times of the various chemical species, making it easy to establish a connection between the response data and the mechanism and kinetics of the process.

The structure of this chapter is as follows. We begin by discussing the lifetime distribution of a chemical species in a complex chemical system and show that simple response experiments involving neutral tracers make it possible to evaluate the lifetime distributions from measurements. We extend the results for disordered kinetics and illustrate the approach by extracting kinetic information from a complex desorption experiment reported in the literature. We then consider more complicated response experiments involving multiple reaction pathways connecting various reaction intermediates and present methods for extracting connectivity, mechanistic, and kinetic information from experimental data. We pay special attention to the effect of different experimental as well as numerical errors on the accuracy of the results. We consider a generalization of the neutral response approach to inhomogeneous, space-dependent systems and illustrate the technique by an application in population genetics.

12.1Lifetime Distributions of Chemical Species

In chemical kinetics the lifetime of an active intermediate is an important variable which characterizes the time scale of the process in which the intermediate is involved. Different techniques have been suggested for the direct experimental evaluation of this parameter; they are used extensively for the analysis of chain reactions [13] such as radical polymerization [14]. These techniques are based on the assumption that the mean lifetime of an active intermediate Xi can be evaluated as the ratio between its concentration xi and its rate of disappearance Wi−:

In general the lifetime of an active intermediate is the time interval that elapses from the generation of the intermediate up to its disappearance. Equation (12.1) is valid only for the particular case of stationary processes for which both xi and Wi− are time-independent. For a time-dependent process, eq. (12.1) is no longer valid. An alternative theory has been suggested for the study of time-dependent systems, based on the use of age-dependent balance equations [1–12]. If the kinetics of a reaction system is known, this theory provides analytical expressions for the probability distributions of the lifetime of the different individuals in the system. Although the lifetime is a random variable obeying probability laws, its statistical properties can be evaluated by using deterministic kinetic equations. This type of approach is usually referred to in the literature as “semi-stochastic” [15]. In the following we give a short summary of the semi-stochastic theory of lifetime distributions.

We consider an open isothermal system made up of many active intermediates Xi , i = 1, 2, . . . , as well as many stable species Ai , i = 1, 2, . . . . The deterministic evolution equations for the concentrations xi , i = 1, 2, . . . , of the active intermediates Xi , i = 1, 2, . . . , are given by

where Wi±(x), i = 1, 2, . . . , are the rates of generation and consumption of the different intermediates, respectively, xi (t) and ai (t) are the concentrations of the active

172 DETERMINATION OF COMPLEX REACTION MECHANISMS

intermediates and stable species, respectively, and x (t) = (xi (t)) and a (t) = (ai (t)) are the corresponding concentration vectors. A similar set of kinetic equations can be written for the concentrations ai (t) of the stable species; in some cases we can assume that the concentration of stable species is kept constant by interaction with a set of buffers.

We denote by ηi (τ ; t) dτ the density of particles of type Xi at time t with a lifetime in the system between τ and τ + dτ . We have

interval t, t + dt; the complementary probability, that a particle Xi survives in a time

Equation (12.5) is derived by writing a balance equation for the number of particles Xi

the lifetime probability density of active intermediates Xi at time t. This probability density can be expressed in terms of the density function ηi (τ ; t) dτ as

from eqs. (12.2)–(12.8), resulting in a system of evolution equations for the lifetime

is the probability that a new Xi intermediate is generated at a time between t and t + dt. An analytical solution of eqs. (12.9)–(12.10) can be derived by using the

method of characteristics [16]. The solutions corresponding to the initial conditions,

ξi (τ ; t = t0) = ξi0 (τ ) , i = 1, 2, . . ., are given by

where h(x) is Heaviside’s step function. In particular, for a stationary state we have λ+i , λ−i and Wi+ = Wi− independent of time, and eqs. (12.12) become

and the average lifetimes are given by eq. (12.1):

∞

< τ >xi = τ ξi (τ ) dτ = xi /Wi− (12.14)

0

The theory can be extended to reaction systems with disordered kinetics, for which the rate coefficients are random. There are two different cases: (1) static disorder, for which the rate coefficients are random variables selected from certain probability laws; and (2) dynamic disorder, for which the rate coefficients are random functions of time. For details see [5]. Here we give the expression of the average probability density of the lifetime for stationary systems with static disorder:

where p (k) dk is the probability density of the vector k of the rate coefficients. Equation (12.15) will be used later for the theoretical analysis of the desorption response experiments of Drazer and Zanette [17].

