188 DETERMINATION OF COMPLEX REACTION MECHANISMS

of these random variable in the particular case of a linear reaction sequence operated in stationary conditions. Within the framework of our approach, the stochastic dependence among the various lifetimes is generally non-Markovian, and a simple relationship between the average transit time θ and the average lifetimes τ1 , . . . , τq+1 similar to Easterby’s equation (12.102) does not exist.

In the particular case of a stationary regime, the stochastic dependence among the random variables attached to the different carriers becomes Markovian and our formalism becomes much simpler, and we can express the transit time corresponding to a state u, θ u , in terms of the lifetimes of the intermediates corresponding to the different pathways:

∞

θ

u

=

τ

uu

+

· · ·

τ

uuq

+ · · · +

τ

uu

(12.104)

u

0

u

q 1 uq

=

0

Equation (12.104) is a generalization of Easterby’s equation (12.102). In the particular case of a linear reaction sequence of the type (12.101), eq. (12.104) reduces to the original Easterby relation (12.102).

In conclusion, we have introduced a “neutral” type of linear response experiment for nonlinear kinetics involving multiple reaction intermediates. We have shown that the susceptibility functions from the response equations are given by the probability densities of the transit time in the system. We have shown that a transit time is a sum of different lifetimes corresponding to different reaction pathways, and that in the particular case of a time-invariant system our definition of the transit time is consistent with Easterby’s definition [23].

12.4Transit Time Distributions, Linear Response, and Extracting Kinetic Information from Experimental Data

In order to extract kinetic information from tracer experiments, we express the response theorem (12.82) in terms of the probability density ϕuu (θ; t) (eq. (12.91)). We have

βu (t) = u S 1	0	∞	ϕuu (θ; t) αu (t − θ) dθ	(12.105)

=

We consider two different types of response experiments: (1) transient experiments, for which the excitation functions αu (t) have the form of a unitary impulse or of a Heaviside step function; (2) frequency response experiments, for which the excitation functions αu (t) are periodic.

An example of the first type of transient response experiment consists in the generation of an impulse in the input channel u0, at different initial times t0 = t0(1), t0(2), . . . :

αu (t) = Au0 δuu0 δ (t − t0)

(12.106)

where Au0 is an amplitude factor. According to eq. (12.105) the response to the excitation (12.106) is

βu (t)|t0 = Au0 ϕuu0 (t − t0; t)

(12.107)

LIFETIME AND TRANSIT TIME DISTRIBUTIONS

189

from which we obtain the following expression for the probability density ϕuu0 (θ ; t):

ϕuu0 (θ ; t) =	βu (t)|t0	=t−θ	(12.108)
	Au0

We notice that, in order to evaluate the probability density ϕuu0 (θ ; t), it is necessary to carry out many response experiments, for which the excitation of the system takes place at different initial times t0 = t0(1), t0(2), . . . . The method simplifies considerably if the system is operated in a stationary regime. In this case, owing to the property of temporal invariance, we have

βu (t)|t0		= βu (t − t0)		(12.109)
and (12.108) reduces to
ϕuu0 (θ) =	βu (θ)	,	independent of t0	(12.110)
	Au0

Here a single set of response experiments is enough for the evaluation of the probability density ϕuu0 (θ ; t) of the transit time.

The generation of an impulse with the shape of a Dirac delta function is a difficult experimental problem. It is usually easier to generate a response having the shape of a

step function of the type
αu (t) = Bu0 δuu0 h (t − t0)	(12.111)

where Bu0 is another amplitude factor and h(t − t0) is the unitary Heaviside step function. The response to the excitation (12.111) is

t−t0

βu(t)|t0 = Bu0 ϕuu0 (x; t)dx (12.112)

0

from which we obtain the following expressions for the probability density ϕuu0 (θ; t):

ϕuu0	(θ; t) = − Bu0		∂t0		*βu(t)|t0			+ t0 =t−θ	(12.113)
	1			∂
In particular, if the system is operated in a stationary regime,									the equations become
simpler:
	ϕuu0 (θ) =		1			∂			(12.114)
							βu(θ)
			Bu0			∂θ

For a typical frequency response experiment the excitation function αu (t) is a periodic function characterized by a single harmonic, corresponding to the carrier u0 :

		αu (t) = * αu0 + αu0 exp(i t)+ δu u0		(12.115)
			and αu are the temporal average and
where is a characteristic frequency, and αu
the amplitude of the excitation function, respectively. The response is given by
βu (t) =		uu0 ( = 0; t) + αu0	uu0 ( ; t) exp (i t)	(12.116)
	αu0

190 DETERMINATION OF COMPLEX REACTION MECHANISMS

where

∞

uu ( ; t) = ϕuu (θ ; t) exp (−i θ) dθ (12.117)

0

is a frequency-dependent susceptibility function given by the one-sided Fourier transform of the transit time distribution ϕuu (θ ; t). The frequency response of the system, expressed by eqs. (12.116)–(12.117), can be rather complicated: for example, if the underlying kinetics of the process generates a stable limit cycle, it is made up of the superposition of two different oscillatory processes, an external oscillation given by the excitation and an internal oscillation, represented by the complex susceptibilityuu ( ; t), which, like the transit time probability density, ϕuu (θ ; t), is a periodic function of time. The interaction between these two oscillations may lead to interesting effects [2].

