Lecture 4

Fluid-solid catalytic reactions: Rate-Limiting Step

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The Rate-Limiting Step

When heterogeneous reactions are carried out at steady state, the rates of each of the three reaction
steps in series (adsorption, surface reaction, and desorption) are equal to one another:

However, one particular step in the series is usually found to be rate-limiting or rate-
controlling. That is, if we could make this particular step go faster, the entire reaction
would proceed at an accelerated rate. Consider the analogy to the electrical circuit shown in
Figure 11.

Fig. 11. Electrical analog to heterogeneous reactions

A given concentration of reactants is analogous to a given driving force or electromotive

force (EMF). The current I (with units of Coulombs/s) is analogous to the rate of reaction,
-r’A (mol/s·g cat), and a resistance Ri is associated with each step in the series. Because the
resistances are in series, the total resistance is just the sum of the individual resistances,
for adsorption (RAD ), surface reaction (RS), and desorption (RD ). The current, I, for a given
voltage, E, is

Since we observe only the total resistance, Rtot, it is our task to find which resistance is much
larger (say, 100 Ω) than the other two resistances (say, 0.1 Ω). Thus, if we could lower the
largest resistance, the current I (i.e., -r’A) , would be larger for a given voltage, E.
Analogously, we want to know which step in the adsorption-reaction-desorption series is
limiting the overall rate of reaction.

The approach in determining catalytic and heterogeneous mechanisms is usually termed the
Langmuir-Hinshelwood approach, since it is derived from ideas proposed by Hinshelwood
based on Langmuir's principles for adsorption. The Langmuir-Hinshelwood approach was
popularized by Hougen and Watson and occasionally includes their names. It consists of first
assuming a sequence of steps in the reaction. In writing this sequence, one must choose among
such mechanisms as molecular or atomic adsorption, and single- or dual-site reaction. Next, rate
laws are written for the individual steps as shown in the preceding section, assuming that
all steps are reversible. Finally, a rate-limiting step is postulated, and steps that are not rate-
limiting are used to eliminate all coverage-dependent terms. The most questionable assumption
in using this technique to obtain a rate law is the hypothesis that the activity of the surface
is essentially uniform as far as the various steps in the reaction are concerned.

Analysis of the rate law suggests that CO2 and N2 are weakly adsorbed, i.e., have
infinitesimally small adsorption constants.
Where Are We Heading?

One of the tasks of a chemical reaction engineer is to analyze rate data and to develop a
rate law that can be used in reactor design. Rate laws in heterogeneous catalysis seldom follow
power law models and hence are inherently more difficult to formulate from the data. To develop
an in-depth understanding and insight as to how the rate laws are formed from heterogeneous
catalytic data, we are going to proceed in somewhat of a reverse manner than what is
normally done in industry when one is asked to develop a rate law. That is, we will postulate
catalytic mechanisms and then derive rate laws for the various mechanisms. The mechanism will
typically have an adsorption step, a surface reaction step, and a desorption step, one of
which is usually rate-limiting. Suggesting mechanisms and rate-limiting steps is not the first
thing we normally do when presented with data. However, by deriving equations for
different mechanisms, we will observe the various forms of the rate law one can have in
heterogeneous catalysis. Knowing the different forms that catalytic rate equations can take, it
will be easier to view the trends in the data and deduce the appropriate rate law. This
deduction is usually what is done first in industry before a mechanism is proposed. Knowing the
form of the rate law, one can then numerically evaluate the rate law parameters and postulate
a reaction mechanism and rate-limiting step that are consistent with the rate data. Finally, we use
the rate law to design catalytic reactors. This procedure is shown in Figure 12. The dashed lines
represent feedback to obtain new data in specific regions (e.g., concentrations, temperature) to
evaluate the rate law parameters more precisely or to differentiate between reaction mechanisms.

Fig. 12. Collecting information for catalytic reactor design

Synthesizing rate law, mechanism and rate limiting step

We now wish to develop rate laws for catalytic reactions that are not diffusion-limited. In
developing the procedure to obtain a mechanism, a rate-limiting step, and a rate law consistent
with experimental observation, we shall discuss a particular catalytic reaction, the
decomposition of cumene to form benzene and propylene. The overall reaction is
A conceptual model depicting the sequence of steps in this platinum-catalyzed reaction is
shown in Figure 13. Figure 13 is only a schematic representation of the adsorption of cumene.

Fig. 13. Sequence of steps in a reaction-limited catalytic reaction

A more realistic model is the formation of a complex of the  orbitals of benzene with the catalytic
surface, as shown in Figure 14.

Fig. 14. -orbital complex on surface

The nomenclature will be used to denote the various species in this reaction is: C = cumene,
B = benzene, and P = propylene. The reaction sequence for this decomposition is shown below.

(R.1)

(R.2)

(R.3)

Reactions (R1) through (R3) represent the mechanism proposed for this reaction.

When writing rate laws for these steps, we treat each step as an elementary reaction; the
only difference is that the species concentrations in the gas phase are replaced by their
respective partial pressures: Cc → Pc
There is no theoretical reason for this replacement of the concentration, Cc, with the partial
pressure, Pc; it is just the convention initiated in the 1930s and used ever since. Fortunately,
Pc can be calculated directly from Cc using the ideal gas law (i.e., Pc = CcRT).

