Activated complex theory (or absolute speeds reactions) was proposed by G. Eyring and M. Polyani (1935).
The main position of the theory of the activated complex chemical reactions: every chemical act proceeds through a transition state or activated complex.
An activated complex is a state of the system in which individual bonds in the original molecules disappear and new bonds of reaction products appear.
In the theory of the activated complex or the theory of absolute reaction rates, two main problems can be distinguished:
1) calculation of the potential surface of the potential energy of an elementary act of a chemical reaction - is associated with calculations using the Schrödinger equation.
2) calculation of the probability of formation and lifetime of the transition complex, estimation of the energy of its formation based on the properties of the reacting molecules.
According to the activated complex theory, the reaction proceeds with the formation of a transition complex:
The change in potential energy of the system follows the minimum of possible values potential energy of the system. However, the state of the activated complex corresponds to the maximum of the minimum potential energy values. The maximum value of potential energy is an unstable, unstable state of the system. The state of the transition complex is the least energetically unstable. Movement in other directions leads to even more unstable options for existence.
Rice. 38. Change in the potential energy of the system along the reaction coordinate A+BC = AB+C during an elementary reaction event
The difference between the potential energy of the starting substances and the potential energy of the activated complex is equal to the activation energy, possessing which, the molecules of the starting substances are able to overcome the potential barrier and transform into the final products:
Distinctive feature activated complex - the presence of an additional degree of freedom, which is expressed in movement along the reaction path. As a rule, towards the reaction products, after some fluctuation in the δ zone. The system passes through section δ in time τ – the lifetime of the activated complex.
Average lifetime of an activated complex:
Where - average speed passage of the activated complex through the potential barrier.
Taking into account the existence of an activated complex, the reaction rate, i.e. number of elementary reaction events per unit volume per unit time:
,
where is the concentration of activated complexes per unit volume, equal to the number emerging AKs during time τ.
This equation is valid if all transition complexes will turn into reaction products.
In other words, the rate of the process is the number of activated complexes crossing the top of the potential barrier along the reaction coordinate per unit time and per unit volume.
Function of the potential energy of atomic nuclei U from their internal. coordinates, or degrees of freedom. In a system of n cores, the number of internal degrees of freedom N = 3n - 6 (or 3n - - 5 if all nuclei are located on the same straight line). The simplest two-dimensional (N = 2) PES is shown in Fig. 1. The reactants and products of the river correspond to areas of relatively low potential energy (valleys), separated by a higher area. energy-potential barrier. The curved line passing along the bottom of the valleys through the barrier is the reaction coordinate. One-dimensional diagrams are often used, depicting a section of the PES, deployed along the coordinate of the district (see Fig. 2). In these diagrams, the top of the potential barrier corresponds to a saddle point, or saddle point. The same concepts are transferred to multidimensional PES with N > 2. The states of reactants and products are stable, they correspond to configurations (i.e., fixed values of coordinates φ), which are minima (or valleys) on the multidimensional PES. Chem. The r-tion is considered as a transition from the configuration of reactants to the configuration of products through the configuration of the saddle point along the coordinate of the r-tion. Configurations of both minima and saddle points are stationary points of the PES, i.e. in them U/q i = 0. 
Modern the conclusion of equation (2), chemically less clear, is based on collision theory. The speed of the reaction is identified with the speed of transition of the reacting chemicals. systems through an (N - 1)-dimensional surface in the space of configurations, separating the areas of reactants and products. In collision theory, this speed is called. flow through the critical surface. The equation in the form (2) is obtained if we carry out a critical analysis. surface through the saddle point is orthogonal to the coordinate of the district and assume that it is critical. surface energetic. the distribution of reagents is equilibrium. The corresponding region of coordinate and momentum space (phase space) is characterized by the same statistical amount This allows for a critical look. surface like many AK configurations. Thus, AK is immediately defined as an object with (N - 1) internal. degrees of freedom and there is no need to enter its extent along the coordinates of the district.