If the experimental data are sufficiently accurate, then from the susceptibility function uu ( ; t) it is possible to evaluate the transit time probability density, ϕuu (θ ; t), by means of numerical inverse Fourier transformation. If the experimental data are not very accurate, then it is still possible to evaluate the first two or three moments and cumulants of the transit time. Since the mean transit time is nonnegative, it follows that ϕuu (θ < 0; t) = 0, and thus the characteristic function of the probability density ϕuu (θ ; t) is identical with the susceptibility function in the frequency domain,

uu ( ; t), experimentally accessible from eqs. (12.116)–(12.117). It follows that the moments < θm(t) >u, m = 1, 2, . . . , and the cumulants θm(t) u, m = 1, 2, . . . , of the transit time can be evaluated by computing the derivatives of uu ( ; t) and ln uu ( ; t) for zero frequency, respectively:

		S	dm
θm (t) u	= im			uu ( = 0; t),	m = 1, 2, . . .	(12.118)
		u =1	d m
		S	dm
θm (t) u	= im			ln uu ( = 0; t),	m = 1, 2, . . .	(12.119)
		u =1	d m

The qualitative and quantitative analysis of the probability densities of the transit time may lead to interesting hints concerning the reaction mechanism and the kinetics of the process. This is an important problem that needs further development. Here we present only a simple illustration based on the study of the response of a time-invariant (stationary) system.

In general, in the response law (12.105) the nonlinearity of the underlying kinetics leads to a nonstationary, time-dependent regime, for which the delay functions χuu (t, t ) depend on two different times. In this case, the investigation of the reaction mechanisms and the kinetics is rather difficult. However, if the chemical process can be operated in a stationary regime, then the response experiment can be described by stationary time series and the delay function χuu (t, t ) depends only on the difference, t − t , of the entrance and exit times:

χuu

t, t = χuu

t − t = ϕuu (θ )

(12.120)

where ϕuu (θ) is the probability density of the transit time θ = t − t . For a stationary process the probability density of the transit time ϕuu (θ ) can be expressed in terms

LIFETIME AND TRANSIT TIME DISTRIBUTIONS

191

of a stationary transport matrix K = |Kuu | , where Kuu is the rate of transport of a molecular fragment from a carrier u to a carrier u . In this case the investigation of reaction mechanisms by means of tracer experiments is much simpler.

The qualitative examination of the shape of the response curve may lead to interesting conclusions concerning the reaction mechanism. A simple procedure consists in the examination of the shape of a log-log plot of the effective decay rate keff (θ) = −∂ ln ϕuu (θ)/∂θ versus the transit time, log keff (θ) versus log θ. The number of horizontal regions displayed by such a log-log plot gives a limit for the minimum number of reaction steps involved in the process. Such a method works only if the time scales corresponding to the different reaction steps have different orders of magnitude. An alternative approach, which works if the time scales of the various chemical steps are

close to each other, is based on the fitting of the delay function χ			t	−	t	=	ϕ
by a superposition of exponential factors:		uu						uu (θ )
	Am exp (−λmθ )
ϕuu (θ) =							(12.121)

m

The minimum number of exponential factors in eq. (12.121) necessary for fitting the experimental data provides a lower limit for the number of chemical steps involved in the process.

Our method makes it possible to use simple linear rules for exploring complicated nonlinear systems. A simple application is the study of connectivity among various chemical species in complicated reaction networks. In the simple case of homogeneous systems with time-invariant structure, the susceptibility matrix χ = [χuu ] = χ (τ ) depends only on the transit time and not on time itself. The matrix elements χuu (τ ) are proportional to the elements Guu (τ ) of a Green function matrix G (τ ) = [Guu (τ )] , which is the exponential of a connectivity matrix K, that is, G (τ ) = exp [τ K] . It follows that from a response experiment involving a system with time-invariant structure, it is possible to evaluate the connectivity matrix, K, which contains information about the relations among the different chemical species involved in the reaction mechanism. The nondiagonal elements of the matrix K = |Kuu | show whether in the reaction mechanism there is a direct connection between two species; in particular, if Kuu = 0, there is a connection from the species u to the species u ; the reverse connection, from u to u, exists if Ku u = 0.

Important information about the reaction kinetics and mechanisms can be obtained by investigating the concentration dependence of the fitted eigenvalues λm from (12.121). This dependence can be extremely complicated. In order to circumvent this complication, we suggest to analyze the concentration dependence of a set of the tensor invariants [24] constructed from the eigenvalues λm:

w =	· · ·	λv1 · · · λvw	(12.122)
v1		vw

The reaction invariants depend on the concentrations of the different species present in the system. In most cases the dependence w = w (concentrations) can be expressed by a polynomial, where the coefficients of the concentrations depend on the rate coefficients of the process. Starting from different assumed reaction mechanisms, we can