12.2Response Experiments and Lifetime Distributions

We consider that the chemical process studied takes place in an isothermal, continuousflow stirred tank reactor (CSTR) operated with a constant total flow rate and with constant input concentrations. We assume that at least one of the chemicals entering the system is available in two different forms, unlabeled and labeled, respectively, and that the kinetic isotope effect can be neglected, that is, the kinetic parameters are the same for the labeled

174 DETERMINATION OF COMPLEX REACTION MECHANISMS

and unlabeled compounds. Although the total input concentrations of chemicals are kept constant, we assume that it is possible to control the proportion of labeled compounds entering the system. Under these circumstances the overall kinetic behavior of the system, expressed in terms of total concentrations, is not affected by the introduction of tracer in the system. Only the proportions of labeled compounds in the reactor depend on the amounts of tracers introduced in the system. By measuring the concentrations of labeled chemicals in the reactor it is possible to evaluate the distributions of lifetimes of different individuals, as we now show.

For a given reaction intermediate, which may exist in an unlabeled form Xi as well as in a labeled form, Xi#, we assume that it is possible to measure the time dependence of the following fractions:

1.The ratio αi (t) between the rate Wi+# [x (t)] of generation of the labeled compound Xi# and the total rate of generation Wi+ [x (t)] of the two forms, Xi and Xi# , labeled and unlabeled:

2. The proportion βi (t) of the active intermediate Xi# :

where xi# (t) and xi (t) are the concentrations of the labeled active intermediate and the total concentration of the active intermediate, labeled and unlabeled.

In order to extract the information about the lifetime distributions from these two functions, αi (t) and βi (t), we derive an evolution equation for the concentration xi# (t) of labeled intermediate in the system:

Balance equations of this type were introduced in chemical kinetics by Neiman and Gál [18]. Equations (12.18) must be supplemented by the kinetic equations (12.2):

For simplicity, in eqs. (12.19) the dependence of the reaction rates on the concentrations of stable species is not displayed explicitly.

By combining eqs. (12.16)–(12.19) we can derive the following relationship between

αi (t) and βi (t):

These are differential equations in βi (t). By considering the initial values βi (t0), their solutions are

Equations (12.21) contain transient terms depending on the initial conditions βi (t0); similar transient contributions, depending on the initial conditions ξi (τ, t0), exist in eqs. (12.12) for the lifetime distributions. In both cases, for large times the contribution of these transient terms is negligible. For simplicity we push the initial times in eqs. (12.12) and (12.21) to minus infinity, t0 → −∞, and compare the resulting equations; under these circumstances the response functions βi (t) can be expressed in terms of the excitation functions αi (t) and the probability densities of the lifetimes attached to the different active intermediates ξi (τ, t):

∞

βi (t) = ξi (τ ; t) αi (t − τ ) dτ (12.22)

0

By requiring that the initial conditions in eqs. (12.12) and (12.21) satisfy the law of mass conservation, it can be shown that eqs. (12.22) are valid for any initial times. Equations (12.22) are linear superposition laws, which can be used for evaluating the lifetime distributions ξi (τ, t) of the active intermediates from response experiments. For nonstationary regimes, it is necessary to carry out repeated response experiments, starting at different initial times. By solving the integral equations (12.22) it is possible to compute the time and space dependence of lifetime distributions. Frequency techniques can also be used; for nonstationary regimes they also involve repeating the response experiments.

For stationary regimes the lifetime distributions ξi (τ, t) = ξi (τ ) depend only on the lifetime and eqs. (12.22) become convolution equations:

where denotes the temporal convolution product. In this case a single response experiment, either in real time or in the frequency domain, is enough for evaluating the lifetime distributions.

The response law (12.22) can be extended to disordered kinetics with static or dynamic disorder [5]. For stationary systems with static disorder the response law is

∞

βi (t) = ξi (τ ) αi (t − τ ) dτ (12.24)

0

where . . . denotes a static average with respect to all possible values of the rate coefficients. For more general cases, see [5].