Application of the theory. According to the theory, the r-tion mechanism is completely determined by the configurations of reactants and products (minima, or valleys, on the PES) and the corresponding ACs (saddle points). Theoretical calculation of these configurations using quantum chemistry methods would provide comprehensive information about the directions and rates of chemistry. districts. Such calculations are being intensively developed; for simple chemical systems containing 10-15 atoms, which belong to the elements of the first two periods of the periodic table, they are practically feasible and quite reliable. Consistent calculation of abs. the speed of the river according to equation (2) is to determine the geom. configurations of reagents and AC (at this stage the height of the potential barrier is also determined) and calculation of moments of inertia and oscillations for these configurations. frequencies, which are necessary for calculating statistical data. sums and finish. definitions. In application to complex situations representing practical interest, complete and reliable implementation of such a program is labor-intensive and often impracticable. Therefore, the molecular constants necessary for calculations using equations (2) and (3) are often found empirically. methods. For stable configurations of reagents, moments of inertia and oscillations. frequencies are usually known from spectroscopic data. data, however, for AK we are experimenting. their determination is impossible due to the short time of his life. If followed Quantum chemical calculations are not available, interpolation calculation schemes are used to estimate these values.
Limitations of the theory and attempts to improve it. The activated complex theory is based on two assumptions. The first is the thermodynamic hypothesis. equilibrium between reactants and AA. According to the second, the rate of disintegration is identified with the rate of decay of AK. Both assumptions cannot be strictly substantiated. This is revealed if we consider the movement of chemicals. systems along the p-tion coordinate all the way from reactants to products, and not just near the top of the potential barrier. Only in rare cases is it correct to consider the coordinate of a district as a straight line, as in Fig. 2. Usually it is a curve in a multidimensional space internally. variables and is a complex combination of elementary movements, the edges are not the same for different types. their areas. For example, in Fig. 1 coordinate of the p-tion is a continuously changing combination of two stretching vibrations.
Equilibrium energy distribution in thermal reagents. r-tions are almost always provided; it is violated only in extremely fast processes. The problem is whether it will remain in AK. Due to curvilinearity, the coordinate of the region cannot be considered an independent degree of freedom. Her interaction with other, transverse movements leads to the exchange of energy between them. As a result, firstly, the initially equilibrium distribution of energy over the transverse degrees of freedom may be disrupted and, secondly, the system may return to the reactant region even after it has already passed through the AC configuration in the direction of the products. Finally, it must be borne in mind that, according to equations (2), (3) and (5), chem. r-tion is considered as a classic. transition; quantum features are ignored, e.g. electronic non-adiabatic processes and tunnel effect. In the early formulations of the theory, the so-called transmission multiplier It was assumed that it collected the influence of the factors listed above that were not taken into account when deriving these equations. Thus, the definition of x goes beyond the activated complex of the theory; Moreover, for p-tions in which x differs significantly from unity, the theory loses its meaning. However, for complex areas the assumption does not contradict the experiments. data, and this explains the popularity of the activated complex theory.
Consistent informal consideration of all these effects is possible only within the framework of dynamic. calculation (see Dynamics of an elementary act). Attempts have been made to take them into account separately. For example, a systematic method was proposed. clarification of the AC configuration, since the choice of a saddle point as such is based on intuitive ideas and, generally speaking, is not necessary. There may be other configurations, for which the error in calculations according to formulas (2) and (3), due to the return of the system to the region of reagents after passing through these configurations, is less than for the saddle point configuration. Using the formulation of the activated complex theory in terms of collision theory (see above), it can be argued that the reverse flow (from products to reactants) through the critical. The surface corresponds to the part of the total direct flow that generates it and is equal to it (from reactants to products). The smaller this part, the more accurate the calculation of the speed of the river according to the activated complex theory. These considerations formed the basis of the so-called. variational definition of AC, according to which the surface that minimizes the forward flow is considered critical. For it, the speed of the river, calculated from equations (2) and (3), is minimal. As a rule, zero-point energies of transverse vibrations change along the coordinates of the region. This is another reason for the displacement of the AK configuration from the saddle point of the PES; it is also taken into account by variational theory.
Means. attention was paid to the development of methods for determining the probabilities of quantum tunneling in chemistry. r-tions. Finally, it became possible to estimate the transmission multiplier within the framework of dynamic models. calculations. It is assumed that with the postulate. By the movement of the system along the coordinates of the region, not all of them interact, but only some of the transverse degrees of freedom. They are taken into account in quantum dynamics. calculation; the remaining degrees of freedom are processed within the framework of the equilibrium theory. In such calculations, corrections for quantum tunneling are also automatically determined.
