182 DETERMINATION OF COMPLEX REACTION MECHANISMS

We notice that, by combining eqs. (12.61)–(12.63), we recover the linear energy relation (12.59), compatible with the experimental data of Drazer and Zanette. We have

E−(j ) = Umax − Usite(j ) = Umax − Uliquid + Uliquid − Usite(j ) = E−0 + ε (12.64)

where

E−0 = Umax − Uliquid = E+ = E+0

(12.65)

In conclusion, the lifetime analysis of the experimental data of Drazer and Zanette provided important clues regarding the kinetics and mechanism of the desorption process. In particular, the knowledge of the fractal exponents of the adsorption isotherm and of the tail of the lifetime distribution was enough to elucidate the shape and structure of the activation energy barrier. We have also shown that the experimental data indicate that the adsorption rate and the adsorption activation energies are constant and only the desorption rate and desorption activation energy are random. The application of the method may involve detailed theoretical developments for different systems; nevertheless we expect that the results are worth the effort.

12.3Transit Time Distributions in Complex Chemical Systems

Response experiments that can be interpreted in terms of lifetime distributions are limited to the study of the response of an excitation of the same species. Because of this constraint the information acquired is local and refers to a single species, which is usually part of a large reaction network. In order to obtain global information about reaction mechanisms and kinetics, we need to involve responses to excitations involving at least two species, preferably more.

We consider a complex chemical system and focus on a set of S species Mu, u = 1, . . . , S, which can carry one or more identical molecular fragments that are unchanged during the process; in the following we refer to these species as “carriers.” For simplicity, in this section we limit ourselves to the case of isothermal, well-stirred, homogeneous systems, for which the concentrations cu = cu (t), u = 1, . . . , S, of the chemicals Mu, u = 1, . . . , S, are space independent and depend only on time. Later on we consider the more complicated case of reaction–diffusion systems. The deterministic kinetic equations of the process can be expressed in the following form:

dc

u

(t)/dt

=

u

−

u

(t)

+

u

;

t)

−

u

;

t) ,

u

=

1, . . . , S

(12.66)

+ (t)

−

ρ+ (c

ρ

− (c

where

u

;

are the rates of formation and consumption of the species Mu,

ρ± (c

t)

respectively, u± (t) are the input and output fluxes of Mu, respectively, and

c (t) = [cu (t)]u=1,...,M

(12.67)

is the composition vector of the system. In general the functions ρu± (c; t) can be found with the mass action law or other kinetic laws. Together with the initial condition

c (t = t0) = c0

(12.68)

the kinetic equations (12.67) determine the time evolution of the concentration vector.

LIFETIME AND TRANSIT TIME DISTRIBUTIONS

183

We denote by zu the number of fragments in the carrier species u. We use the notation

Fu = F (Mu) , u = 1, . . . , S

(12.69)

for a molecular fragment in the carrier Mu. All fragments Fu in different carriers Mu have the same structure; the label u means that a fragment belongs to a given carrier. The concentrations fu (t), u = 1, . . . , S, of the fragments Fu, u = 1, . . . , S, which belong to different carriers are given by

fu (t) = zucu (t) , u = 1, . . . , S (12.70)

We express the kinetic equations (12.66) in terms of the concentrations fu (t), u = 1, . . . , S, of the fragment Fu, u = 1, . . . , S, in the carrier u, resulting in

df

u

(t)/dt

=

J

u

(t)

−

u

+

u

(c

;

t)

−

R

u

;

t) ,

u

=

1, . . . , S (12.71)

+

J

− (t)

R+

− (c

where

;

=

;

=

u

(c

t)

z

u

ρ

u

(c

t) ,

u

1, . . . , S

(12.72)

R±

±

are the rates of formation and consumption of the fragment Fu,

in the carrier u =

1, . . . , S, and

J

±

(t)

=

z

u

±

(t)

(12.73)

u

u

are input and output fluxes of fragment Fu, in the carrier u = 1, . . . , S, respectively.