The lifetime distributions contain kinetic information. For illustrating the usefulness of our approach we consider the experiments of Drazer and Zanette on disordered

176 DETERMINATION OF COMPLEX REACTION MECHANISMS

desorption kinetics [17]. These authors used nonconsolidated packings of relatively uniform, spherical, activated carbon grains obtained from apricot pits, with an average radius of d = (0.13 ± 0.01) cm. The carbon grains were packed in a 30 cm high, 2.54 cm inner diameter cylinder. In the experiments, the porous medium was initially filled up with aqueous 0.1 M NaI solution tagged with I131. The authors performed measurements of tracer adsorption and dispersion, in which a stepwise variation of the concentration of I131 is induced at time t = 0 and kept constant thereafter. Two different types of experiments, each with a total constant flow rate, were carried out, by using different displacing fluids. In the first set of experiments the system was flushed with an untagged NaI solution having the same concentration as the initial, labeled, solution (exchange experiments), whereas in the second set of experiments the untagged NaI solution was replaced by distilled water (desorption experiments).

The results of the exchange experiments show that the replacement of the radioactive isotope I131 obeys a classical adsorption–desorption and dispersion mechanism commonly encountered in chemical engineering. In this case the observed concentration profiles can be reproduced theoretically by using a classical reaction–convection– diffusion equation. The time dependence of the concentration of I131 is described by a set of kinetic curves, which decay very quickly to zero. In the case of desorption experiments the concentration profiles I131 also decrease to zero for large times, but this decrease is much slower than in the case of exchange experiments. In the case of desorption for large times the concentration profiles converge toward a long time tail of the negative power law type, C (t) t−µ, t 0, characterized by a fractal exponent µ = 0.63. Such long tails cannot be explained by assuming a classical reaction–diffusion mechanism.

The experimental results of the authors suggest that the displacement of the radioactive isotope I131 involves a very slow, dispersive (Montroll) diffusion processs [19]. The qualitative physical picture suggested by the authors is the following: the motion of an atom of the radioactive isotope along the column can be represented by a hopping mechanism involving a succession of desorption and readsorption processes, which is basically a random walk in continuous time (CTRW [19]). We developed a transport theory [7] for the Drazer–Zanette experiment which shows that the combination of hydrodynamic and rate components leads to a linear response law and that for desorption experiments the probability density of the lifetime is proportional to the concentration profile, where the time is replaced by the lifetime, ξ (t) C (τ ) τ −µ, a result that is consistent with the Drazer and Zanette interpretation of their experiment. In another publication, we focused on the chemical interpretation of their result and suggested a kinetic model for their experiment [6].

Following Drazer and Zanette, we assume that, although the whole system is not at equilibrium, in different local areas of the column there is a local equilibrium, We invoke the homottatic patch approximation in heterogeneous catalysis [20], according to which the overall adsorption isotherm is made up of a contribution of local Langmuir adsorption isotherms:

where θlocal is the local coverage of the surface, C is the local concentration of the chemical in the solution, and

are local adsorption and desorption rates, respectively, E± are activation energies, and

is the local change of the enthalpy during the adsorption process, ε is the adsorption heat, kB is Boltzmann’s constant, and T is the temperature of the system. According to the patch approximation, different patches of the surface are characterized by different adsorption enthalpies and their statistical distribution can be described in terms of a probability density:

p (ε) dε with p (ε) dε = 1 (12.29)

The total adsorption isotherm can be expressed as an average of the local isotherm (12.25) with respect to the adsorption heat:

In addition, we assume that the activation energies E± of the adsorption–desorption processes and the local adsorption heat, ε = − H , are related to each other by means of the linear relations:

where β± are proportionality (scaling) coefficients. The existence of the linear relationships (12.31) for adsorption kinetics has been well documented in the literature of heterogeneous catalysis, both experimentally and theoretically [21]. The relations (12.31) are referred to as Polanyi relations or “linear free energy relations,” even though they involve the adsorption enthalpy rather than the Gibbs or Helmholtz free energy. They are equivalent to linear free energy relations only if the entropy factors of the adsorption–desorption process, both thermodynamic and kinetic, are the same for all active sites on the surface (eq. (12.21)). From eqs. (12.28) and (12.31) it follows that the scaling coefficients β± are related to each other by means of the relationship

An important consequence of the linear energy relations (12.31) is that the rate coefficients klocal± are deterministic functions of the adsorption heat ε = − H :

and thus, both for thermodynamic and kinetic variables, the global averages can be expressed in terms of the probability density p (ε) of the adsorption heat.