The mentioned improved methods for calculating abs. chemical speed r-tions require serious calculations. efforts and lack the universality of the activated theory complex.
===
Spanish literature for the article "ACTIVATED COMPLEX THEORY": Glesston S, Leidler K., Eyring G., Theory of absolute reaction rates, trans. from English, M., 1948; Leidler K., Kinetics of organic reactions, trans. from English, M., 1966: Thermal bimolecular reactions in gases, M., 1976. M. V. Bazilevsky.
Activated Complex Theory (ACT).
Theory of the Activated Complex - Theory of the Transition State - Theory of Absolute Rates of Chemical Reactions... All these are names of the same theory, into which, back in the 30s, attempts took shape to represent the activation process with the help of fairly detailed, and at the same time still quite general models built on the basis of statistical mechanics and quantum chemistry (quantum mechanics), combining them and creating the illusion of an individual analysis of a specific chemical transformation already at the stage of restructuring the electron-nuclear structure of the reagents.
The task itself seems very complex, and therefore quite a lot of logical ambiguities inevitably formed in SO... Nevertheless, this is the most general and fruitful of the theoretical concepts by which elementary processes are currently described, and its capabilities are not limited to the framework of the elementary chemical act. The development of modern chemical kinetics turned out to be closely related to it. The latest algorithms and graphical techniques of computer chemistry are attached to it, and on its basis the orbital theory of chemical reactivity is rapidly developing...
And that's not all! On the basis of SO, it turned out to be possible to uniformly analyze many physicochemical phenomena and many macroscopic properties of substances, which, at first glance, seem to be the lot of only scientific empirics, which would seem hopelessly inaccessible to theoretical understanding. The reader will find a number of such situations in the magnificent, albeit old, book of Glesston, Eyring and Laidler “The Theory of Absolute Velocities”, written by the creators of this theory...
As elementary reactions in the gas phase, trimolecular collisions are not common, since even in chaotic Brownian motions the probability of simultaneous collisions of three particles is very low. The probability of a trimolecular stage increases sharply if it occurs at the phase boundary, and fragments of the surface of the condensed phase become its participants. Due to such reactions, the main channel for the removal of excess energy from active particles and their disappearance in complex transformations is often created.
Let's consider a trimolecular transformation of the form:
Due to the low probability of trimolecular collisions, it is advisable to introduce a more realistic scheme using a symmetrized set of bimolecular events. (See Emanuel and Knorre, pp. 88-89.)
4.1. Qualitative model of successive bimolecular collisions:
The main assumption is based on the detailed equilibrium in the first stage:
Quasi-equilibrium regime of formation of bimolecular complexes
The resulting rate constant should take the form:
Let us consider the elementary provisions of the theory of the activated complex, including:
- kinetic scheme of activation through an intermediate transition state,
- quasi-thermodynamics of activation through the formation of an activated complex,
is the dimension of the second-order reaction rate constant in SO.
The simplest kinetic model of activation in SO:
(6.1)
The first stage of the activation mechanism is bimolecular. It is reversible, an activated complex is formed on it, and it further decomposes along two routes: a) back into the reagents with which it is in equilibrium, and for this process an equilibrium constant should be introduced, b) into the reaction products and this final process is characterized by some mechanical decay frequency. By combining these steps, it is easy to calculate the reaction rate constant. It is convenient to consider the transformation in the gas phase.
The equilibrium constant of the reversible stage can be expressed in the following way.
If the standard states in the gas phase are chosen according to the usual thermodynamic rule, and the partial pressures of the gaseous participants in the reaction are standardized, then this means:
Attention! This implies an expression for the rate constant of a bimolecular reaction in SO, which does not raise doubts about the dimensionality of the rate constants of bimolecular reactions:
In textbooks, a less transparent expression is most often given, based on a different standardization of states - the concentration is standardized, and as a result, the dimension of the rate constant appears, which outwardly corresponds to a mono- rather than a bi-molecular reaction. The dimensions of concentrations appear to be hidden. Eyring, Glesston and Laidler - the creators of SO themselves - have an analysis in the book “The Theory of Absolute Rates of Reactions”, which takes into account the standardization of states by pressure. If we consider the standard state with single concentrations of reagents and products, then the formulas will be slightly simplified, namely:
This implies the expression for the rate constant usually presented in textbooks according to SO: 
(6.3)
If the role of the standard state is not highlighted, then the theoretical rate constant of a bimolecular transformation may acquire an alien dimension, inverse to time, which will correspond to the monomolecular stage of decomposition of the activated complex. The activation quantities S#0 and H#0 cannot be considered ordinary thermodynamic functions of the state. They are not comparable with the usual characteristics of the reaction path simply because there are simply no methods for their direct thermochemical measurement... For this reason, they can be called quasi-thermodynamic characteristics of the activation process.
