Thermodynamic rush

When the thermodynamic rushes of the reaction groups are equal to one another, h = hj, for aU feasible reactions ij in the system, the system is in the stable thermodynamic equilibrium (A y = 0), and, therefore, changes in chemical variables and overall rates of aU the reactions are naturally equal to zero. 

Figure 1.2 gives the comparative graphical interpretations of an elemen tary chemical reaction in commonly accepted energetic coordinates and in the thermodynamic coordinates under the discussion. Note that the traditional energetic coordinates are always related to the fixed (typically, unit) reactant concentrations and, therefore, identify the behavior of standard values of the plotted parameters. As for the thermodynamic coordinates, they illustrate the process that proceeds under real conditions and are not restricted by the standard values of chemical potentials or thermodynamic rushes of the reac tants. The thermodynamic (canonical) form of kinetic equations is conve nient for a combined kinetic thermodynamic analysis of reversible chemical processes, especially for those that proceed in the stationary mode. 

Figure 1.2 Schematic representation of the pathway of elementary reaction ij in the traditional energetic coordinates with the activation barrier (a) and in the coordinates of thermodynamic rushes h of reactants (b). in the latter case, the reaction can be represented as a flow of a fluid between two basins separated by a membrane with permeability e-,j the examples are given for the left-to-right and right-to-left reactions (cases 1 and 3, respectively) case 2 illustrates the thermodynamically equilibrium system.

The simplest example is a stepwise process that proceeds via an arbitrary combination of elementary monomolecular transformations of inter mediates that all exist in their stationary states. When this is the case, the stationary rate of the stepwise process appears to be proportional to the difference between thermodynamic rushes of the initial and final reaction groups—in other words, the stepwise process can be considered from the viewpoint of both kinetics and thermodynamics as a single effective elementary reaction. 

Here, Ry and ay are the active resistance and the corresponding electric conductance of the circuit fragment between points i and j. Note that in kinetic equation (1.31), thermodynamic rushes fr of the reactant groups behave as electric potentials in the points, while parameter Ey is equivalent to electric conductance ay. 

Apparently, thermodynamic rushes for the series of transformations under consideration and, as a result, chemical potentials of the intermediates in their stationary states must decrease progressively while passing from one intermediate to another. When the stepwise transformation may follow sev eral parallel pathways of consecutive elementary transformations, the said relationship between the stationary chemical potentials of the intermediates must be met for each of the possible pathways of the stepwise reaction. 

Figure 1.4 A schematic diagram of chemical potential changes at the stationary occurrence of a stepwise reaction R Yq Y2 P, where R and P are the initial reactant and final product of the reaction, while Yq and Y2 are thermalized Intermediates. The minimums in the traditional potential energy profile relate to the standard chemical potentials of thermalized external reactants and intermediates. However, actual chemical transformations of the intermediates occur at stationary values Pyi and pvz (bold lines), the rates of these transformations being dependent on the difference of the corresponding thermodynamic rushes and the values of truncated rate constants e-,j (the latter are functions of standard chemical potentials of the transition states only).

The thermodynamic form of kinetic equations is helpful for providing the kinetic thermodynamic analysis of the effect of various thermodynamic parameters on the stationary rate of complex stepwise processes. Following are a few examples of such analyses in application to the noncatalytic reac tions. The analysis of the occurrence of catalytic transformations is more specific because the concentrations and, therefore, the chemical potentials and thermodynamic rushes of the intermediates are usually related to one another in the total concentrations of the catalyticaUy active centers. (Catalytic reactions are discussed in more detail in Chapter 4.)... 

In this case, Sj is determined by the standard thermodynamic parameters of both the transition states and the initial and final reactants but not of the thermalized intermediates. Therefore, equations (1.36) and (1.40) con elude that the stationary rate of a complex stepwise reaction comprising an arbitrary combination of intermediate linear transformations is indepen dent of the standard values of thermodynamic parameters of the intermedi ates. It is determined only by the difference of thermodynamic rushes of the initial reactant and the product, as well as by the standard thermo dynamic parameters of the transition states between various intertrans forming reaction groups. 

