2430 J. Phys. Chem. B, Vol. 109, No. 6, 2005
Reyes et al.
at zero frequency, i.e., H0. The latter quantity is a function only
of equilibrium properties of the unperturbed vapor phase at the
beginning of a run, and is directly obtainable from the theory
via eq 40. Although H0 depends on KS, the dependence is very
weak for the range of KS values characteristic of the system,
and for all practical purposes, H0 is determined solely by the
parameters γ and R. Therefore, very good estimates of H0 can
be obtained without any information on the values of KS
obtained by the fitting of the frequency response data. The fact
that such estimates yield an excellent prediction of the transfer
function at low frequencies provides support for the conclusion
that the pronounced attenuation of the magnitude of the transfer
function over the range of frequencies investigated is due to
the interconversion of methylcyclohexane, toluene, and hydro-
gen over the catalyst, rather than to some other process. We
call attention to the fact that eq 40 for H0 is derived strictly
from a consideration of the equilibria of the system.
As can be seen from the values of the parameters given in
Table 2, the values of kII are much smaller than those of kI.
Therefore, values of kR are included in Table 2. This indicates
that step II of the proposed two-step process for the intercon-
version of the reaction species is a much slower process than
the process designated as step I. It is therefore reasonable to
assume that step I is equilibrated for all values of the angular
frequency ω smaller in magnitude than kII, and for values of ω
substantially larger than kII as well, so long as the latter values
of ω remain sufficiently small compared to kI. Equations 52
and 58 are then applicable, and, because they simplify the
analysis, are very useful in the analysis of the data over a certain
range of frequencies. For values of ω comparable in magnitude
to kI, or for all larger values, eqs 59 and 60 become applicable
and find use in the analysis of data at the higher frequencies.
net rate of reaction, i.e., the difference in the rates of the
processes occurring in the two directions. Sufficiently close to
equilibrium, the net rate is first order in the distance from
equilibrium. The rate parameter characterizing this proportional-
ity is a relaxation frequency obtainable by a frequency response
method involving a very small volume perturbation of the
system. Since the perturbation frequency was variable over a
sufficiently wide range, it was possible to obtain relaxation
frequencies for both of the steps comprising the postulated two-
step process. From features observed in plots showing the
frequency dependence of the magnitude of the transfer function
and the phase angle obtained from the frequency response
measurements, a direct indication of the relative rates of the
two steps is obtained from simple inspection of the plots. This
illustrates the power of the approach, since such direct informa-
tion is not obtainable from conventional kinetic studies. In effect,
the frequency response measurements provide us with a
spectrum of relaxation frequencies. In the particular reaction
described in this paper, two characteristic relaxation frequencies
differing by almost 2 orders of magnitude were identified. Via
a postulated two-step sequence for the reaction, they were
identified with a very rapid hydrogenation-dehydrogenation step
and a much slower toluene adsorption-desorption step.
An important feature of the present investigation is the
development of a phenomenological theory describing the
frequency dependence of the transfer function and phase angle
obtained from the frequency response measurements. No
consideration is given to details of the kinetics of the individual
steps in terms of rate constants, adsorption equilibrium constants,
surface coverages, etc. Herein lies a limitation, but at the same
time also the strong point, of the theory. The theory is based
entirely on rates expressed in terms of relaxation frequencies
of the two steps and differences between concentrations and
their equilibrium values as a function of time during the
perturbation of the system. Thermodynamic properties including
the equilibrium composition of the unperturbed vapor phase and
a parameter relating to the toluene adsorption equilibrium are
the other quantities entering into the theory. While further
consideration could have been given to kinetic details and the
relation of relaxation frequencies to rate constants of the
individual steps, the determination of rate constants would have
required extensive further investigation of the variation of the
relaxation frequencies as the initial equilibrium composition of
the unperturbed vapor phase was changed over a suitably broad
range. Such studies fell outside the scope of what we intended
to accomplish in the investigation, namely, the demonstration
of the feasibility of a kind of “relaxation frequency spectros-
copy” providing information of a kind different from that
commonly obtained in studies of the kinetics of chemical
transformations on surfaces.
Concluding Remarks
In heterogeneous catalysis it is frequently observed that the
kinetics of a reaction can be treated satisfactorily by treating
the reaction as if it occurred in two steps via a single surface
intermediate, any other intermediates on the active sites being
present at such low concentrations that they may be ignored.
Boudart has referred to such an intermediate as the most
abundant reaction intermediate, the mari.40 It is important to
realize that this intermediate may not be the most abundant
surface species, since there may well be species that do not
participate in the reaction.
Kinetic data in heterogeneous catalysis are commonly ob-
tained in a steady state flow reactor at very low conversion levels
so that reactant concentrations are essentially constant through-
out the reaction zone. If the extent of conversion that is possible
is limited by the reaction equilibrium to any significant degree,
obtaining data at low conversions also serves to minimize any
complication due to the reverse reaction, often eliminating the
need to consider it at all. The rate of the reaction is commonly
investigated as a function of concentrations or partial pressures
of the reactants, with the objective of obtaining one or more
rate constants, and possibly an adsorption equilibrium constant,
in a rate equation representative of a postulated two-step
sequence. From data on the temperature dependence of these
parameters, information on activation energies and heats of
adsorption are obtainable.
Acknowledgment. The authors are delighted to be contrib-
uting in this issue of The Journal of Physical Chemistry B to a
collection of papers honoring Michel Boudart on the occasion
of his 80th birthday. One of us (J.H.S.) has had a continuing
dialogue with Michel on kinetics and catalysis for half a century,
and wishes to acknowledge the great enrichment and stimulation
that has been provided by this interaction.
References and Notes
The investigation described in the present paper is very
different in kind, with the objective of obtaining different kinds
of information regarding the dynamics of the reaction. In place
of studying the reaction far from equilibrium, we investigate
the reaction very near equilibrium with attention directed to the
(1) Debye, P. Polar Molecules; Reinhold Publishing Corporation: New
York, 1929; pp 83-108 (Reprint, Dover Publications: New York).
(2) Gorter, C. J. Paramagnetic Relaxation; Elsevier: New York, 1947.
(3) Dekker: A. J. Solid State Physics; Prentice-Hall: Englewood Cliffs,
NJ, 1957; pp 150-154, 498-521.