C. 1. Arnot
Bishop's Universily Lennoxville, Quebec, Canada
Activated Complex Theory of Bimolecular Gas Reactions
Derivations from activated-complex theory of an expression for the bimolecular rate constant are based on the original proposals of Eyring (1) in which em~hasisis placed on the rate at which activated compl&es move through a transition state; the rate of forward reaction is expressed as the ratio of the concentration of complexes and their mean life. The basic assum~tionis that eauilibrium exists between activated cdmplexes and reactants. The nature of this assumption has been discussed frequently (3-6). An alternative approach is discussed below. The rate of formation of complexes is considered, and some of the subtleties of activated complex theory are exposed. The direction of the reaction of an activated complex has, of course, been recognized as one of its fundamental properties. Complexes formed from reactants almost invariably dissociate to products, and motions in opposite directions through the transition state are independent.
1B
,kb
>
L
Xb'
kb
I
Remtanh
cComplexet disk
Activated Complex
Rod-
REACTION CO-ORDINATE Figure 1. Energy change along the remstion so-ordinate in a bimoleculor gar reaction. Activated complex represented as one configuration.
XI* and XB*,although having the same configurations, do not interconvert. At the conditions of overall equilibrium the concentrations of XI* and Xa* are equal, and in usual discussions rXf*] is equated with one half of the total concentration calculated from equilibrium theory. This point is discussed below. If the forward reaction occurs at conditions of negligible back reaction, an actual equilihrium between reactants and XI* is not possible, but since the opposing motions through the transition state are iudependent, the rate of formation and the steady state concentration of XI* are identical with the corresponding values at overall equilibrium and the same concentrations of reactants. Reactants will at all conditions be in equilibrium with unstable collision complexes Y, which are incapable of dissociation to products (Fig. 1). XI* differ from Y by the nature of thc dissociation products, hut the rate and rate constant of the formation of X,* do not depend on the fate of these complexes, and these quantities have precisely the values they would have if XI*, like Y, reverted to reactants. This argument has been used by Bishop and Laidler (7) in a nonequilibrium derivation of an expression for the bimolecular rate, and a similar argument was used by Slater (8) to express the rate of activation in his theory of unimolecular reactions. The forward bimolecular rate constant is therefole given by k, = kS*K* (11 where ko* is the rate constant for the hypothetical dis480
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Journal of Chemical Education
sociation of XI* to reactants, and is equal to the rate constant for the real dissociation of Xn+. K* is the constant for the hypothetical equilibrium A + BdXl*
and can be expressed by the usual statistical expression, so that
where Q represents the appropriate molecular partition function. Overall equilibrium has not been specified in order to derive this equation. Xf* and Xb*are represented as single configurations corresponding to the top of the energy barrier in Figure 1. The actual steady-state concentration of XI* is obtained by equating the rates of its formation and dissociation kt[Al IBI = k,*[X~*l (3 1 On the assumption that kl* and kk are equal, or that X,+ and Xa* have equal mean lives, the steady state concentration of XI* is that given by the equilibrium constant K*. Several interesting conclusions follow from eqn. (2). k,* can perhaps be expected to be of the order of magnitude of the stretching frequencies of normal bonds, loLas-1, but an exact value in any reaction will depend on the nature of the bond, or bonds, that break in the dissociation of a complex, and on the degree to which
the bond is weakened. An additional kinetic isotope effect is suggested if kb* varies with isotopic substitu-tion. Comparison of eqn. (2) with the usual activated complex theoretical equation
where Q* is deficient in a factor for motion through the transition state, shows that the temperature dependences of k, differ in the two equations. Since Qx,+ is 'a complete partition function, k, can be given a thermodynamic formulation without the usual reservation concerning the missing factor in Q*. An indication of the sizes of kB*is obtained by equating Eo approximately with the Arrhenius activation energy, and identifying the empirical frequency factor with
Bimolecular frequency factors are typically 10'O1014 ems mole-' s-', or 10-'~10-'0 cm3 molecule-' s-1, and molecular part,ition functions are generally of the order 1025-1027~ m at - ~normal temperatures. The larger frequency factors can be associated with reactants of smaller Q, so that kb* is given generally as approximately 1013s-' if Qx,* has a normal value. Calculation of Qx,* is made difficult by the uncertainty of the contributions of the vibrations of the complex, particularly the vibration leading to its dissociation. (For simplicity, it is assumed that only one vibrational mode dissociates the complex; for example, an antisymmetric stretching mode of a linear triatomic complex. By the hypothetical nature of K*, this vibrational partition function corresponds to that for the actual dissociation of Xb*.) The usual statistical expression for the partition function of a harmonic oscillator cannot be used; the dissociative motion is neither harmonic nor oscillatory. A direct assumption that the dissociative vibration contributes an approximate factor of unity to the total partition function would lead to a normal value for Qx?. Alternatively, following Eyring's derivation (I), this vibration can be described as a one-dimensional translation through a transition state of length 6 (Fig. 2). The transition state is then a succession of configurations of equal energy along the reaction coordinate, and the total concentration of complexes XI* will depend on the size of 6. ka* is now the rate constant for dissociation of the limiting configuration on the reactant side of the transition state. The dissociating vibrationof the limiting XD*can be considered to have been a translation across the transition state, and its partition function expressed by
q approximates to unity for complexes of molecular weight of the order 10%and at normal temperature if 6 is about cm. This may not be an unreasonable order of magnitude for possible flat tops of energy barriers, but since 6 is generally unknown, this procedure does not help the evaluation of Qx?. However, it does provide a device for changing eqn. (2) to the form of the usual activated complex equation (eqn.