In the system a fragment is transferred from one carrier to another. These transfer processes involving fragments Fu, u = 1, . . . , S, among different carriers can be formally represented as:

Ru u	, u, u = 1, . . . , S	(12.74)
Fu −−−−−−→ Fu , u = u
←−−−−−−

Ruu

where Ru u = Ru u (c; t) is the rate of transport of the fragment Fu from a carrier Mu into a carrier Mu . The rates Ru u are related to the formation and consumption rates Ru± (c; t) of the fragment Fu in the carrier Mu by means of the balance equations

S

S

u

(c

;

uu

(c

;

u

;

=

u u

(c

;

=

R+

t)

R

t) ,R− (c

t)

R

t) ,

u

1, . . . , S

(12.75)

u =u

u =u

If the mechanism and kinetics of the process are known, then the transformation rates Ru u = Ru u (c; t) can be evaluated by using the mass action law or other kinetic laws.

Now we consider a kinetic tracer experiment by assuming that a fraction αu, u = 1, . . . , S, of the “in” flux Ju+ (t) of the fragment Fu is replaced by a labeled fragment Fu, and assume that there is no kinetic isotope effect, that is, the rates of the processes involving labeled compounds are the same as the rates of the processes involving unlabeled chemicals. We assume that the fractions βu, u = 1, . . . , S, of the labeled fragments in the output fluxes Ju− (t) , u = 1, . . . , S, can be measured experimentally. We are interested in answering the following questions:

1.If the mechanism and kinetics of the process are known, and if the kinetic isotope

effect can be neglected, what is the general relationship between the input fractions αu, u = 1, . . . , S, of labeled fragments and the output fractions βu, u = 1, . . . , S, of labeled fragments?

184DETERMINATION OF COMPLEX REACTION MECHANISMS

2.If the kinetics of the process is known, what are the lifetime probability densities of the fragments from different carriers present in the system and the probability densities of the time necessary for a fragment to cross the system?

3.If the kinetics and the mechanism of the process are unknown, what useful mechanistic and kinetic information can be extracted from the response experiments?

Although basically similar to the single species case presented before, the derivation of a response law for many species is more complicated. In the following, we give only the main balance equations resulting from the application of the assumption that the kinetic properties of the labeled and unlabeled species are the same (neutrality condition), and the resulting response law.

We introduce the specific rates of transport of a fragment from one carrier to another and in and out of the system:

ωuu (t) = Ruu (t)/fu	(12.76)
u± = Ju± (t)/fu	(12.77)

Here ωuu (t) dt = dtRuu (t)/fu can be interpreted as the infinitesimal probability

of transport of a fragment from the carrier Mu to the carrier Mu at a time between t and t + dt; similarly, ±u dt = dtJu± (t)/fu can be interpreted as the infinitesimal probability that a fragment in the carrier Mu enters or leaves the system at a time between t and t + dt, respectively. If the kinetic isotope effect is missing, the rate of exchange of the labeled fragments in the system can be completely expressed in terms of these infinitesimal probabilities. We assume that the time dependences of the total rates Ru u = Ru u (c; t) and Ju± (t) = zu (±u (t) attached to the total amounts of fragments from different carriers, labeled and unlabeled, are not changed during the process. We use the notations fu (t) , u = 1, . . . , S, for the concentrations of labeled fragments and Ju± (t) , u = 1, . . . , S, for the input and output fluxes of labeled fragments, respectively. We use again the kinetic isotope approach in the form suggested by Neiman and Gál [18], from which we can derive the balance equations:

S

u

=

u

−

u

+

uu

(c

;

−

u u

;

df (t)/dt

J

+ (t)

J

−

(t)

R+

t)

R− (c

t)

u =u

S

=

+

− (t) f

(t) f (t)

J

(t)

−

(t)