An important consideration is the variations in thermodynamic rushes of the reaction groups in elementary reactions that are not in the partial... 

At the same time, the rate determining step features the maximum dif ference of thermodynamic rushes (or the maximum difference of chemical potentials) between the reacting partners in the process under consider ation. Indeed, in the consecutive transformations, the stationary rates are identical through aU of the reaction channels ... 

Therefore, the difference of thermodynamic rushes at elementary step i is equal to... 

The maximum difference of thermodynamic rushes is, obviously, corre spondent to the occurrence of the bottleneck in the stepwise reaction. Therefore, the rate limiting step in a sequence of chemical transformations is naturally to define as some elementary step with the maximum differ ence of thermodynamic rushes (or that has nearly the same chemical potentials) of the reaction groups involved in the transformation. Horiuti first mentioned this specificity of the stepwise reaction bottleneck. 

Figure 1.7 An example of the interrelation of the bottleneck created by the rate-limiting step of a consecutive set of monomolecular transformations with the decrease in thermodynamic rushes of consecutive thermalized in intermediates Y. The stationary rate of the overall stepwise reaction here should be VE = 2(R P).

Thus, the rate-controlling parameters here are truncated rate constant 82 and, consequently, the energy of the transition state for elementary reaction Yq Y2, as well as thermodynamic rushes of initial reactant R and final product P. 

Hence, practically aU the difference of thermodynamic rushes in the trans formation chain falls on only one rate limiting step. When the stepwise reaction is kineticaUy irreversible (i.e., R P),... 

In the left-to-right reaction, the necessity of decreasing thermodynamic rushes as well as the existence of partial equilibrium for transformations B 2 C give... 

In addition, as shown in Section 1.3, the fluxes of the concentration of chemical components are determined generally by the differences in the thermodynamic rushes of corresponding reaction groups rather than by true thermodynamic forces (chemical affinities of the reactions). Consider the simplest pathway of cocurrent transformations, which includes two parallel channels of independent reversible transformation of initial reactant R into products Pi and P2 ... 

We wiU treat the spontaneous transformation of R to Pi as the main process route (process Zl). This transformation is possible when chemical potential ip (and, consequendy, thermodynamic rush R = exp(dR/RT)) of compound R is higher than potential (correspondingly, higher than the thermodynamic rush Pi = exp(ppj/RT)) of compound Pr An analogous relation between Pr and Pp is necessary for the transformation of compound R to the by product P2 (Figure 2.4A). 

The discussed chemical transfomiations can be visualized in the coordi nates of current chemical potential values (see Figure 2.4a), where chemi cal potentials of components R, Pj, and P2 are the external parameters. While this is the case of the absence of common intermediates, an indirect influence of the second reaction on the first reaction is possible only when chemical potential iTp2 of compound P2 (its thermodynamic rush P2) is higher than the chemical potential of compound R (its thermodynamic rush R). When so, the consecutive reaction P2 — R Pi becomes pos sible, and this compound P2 can be involved as an additional substrate into the main reaction R — Pi. In the considered example, the phenomenon of conjugation of chemical processes is not observed. The conjuga tion takes place only in the presence of common intermediates for both channels of the transformations. 

Equations of type (2.17) for the interrelation of the rates of conjugate stepwise reactions are valid for any intermediate linear transformation pathways (including catalytic reactions). The value of A may be expressed by relations that are much more complicated than (2.15) and depends not only on parameters Sy but also on thermodynamic rushes of some external reactants of the stepwise reactions (see Section 2.3.5 for exam pies). At the same time. A > 0 always. However, the relationship between the cross coefficients Ay and Aj may be more intricate than that in the traditional Onsager equations. 