(3)). Following the Eyring procedure, the rate of crossing through the transition state is the ratio of [XI*],, the steady state concentration of complexes with the limiting configuration on the reactant side, and the mean time required for XI* to cross. The mean time is the ratio of 6 to the mean velocity of XI*. Using the gas kinetic expression for the mean absolute speed of gas molecules in one dimension, the rate of crossing is
I n the steady state
where [X,*], is the concentration of complexei in the limiting configuration that dissociates to products.
If
there will be a unifo~m distribution of complexes through the configurations of the transition state. Assuming kb*is also given by eqn. (€9,which is equivalent to assuming 6 cm, and expressing the partition function for the dissociative vibration by eqn. ( 5 ) , eqn. (2) becomes
-
where Q* is deficient in the factor corresponding to dissociation of the complex. This deficiency is seen to be a mathematical consequence of the expression of kb* by eqn. (8) and the subsequent elimination of the unknown 6, and is not fundamental to the theory. Equation (9) has a factor of two in the numerator that is absent from eqn. (4), the usual theoretical equation. Eyring (1) expressed the rate of crossing through the transition state as the product of the total concentration of complexes and a forward crossing frequency calculated from an amended gas kinetic mean velocity for forward motion. Positive velocities of gas
REACTION CO-ORDINATE Figure 2 . Activated complexes represented as configurations within o di9tance 6 along the reoction co-ordinate.
Volume 49, Number 7, July 1972
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481
molecules were summated, and division was made by twice the number of molecules with positive velocities. The rate of crossing was expressed But for simultaneous steady states of XJ* and Xa* Overall equilibrium was implicitly assumed therefore. Since [XI*] = [X*] at equilibrium, assuming kl* = k,*, this procedure might seem to be equivalent to that used to obtain eqn. (6). However, at overall equilibrium, the total concentration of complexes of one configuration cannot be expressed by the usual statistical equation. Due to the unidirectional character of XJ* and Xb*, the total concentration [X*] at overall equilibrium is twice that given by the equation
k,[A[[B]
A+B=X*=P+Q
at overall equilibrium kf [A] [B]
482
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=
kf*[X*]
=
ka*[X*l
Journal o f Chemicol Education
=
[@I
kf*[Xj*]
=
ka*[Xa*]
=
kdPI [Ql
at overall equilibrium; and, assuming kl+ [XI*]
=
[Xs*]
=
' *
k [A1 [Bl kf
=
ka
=
k6
[PI IQI
(13)
The factor of two in eqn. (9) would therefore seem to be correct. Literature Cited
. .
Thus, considering a single configuration for the transition state (Fig. I), if X* could dissociate in either direction
=
.
.
..
London, 1962. (3) Erxma. H., *so EYRINO, E. M., "Modern Chemical Kinetics." Reinhold Publishing Co.. N e w Y o r k , 1963, p.49. (4) L * I D L E ~ K. , J., "Chemical Kinetics" (2nd ed.), MoGmw Hill Book Co.. N e r v Y o r k . 1 9 6 5 , ~7. 3 . . (5) L A r n m n , I