+

ω

uu

−

ω

u u

(t) f

(t)

u

u

u

u

u

u =u

(12.78)

and

J

−

(t)

=

−

(t) f

(t)

(12.79)

u

u

u

Our purpose is to derive a relationship between the fractions of labeled fragments in the input fluxes:

α	u	(t)	=	J	+	(t)/J	+	(t)	(12.80)
					u		u

and the fractions of labeled fragments in the output fluxes:

β	u	(t)	=	J	−	(t)/J	−	(t)	(12.81)
					u		u

LIFETIME AND TRANSIT TIME DISTRIBUTIONS

185

From eqs. (12.78)–(12.79) we can derive the following response laws for multiple species:

S

t

βu (t) = u =1

−∞ χuu

t; t

αu

t

dt

(12.82)

;

where χuu

t

t

are nonnegative susceptibility functions that fulfill the normalization

condition:

S

t

u =1

−∞ χuu

t; t

dt = 1

(12.83)

t .

Two types of expressions have been derived for the susceptibility functions χuu

t

A first type depends on a set of direct Green functions Guu t, t

=

G t, t

; uu ,

which are the solutions of the matrix differential equation

d

G t, t = K (t) G t, t

with G

t = t , t = I

(12.84)

with

dt

K (t) = (1 − δuu ) ωuu (t) − δuu

u− (t) +

ωu u (t)

1,...,S

(12.85)

u =u

u,u

=

The expressions for the susceptibility functions are:

;

χuu

t; t =

u

fu

(t)

(12.86)

J +

t

Guu

t

t

A second type of expressions depends on a set of adjoint Green functions G

* ˆ

+uu

, which are the solutions of the matrix differential equation

ˆ uu

t, t =

d

G t, t

Gˆ t, t = Kˆ (t) Gˆ t, t

with

Gˆ t

= t , t = I

(12.87)

with

dt

Kˆ (t) =

(1 − δuu ) ωˆ uu (t) − δuu

u+

(t) +

ωˆ u u

(t)

(12.88)

=

u =u

u,u

1,...,S

The expressions for the susceptibility functions are

u+

χ

uu

t, t

=

ˆ uu

;

t

(12.89)

G

t

t

The physical interpretation of the response equations (12.82) is related to a generalization of the lifetime probability densities, the transit time probability densities. We define these probability densities in terms of the following density functions:

ζuj (θ; t) dθ	with	j	0	∞ ζuj (θ; t) dθ = fu (t)	(12.90)

ζuj (θ; t)dθ is the concentration of fragments in the carrier Mu present	in the system
at time t which have entered the system in a carrier molecule of type	j and have a

186 DETERMINATION OF COMPLEX REACTION MECHANISMS

transit (residence) time in the system between θ and θ + dθ. In terms of these density functions, we can introduce various probability densities of the transit time (see [9]). In the following we focus on only one probability of the transit time, which has a direct connection with the response law (12.82):

ζuu

(θ

;

t) dθ

ζuu (θ

;

t) dθ

ϕuu (θ t)dθ

(12.91)

(12.91)

;

=

=

fu (t)

u

0∞ ζuu (θ; t) dθ

with the normalization condition

∞ ϕuu (θ; t) dθ = 1

u

0

(12.92)

ϕuu (θ ; t) dθ is the probability that a fragment that is present in the system at time t, in a known carrier of type u, has a random transit (residence) time between θ and θ + dθ and entered the system in a carrier of random type u .