Let us prove the GlansdorGPrigogine theorem with an example of an arbitrary spatially homogeneous chemical reactive system. The internal parameters for such a system are the concentrations of intermediates of the stepwise chemical transformations. Any spontaneous changes of the system (and, as a result, changes in internal driving forces) relate namely to changes in the intermediate concentrations. Therefore, the partial force differential dxP may be substituted for by its full analogue related thermodynamic rushes (concentrations) of intermediates (a 1,. .., k) of the stepwise transformations ... 

Here, ( Aq[ ) symbolizes a function of independent variables that are thermodynamic rushes of aU hnearly independent components in the system. In an open system, obviously, the component rushes may either be controlled by external conditions due to the fixed concentrations of ini tial reactants and, often, final reaction products (if so, aU serve as the external variables) or be determined by spontaneous internal processes... 

Here, index i and parameter are substituted for by index a and parame ter Ao(, which is always allowed for monomolecular processes. Thus, the functional O derivative with respect to thermodynamic rush of an inter mediate is proportional to the rate of this intermediate concentration changes everywhere, even far from thermodynamic equihbrium. For this reason, in the state stationary with respect to the intermediate concentra tion, this derivative turns zero (cf the case of the Rayleigh Onsager functional). 

In conditions of a stable thermodynamic equilibrium, thermodynamic rushes of all of the involved interacting reaction groups are equal to one another and, therefore, O equals zero for any chemically reactive system. 

Suppose that aU thermodynamic rushes hj and hj are independent of one another in that the thermodynamic rushes of one or several inter mediates (i.e., reactants whose concentrations are not fixed by external conditions but determined by the stationary mode) if they are included in rush hi, are absent in aU of the rest of rushes hj (j i). Formally, this is proved by expression... 

However, equation (3.16) is not always positive. This implies a not necessarily stable stationary state of the stepwise reaction under consideration. Indeed, it was shown previously that autocatalytic reactions at certain ratios of internal parameters the reactant concentrations (thermodynamic rushes) are a spectacular example of processes with the unstable stationary state. 

Figure 3.1 Dependencies of the stationary thermodynamic rushes of intermediate Y (A) and the reievant rate of the stepwise process (B) on the vaiue of the controiiing parameter for autocataiytic scheme (3.21)—thermodynamic rush of initiai reactant R. Point Rcr = 2/ 1 is the point of bifurcation of stationary states. Symbois i and ii indicate different branches of the stationary states.

The increment of the dissipation energy upon emergence of a small fluctuation in the stationary concentration thermodynamic (rush) of this intermediate is described by equation... 

Let us analyze the stability of these stationary states using the kinetic method by considering the relaxation of a minor fluctuation of thermodynamic rush Y about stationary solution Y. Let Y = Y, -Ly, with y as the minor fluctuation. When retaining only linear terms in respect to y, from equation (3.25) we receive the kinetic equation for the y evolution ... 

Figure 3.2 The stationary thermodynamic rushes of intermediate Y in the autocatalytic Schloegl scheme in respect to the value of controlling parameter R. See the text for the explanation.

Thus, the system, once it reaches the stationary state of the branch Y2, turns out to be unstable, and any minor fluctuation makes Y transfer to either branch Yi or branch Y3. Such behavior resembles an electronic switch (trigger), and, for this reason, the system under discussion is referred to as "trigger system. A thermodynamic rush of initial reactant R may be taken here, similar to Example 7, as the controlling parameter. 

Let us say that the system has only one independent variable Y for example, the concentration (thermodynamic rush) of some intermediate. In this case, the evolution criterion d P < 0 can be expressed in the form of total differential (3.4)... 

Figure 3.5 Typical diagrams of the stationary magnitude of some internal parameter Y (e.g., the concentration or thermodynamic rush of an intermediate) (A) and corresponding changes in the energy dissipation rate P (B) as the functions of a controlling parameter a upon deviation from point Y( o) of the initial stable stationary state and upon crossing the bifurcation point a 1 is the...