178 DETERMINATION OF COMPLEX REACTION MECHANISMS

We formally represent the adsorption–desorption process as a chemical reaction:

and denote by [X] = C the concentration of chemical in the solution, by [Y (ε)] = y (ε) dε the surface concentration of free sites with an adsorption heat between ε and ε + dε, and by [Z (ε)] = z (ε) dε the corresponding surface concentration of occupied sites. We also introduce a joint density function for the adsorption heat and the trapping time:

ηZ (τ, ε) dεdτ with z (ε) dε = dε ηZ (τ, ε) dτ (12.35)

Here ηZ (τ, ε) dεdτ is the surface concentration of occupied sites with an adsorption heat between ε and ε + dε and which have trapped a molecule for a time interval between τ and τ + dτ . By generalizing the theory of lifetime distributions we can derive the balance equations:

We introduce the conditional probability density

of the trapping time corresponding to a given value of the adsorption heat. Finally, the unconditional probability density of the trapping times can be evaluated by averaging over all possible values of the adsorption heat:

Next we use the Zeldovich–Roginskii model [22], for which the adsorption heat can take any positive value between zero and infinity and the probability density p (ε) is exponential:

The physical meaning of the Zeldovich–Roginskii model has been discussed in detail in [22]: it corresponds to a canonical distribution “frozen” at a temperature T > T where

Tis the current temperature of the system. The value of the characteristic temperature

T> T provides information about the history of processing the surface.

At local equilibrium the adsorption rate equals the desorption rate, resulting in:

In addition the total number of sites, occupied and free, is conserved, and thus we have the balance equation:

where u (ε) dε is the total number of sites, free and occupied, with an adsorption heat between ε and ε + dε. By solving eqs. (12.40)–(12.41) we arrive at

By integrating eq. (12.42) over all possible values of the adsorption heat we arrive at

where

is the total surface concentration of active sites, free and occupied. The probability density p (ε) dε is expressed as a ratio between the surface concentration u (ε) dε of active sites with the adsorption heat between ε and ε + dε, and the total surface concentration U of active sites:

We insert eq. (12.39) into eq. (12.44) and carry out the integral. After a number of transformations we obtain

are the complete and incomplete beta functions, respectively, and α = T /T , 1 > α > 0, is a scaling exponent between zero and unity. The limit C → 0 in eq. (12.48)

180 DETERMINATION OF COMPLEX REACTION MECHANISMS

corresponds to the Freundlich adsorption isotherm which describes the experimental data of Drazer and Zanette.

is the incomplete gamma function. In eq. (12.52), for physical consistency, we have to assume that

These restrictions must be introduced in order to ensure the nonnegativity and the normalization to unity of the probability density of trapping times. We shall see later that restrictions (12.55) are fulfilled by the experimental data reported by Drazer and Zanette.

By applying eqs. (12.4) and (12.53)–(12.54) we can evaluate the numerical values of the scaling exponents β± entering the linear energy relations (12.31):

It follows that the adsorption rate is constant for all adsorption sites, irrespective of the value of the adsorption heat. The desorption rate depends on the adsorption heat in a simple way: up to a constant additive factor, the activation energy of the desorption process is equal in magnitude to the adsorption heat of the site considered. The activation energy profiles for the adsorption–desorption process have the form represented schematically

Fig. 12.1 Schematic representation of the activation energy profiles for the adsorption-desorption process studied in the experiments of Drazer and Zanette [17]. The different activation energy profiles correspond to adsorption sites characterized by different adsorption heats. All energy profiles

start from the same value Uliquid , which corresponds to the liquid phase,

increase up to the same maximum value Umax , and then decrease to

various final values Usite(1), Usite(2), Usite(3), . . ., corresponding to surface

sites characterized by various adsorption heats. (From [6].)

in fig. 12.1, which represents different energy profiles for the adsorption–desorption process on sites characterized by different adsorption heats. All energy profiles start from the same energy value Uliquid corresponding to a molecule in the liquid phase, and go up to the same maximum value Umax . The descending parts of the activation energy profiles are different for sites with different adsorption heats. The difference between the energy Usite(j ), corresponding to the surface site of type j , and the characteristic energy Uliquid corresponding to a molecule in the liquid phase, is equal to the variation of enthalpy during the adsorption process on the site j :

The corresponding adsorption heat is equal to the variation of enthalpy with changed sign:

The activation energy of the adsorption process is equal to the difference between the maximum value of the energy, Umax , corresponding to the top of the various energy profiles, and the characteristic energy Uliquid , corresponding to a molecule in the liquid:

Similarly, the activation energies E−(j ) of the desorption processes corresponding to the various sites of the surface are given by