When a particle of an activated complex is formed from two initial particles, 
, and the result is
The dimension of the rate constant is usual for a second-order reaction:
Empirical activation energy according to Arrhenius and its comparison with similar ones
similar activation parameters (energies) TAS and SO:
The basis is the Arrhenius equation in differential form: 
1) in TAS we get:
2.1) SO. Case 1. (General approach subject to standardization of concentrations)
substitution into the Arrhenius equation gives
2.2) SO. Case 2. (A special case of the bimolecular activation stage 
).
Arrhenius activation energy for a bimolecular reaction:
Attention!!! We believe most often 
2.2) Based on the standardization of pressure, we obtain the activation energy:
(6.7)
2.3) The same is obtained for a bimolecular reaction and when standardizing the concentration:
in the bimolecular act of activation n#= -1, and 
(6.10)
Result: The formula relating the Arrhenius activation energy to the quasi-thermodynamic activation functions of the transition state theory does not depend on the choice of the standard state.
3. Adiabatic potentials and potential surfaces.
Example. The reaction of the exchange of one of the atoms in a hydrogen molecule for deuterium
(This is the simplest possible example)
As the deuterium atom approaches the hydrogen molecule, loosening of the old two-center chemical H-H connections and the gradual formation of a new H-D bond, so that the energy model of the deuterium exchange reaction in a hydrogen molecule can be constructed as a gradual movement of the initial triatomic system to the final one according to the scheme:
Potential surface simplest reaction- adiabatic potential of the reacting system, sections and singular points.
The potential energy surface (potential surface) is a graphical representation of a function called the adiabatic potential.
The adiabatic potential is the total energy of the system, including the energy of electrons (kinetic energy and potential energy of their attraction to nuclei and mutual repulsion), as well as the potential energy of mutual repulsion of nuclei. The adiabatic potential does not include the kinetic energy of nuclei.
This is achieved by the fact that in each geometric configuration of the nuclear core the nuclei are considered to be at rest, and their electric field is considered as static. In such an electrostatic field of a system of nuclei, the characteristics of the main electron term are calculated. By changing the relative position of the nuclei (the geometry of the nuclear core), for each of their relative positions the calculation is again made and thus the potential energy surface (PES) is obtained, the graph of which is presented in the figure.
The figurative point represents a reacting system consisting of three HHD atoms and moves along the potential surface in accordance with the principle of minimum energy along line abc, which is the most probable energy trajectory. Each point lying in the horizontal coordinate plane corresponds to one of the possible combinations of two internuclear distances 
, whose function is the total energy of the reacting system. The projection of the energy trajectory abc onto the coordinate plane is called the reaction coordinate. This is the a’b’c’ line (it should not be confused with the thermodynamic reaction coordinate).
The set of horizontal sections of the potential surface forms a map of the potential surface. It is easy to trace the reaction coordinate on it in the form of a curve connecting the points of maximum curvature of the horizontal sections of the adiabatic potential (APE) graph.
Rice. 12-14. Potential surface, its energy “map” and its “profile” section along the reaction coordinate H3 + D HD + H
By unfolding on the plane a fragment of the cylindrical surface abcb’c’a’, formed by the verticals placed between the coordinate plane and the PES, we obtain the energy profile of the reaction. Note that the fairly symmetrical appearance of the potential surface and, accordingly, the energy profile of the reaction is a feature of this particular reaction, in which the energy electronic characteristics of the reactant particles and product particles are almost the same. If the sets of particles entering the reaction and those formed are different, then both the potential energy surface and the energy profile of the reaction lose symmetry.