This corresponds to the situation "center" by Fyapunov (see Figure 3.4) and to a soiution with the thermodynamic rushes (concentrations) of intermediates Y and Z osciiiating... 

Figure 3.7 Typical kinetic curves (A, C, E) and the corresponding phase trajectories (B, D, F) of the evolution of thermodynamic rushes of intermediates Y (solid line) and Z (dashed line) in the the Lotka-Volterra scheme (3.30). The calculations are given for the cases of constant values R = 1, = 82 = 1, and 83 = 5 at different...

Figure 3.8 Typical kinetic curves (A, C, E) and the corresponding phase trajectories (B, D, F) of the evolution of thermodynamic rushes of intermediates Y (solid line) and Z (dash line) in scheme (3.35) with damped oscillations. Calculations are given for the cases of constant values Sq = 2, S2 = 0.5, and S3 = 5 and starting condition Ya =2, Za = t at different values of the controlling parameters R R = 1 in diagrams A and B (stationary state at Y = 10.4, Z = 0.4) R = 30 in diagrams C and D (stationary state at Y — 22, Z — 12), and R = 50 in diagrams E and F (stationary state Y = 30, Z — 20). There is a bifurcation value of the controlling parameter R = 25 for all the other external parameters.

The expressions of the derived type (4.10) are valid for the intermediate linear schemes of catalytic stepwise reactions over a wide range of condi tions. Therefore, the stationary rate vj of the overall stepwise process can be treated here as proportional to the difference of thermodynamic rushes of the initial reactants and final products of the catalytic reaction. This is in line with the Horiuti Boreskov relation (1.36) (see Section 1.3.2) that describes the dependence of the stationary rates of many catalytic processes on the degree of their thermodynamic nonequilibricity. 

It is evident here that the rate-limiting stage (the reaction "bottleneck") that is the step with the largest difference of the stationary values of chemical potentials (thermodynamic rushes) of the interacting reaction groups is the one with minimal s,. This is identical to the case of the chain of noncatalytic transformations (1.54). 

It is essential, however, that expression (4.6) comprises a dependent "internal" parameter that is the thermodynamic rush K of the free form of the active center of the catalyst. At the same time, for making the kinetic-thermodynamic analysis, one must have only "external" parameters in the final expression for Vj. In the case of the active centers of a catalyst, these are their full concentration. Let us denote this concentration by [K]o. It is easy to find the relationship between [K] and [K]o based on the balance between the different forms of active centers. In scheme (4.4), this corresponds to the simple equality... 

Here, Kq is the thermodynamic rush of the free form of the active centers of the catalyst in the case when all of the active centers remain free in other words, in the state with 0K = 1, Hk and Hk, are standard chemical potentials of the free form of active center K and that of intermediate complex Ki, respectively, Xk, s exp ( tK Hk, )/0Tj. 

Hence, at low coverage of the active centers by the intermediate, the stationary rate, v, of the catalytic reaction under consideration, like that of noncatalytic transformations, is independent of the standard thermodynamic parameters of the thermalized intermediate. In addition to the linear dependence of the stationary rate on the difference between thermodynamic rushes of the initial reactant and the final product, there is also proportionality of the rate on the thermodynamic rush of the free form of the active center and the dependence on the standard thermodynamic parameters of transition states of the elementary reactions in scheme (4.4). 

A specific feature of catalytic schemes that imply the type (4.1) balance is the potential involvement of independent internal variables Hke Ki/K and 0Ki/6K instead of, for example, thermodynamic rushes of inter mediates K,. This is due to the existent balance relations. 

The Ki/K ratio ( reduced thermodynamic rush of catalytic intermedi ate Kj ) should be treated here as an independent internal variable that describes the spontaneous evolution of the system in tending to its station ary state. In fact. 

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