The density functions ζuj (θ; t) obey the following balance equations:

∂t

+ ∂θ

ζuj (θ ; t) = u =u ωuu (t) ζu j (θ ; t) − u−

(t) + u =u ωu u (t) ζuj (θ; t)

∂

∂

(12.93)

with the boundary conditions

ζ

uj

(θ

=

0

;

t)

=

δ

uj

J +

(t)

(12.94)

u

Equations (12.93)–(12.94) are similar to eqs. (12.9)–(12.10). The computation of the probability densities of the transit time is similar to the computation of lifetime probability densities. We integrate eqs. (12.93)–(12.94) along the characteristics with the initial conditions

			ζuj (θ ; t = t0) = ζuj(0) (θ)								(12.95)
and use eqs. (12.91)–(12.92). After lengthy manipulations we arrive at
	G	uu	(t, t	−	θ ) J	+ (t	−	θ )
ϕuu (θ ; t) =						u			= χuu	t; t = t − θ	(12.96)
				fu (t)

It follows that the probability density ϕuu (θ; t) of the transit time is equal to the sus-

;

;

ceptibility function χuu t

; t

evaluated for an initial time t = t − θ.

Conversely, the

susceptibility function χuu

t

t

is equal to the probability density ϕuu (θ

t) evaluated

for a transit (residence)

time equal to θ

=

t

−

t

:

χuu

t; t = ϕuu

θ = t − t ; t

(12.97)

Equation (12.97) establishes a relationship between the susceptibility function χuu

t; t

and the probability densities ϕ

uu (θ ; t) of the transit time of a fragment

crossing the

system.

In order to clarify the physical meaning of the transit time probability densities ϕuu (θ ; t) , we introduce an additional random variable, the number of transport events

LIFETIME AND TRANSIT TIME DISTRIBUTIONS

187

q from one carrier to another of a given fragment present in the system. We consider a succession of q transport events of a fragment:

u0 → u1 → · · · → uq → u

(12.98)

To each transport event we attach a lifetime, for the transition u0 → u1, the lifetime τ1 of a fragment in the carrier u0, for the transition u1 → u2, the lifetime τ2 of a fragment in the carrier u1, . . . , and finally the lifetime τq+1 of a fragment in the carrier u . The total transit time θ attached to the process (12.98) is simply the sum of all lifetimes attached to the different processes (12.98):

θ = τ1 + · · · + τq+1

(12.99)

It follows that the probability density ϕuu (θ; t) of the trantit time θ at time t can

0

→

1

→ · · · →

q

→

θ −

q+1

taken over

be expressed as a multiple average of a delta function δ

q =1 τq

all possible transitions u

u

u

u, all

*possible

intermediate states

+

u1, . . . , uq , all possible lifetimes τ1, . . . , τq+1, and all possible numbers q of transport events:

ϕuutr. (q, θ; t) =	,δ	θ −	q+1	-	(12.100)
			τq

q =1

Thus, the probability density of transit times can be interpreted as a path sum, which is a discrete analog of a path integral, and which expresses the individual contributions of various successions of transformations of different lengths of the type u0 → u1 →

· · · → uq → u, to the transport of a fragment within the system. Similarly, a random realization of the transit time θ is the sum of the various lifetimes τ1, . . . , τq+1 of the

states u1, . . . , uq , u.

Equation (12.99) looks similar to a relationship derived by Easterby [23] for a linear reaction sequence of n intermediate species (metabolites), I1, . . . , In, operated in stationary conditions:

S → I1 I2 · · · In → P

(12.101)

for which the average transit time θ can be expressed as a sum of average lifetimes τ1 , . . . , τn corresponding to the intermediates I1, . . . , In:

					st
					J
		=		=
	θ	n	τj	n	*Ijst+	(12.102)
		j =1		j =1

where I1st , . . . , Inst are the stationary concentrations of the species I1, . . . , In, J st is the stationary value of the reaction flux along the linear reaction sequence (12.101), and

τ1 , . . . , τn are the average lifetimes of the species I1, . . . , In:

* +.

τj = Ijst J st , j = 1, . . . , n (12.103)

Despite the apparently similar structure between eqs. (12.99) and (12.102), there is a fundamental difference: in eq. (12.99) all quantities are fluctuating; they are random realizations of various different random variables, which are generally correlated, whereas Easterby’s equation (12.102) is a deterministic relation connecting the average values