) TST for Reversible Elementary Reactions: Activated Complex in Quasi-Equilibrium Consider now the following elementary reversible reaction (not involving any intermediate species): 1 2 1 2 * * * * - - A + B [A B] P + Q    k k k k (52) 26 TST can be applied to the forward and backward reactions, assuming that the activated complex is in quasi-equilibrium with the reactants and products, respectively, and the following expressions for the corresponding rate constants are derived: o o ,f ,f f oB f f 1 = ( )      S H R RTn k T k c e e h (53) o o ,b ,b b oB b b 1 = ( )      S H R RTn k T k c e e h (54) From eqs 53 and 54, we obtain for the equilibrium constant of the reversible reaction (referred to concentrations): b fof f c b b o o = = ( ) S H n n R RT k K c k e e     (55) the standard reaction enthalpy and entropy being related to the activation parameters corresponding to the forward and backward reactions by: o o o ,f ,b = H H H     (56) o o o ,f ,b = S S S     (57) In addition, the equilibrium constant referred to activities is related to that referred to concentrations by the equation: 27 f b if b o a c i,e i=1 = ( ) n n n n K K c     (58) where i,e are the activity coefficients of reactants and products at equilibrium, and i the corresponding stoichiometric coefficients (i < 0 for reactants and i > 0 for products). Assuming that the reacting mixture is dilute enough (i,e ≈ 1), eqs 43, 55, and 58 lead to: f a b o o = S H R RTK e e     (59) The latter equation is not consistent with the well-known thermodynamic expression for the equilibrium constant unless we assume that f  = b  . This assumption might be too far- reaching because the chemical bonds that must break down for the reaction products to be generated from the activated complex are different from those involved for the generation of the reactants. Moreover, if the activated complex is in quasi-equilibrium with both reactants and products of the reversible reaction, we can write: * 1 ,1 * -1 [A B] = [A] [B] = [A] [B] k K k   (60) * -2 ,2 * 2 [A B] = [P] [Q] = [P] [Q] k K k   (61) where ,1K and ,2K are the equilibrium constants (referred to concentrations) for the formation of the activated complex from the reactants and from the products, respectively. 30 We can see that eq 67 does not raise any compatibility problem as eqs 60 and 61 do. Assuming that the activated complex is present in steady state rather than in quasi- equilibrium, there is no need to invoke the existence of two activated complexes: one crossing the energy barrier in the forward direction and the other in the backward direction. The latter hypothesis seems to be incompatible with the very nucleus of TST itself: this theory admits that the activated complex can go in the two directions since, even in the case of an elementary irreversible reaction, the formation of the activated complex from the reactants is assumed to be a reversible process (see eq 38). Simplification of the Modified Eyring Equation Given that the vibrational energy required to break the activated complex is rather low (only weak chemical bonds must be broken to go forward to the reaction products or backward to the reactants), eq 51 can be simplified by assuming that o 0  : o o oB 1 = (1 ) ( ) S H R RTn k T k c e e h         (68) where, according to the definition of transmission coefficient, it has been taken into account that f  =  . Thus, the equation obtained assuming that the activated complex is in steady state (eq 68) is identical to the one given by conventional TST (eq 37) except for the factor 1  . Therefore, the value of the rate constant now obtained shows the same temperature dependence than that obtained from conventional TST but it is numerically lower (since 1  < 1). This is indeed an expected result because the quasi-equilibrium approximation can be 31 inferred from eq 42 simply by assuming that * -1k >> * 2k . This situation would correspond to an activated complex being simultaneously in steady state and in quasi-equilibrium with the reactants. Thus, conventional TST tends to ignore the decrease in the concentration of the activated complex caused by its decomposition into the reaction products (contribution of parameter * 2k ), considering only the one caused by its decomposition into the reactants (contribution of parameter * -1k ). The value of parameter * 2k might not be negligible after all against that of * -1k as required by conventional TST. In other words, although the new factor 1  would be unimportant provided that 1  ( * *-1 2 >> k k in eq 42), this condition might be difficult to reach in most cases. Boltzmann Distribution Approach Some textbooks 18 avoid the application of the quasi-equilibrium condition by assuming that the energies of the activated complex quasi-molecules and of the reactant molecules follow a Boltzmann distribution. However, this solution might be misleading. In fact, the final equations obtained from the quasi-equilibrium approach and from the Boltzmann distribution approach are identical (both converge into eq 37). Thus, the Boltzmann approach would also lead to eq 59 for the equilibrium constant of an elementary reversible reaction, in clear incompatibility with the prediction made by chemical thermodynamics. The apparent reason is the fact that the Boltzmann distribution of energies would require that the system composed by the reactant molecules and activated complexes be, not only in thermal equilibrium, but also of fixed composition. The latter condition requires a chemical equilibrium between the reactant molecules and activated complex quasi-molecules. The existence of a process 32 converting the activated complex into the reaction products (contribution of * 2k in eq 42) renders this condition impossible to reach in the case of elementary irreversible reactions. Convergence of the Three Different Approaches Only for an elementary reversible reaction and only once the chemical equilibrium state is reached, application of both the quasi-equilibrium hypothesis and the Boltzmann distribution law to the activated complex would give completely satisfactory results. Effectively, application in this case of eq 67 at t = ∞ leads to: e e e e * * 1 -2 e * * -1 2 [A] [B] + [P] [Q] [(A B) ] = + k k k k  (69) and, since the equilibrium constant referred to concentrations associated to the elementary reversible reaction considered (eq 52) can be expressed as: e e * * 1 2 c * * e e-1 -2 [P] [Q] = = [A] [B] k k K k k (70) by combining eqs 69 and 70: * * 1 -2 e e e e e* * -1 2 [(A B) ] = [A] [B] = [P] [Q] k k k k  (71)