The potential surface method is currently one of the most common methods for theoretical study of the energy of elementary processes occurring not only during chemical reactions, but also in intramolecular dynamic processes. The method is especially attractive if the system has a small number of mechanical degrees of freedom under study. This approach is convenient for studying internal molecular activated movements using chemical kinetics techniques. As an example, we can cite the adiabatic potential of internal rotations in the radical anion, constructed on the basis of quantum chemical calculations of the LCAO MO in the MNDO approximation, which is
is a periodic function of two angular variables. A repeating fragment of the PES is shown in Figure 15. The variable corresponds to rotations of the phenyl ring relative to the C(cycle)-S bond, and the variable corresponds to rotations of the CF3 group relative to the S-CF3 bond. Even a quick glance at the potential surface is enough to see that the energy barrier for rotation of the CF3 group relative to the sulfonyl fragment is significantly lower than the barrier for rotation of the phenyl ring relative to the SO2 group.
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The theory of the activated complex is undoubtedly a development of the ideas of Arrhenius. According to this theory, an elementary reaction proceeds continuously from the initial to the final state and passes through a transition state characterized by maximum energy. The complex corresponding to this state is said to be an activated complex. The change in energy during a reaction can be represented by an energy diagram, as in Fig. 5-1. In such a diagram, various energy values can be plotted along the ordinate axis. Since we are looking at reactions in solutions, where the energy difference is measured at constant pressure, it makes sense to use enthalpy. Then the difference between the molar enthalpies of the initial and final states is the enthalpy of the reaction, and the corresponding difference for the initial and transition states is the enthalpy of activation (To denote quantities related to the activated complex, a superscript in the form of a double cross is used.) The abscissa corresponds to the reaction coordinate, which represents the depth of occurrence reactions. It should be noted that there are several problems associated with physical meaning such energy diagrams. The problem arises from the confusion between the microscopic and macroscopic behavior of matter. Obviously, the reaction coordinate corresponds to the path of an individual molecule, and not to the combined behavior of all particles present in the reaction vessel. If all reacting particles simultaneously overcome the energy barrier, this would be inconsistent with the second law of thermodynamics. At the same time
Rice. 5-1. Energy diagram for an endothermic elementary reaction.
energy coordinate reflects thermodynamic properties reacting system, i.e. average changes in the energy of large particle accumulations. However, this problem is rather formal. For example, it can be circumvented by considering a “borderline” number of particles, say which, on the one hand, is so small that the second law of thermodynamics is not violated, since deviations from thermodynamic equilibrium, the so-called fluctuations, are more likely when the number of observed particles decreases. On the other hand, this number of particles turns out to be sufficient for the application of thermodynamic quantities.
It is assumed that the activated complex is in equilibrium with the initial reagents, and the second-order elementary reaction equation is
can be represented in the form
Then, by analogy with the equation, the reaction rate constant can be written as
where K is the equilibrium constant. The rate constant k can be considered as the frequency of decay of the activated complex, due to which the product is formed. Eyring proposed to consider k equal from here
where are the Boltzmann and Planck constants, respectively. The equilibrium constant is related to the change in Gibbs free energy:
The Gibbs free energy of activation can be expressed in terms of the enthalpy of activation and the entropy of activation, i.e.
Substituting (5-11) and (5-10) into leads to the Eyring equation
or, in logarithmic form,
which can be conveniently converted to the form
Accordingly, a dependence on is plotted on the Eyring diagram, and a straight line is obtained with a slope equal to - Knowing this value, they are derived from equation (5-14) using various sets of values. It should be noted that the rate constant in the Eyring equation has the dimension Therefore, strictly speaking, the activation entropy can only be calculated for first order reactions. [The dimension of concentration is lost in the transition from to (5-10), since it is measured in whereas the quantity Of course, Eyring's equation can be applied to reactions of any order, but it should be borne in mind that the absolute values strongly depend on the choice of standard states. Therefore, caution is necessary when comparing the activation entropies of reactions that have different orders. As Melvin-Hughes showed, according to collision theory, the activation entropy of a bimolecular reaction at is more positive than the activation entropy calculated using the Eyring formula. It still needs to be proven whether this is related to the observation that only the indicated difference in activation entropies was experimentally found for two reactions, one of which is first and the other second order; otherwise, it remains to be assumed that these reactions occur according to the same “internal mechanism”.