The rate expression for the adsorption of cumene as given in Equation (10-22) is

Rearranging,
 (1)

If rAD has units of (mol/g cat·s) and Cc·s has units of (mol cumene adsorbed/g cat), then typical
units of kA, k-A, and KC would be

The rate law for the surface reaction step producing adsorbed benzene and propylene in the
gas phase,

is

(2)
with the surface reaction equilibrium constant being

Typical units for ks and Ks are s-1 and kPa, respectively.

Propylene is not adsorbed on the surface. Consequently, its concentration on the surface is zero.

The rate of benzene desorption [see Reaction (R3)] is

(3)
Typical units of kD and KDB are s-1 and kPa, respectively. By viewing the desorption of benzene,

from right to left, we see that desorption is just the reverse of the adsorption of benzene.
Consequently, as mentioned earlier, it is easily shown that the benzene adsorption equilibrium
constant KB is just the reciprocal of the benzene desorption constant KDB:

and Equation (3) can be written as

(4)
Because there is no accumulation of reacting species on the surface, the rates of each step
in the sequence are all equal:
 (5)

For the mechanism postulated in the sequence given by Reactions (R1) through (R3), we
wish to determine which step is rate-limiting. We first assume one of the steps to be rate-
limiting (rate-controlling) and then formulate the reaction rate law in terms of the partial
pressures of the species present. From this expression we can determine the variation of
the initial reaction rate with the initial total pressure. If the predicted rate varies with pressure
in the same manner as the rate observed experimentally, the implication is that the assumed
mechanism and rate-limiting step are correct.

Is the Adsorption of Cumene Rate-Limiting?

To answer this question we shall assume that the adsorption of cumene is indeed rate-
limiting, derive the corresponding rate law, and then check to see if it is consistent with
experimental observation. By postulating that this (or any other) step is rate-limiting, we are
assuming that the reaction rate constant of this step (in this case kA) is small with respect
to the specific rates of the other steps (in this case kS and kD). The rate of adsorption is

(1)
Because we can measure neither Cv or CC·S, we must replace these variables in the rate law with
measurable quantities for the equation to be meaningful.

For steady-state operation we have

(5)
For adsorption-limited reactions, kA is very small and kS and kD are very large. Consequently,
the ratios rs/kS and rD/kD are very small (approximately zero), whereas the ratio rAD/kA is
relatively large.

The surface reaction rate law is

(6)

Again, for adsorption-limited reactions, the surface-specific reaction rate kS is large by

comparison, and we can set

and solve Equation (6) for CC·S:

(7)
To be able to express CC·S solely in terms of the partial pressures of the species present,
we must evaluate CB·S. The rate of desorption of benzene is
(7a)
However, for adsorption-limited reactions, kD is large by comparison, and we can set
and then solve Equation (4) for CB·S:
(8)
After combining Equations (7) and (8), we have

(9)
Replacing CC·S in the rate equation by Equation (9) and then factoring Cv, we obtain

(10)
We observe that at equilibrium rAD = 0 and Equation (10) rearranges to

We also know from thermodynamics that for the reaction

also at equilibrium (-r’C = 0) we have the following relationship for partial pressure
equilibrium constant KP:

Consequently, the following relationship must hold

(11)
The equilibrium constant can be determined from thermodynamics data and is related to the
change in the Gibbs free energy, Go, by the equation:
(12)
where R is the ideal gas constant and T is the absolute temperature.

The concentration of vacant sites, Cv, can now be eliminated from Equation (10) by utilizing the
site balance to give the total concentration of sites, Ct which is assumed constant:

Total sites= Vacant sites+ Occupied sites

Because cumene and benzene are adsorbed on the surface, the concentration of occupied sites is
(CC·S + CB·S), and the total concentration of sites is
(13)
Substituting Equations (8) and (9) into Equation (13), Solving for Cv, we have

(14)
Combining Equations (14) and (10), we find that the rate law for the catalytic decomposition
of cumene, assuming that the adsorption of cumene is the rate-limiting step, is
 (15)

We now wish to sketch a plot of the initial rate of reaction as a function of the partial
pressure of cumene, PC0· Initially, no products are present; consequently, Pp = P B = 0. The
initial rate is given by

(16)
If the cumene decomposition is adsorption rate limited, then the initial rate will be linear
with the initial partial pressure of cumene, as shown in Figure 15.

Fig. 15. Adsorption-limited reaction

Before checking to see if Figure 15 is consistent with experimental observation, we shall

derive the corresponding rate laws for the other possible rate-limiting steps and then develop
initial rate plots for the case when the surface reaction is rate-limiting and then for the case
when the desorption of benzene is rate-limiting.

Is the Surface Reaction Rate-Limiting?

The rate of surface reaction is

Since we cannot readily measure the concentrations of the adsorbed species, we must utilize
the adsorption and desorption steps to eliminate CC·S and CB·S from this equation.

From the adsorption rate expression in Equation (1) and the condition that kA and kD are
very large by comparison with kS when surface reaction is limiting (i .e. r AD/kA ≈ 0), we
obtain a relationship for the surface concentration for adsorbed cumene:

In a similar manner, the surface concentration of adsorbed benzene can be evaluated from the
desorption rate expression [Equation (3)] together with the approximation:

Substituting for CB·S and CC·S in Equation (2) gives us

where the thermodynamic equilibrium constant was used to replace the ratio of surface reaction
and adsorption constants, i.e.

The only variable left to eliminate is Cv:

Ct = Cv + CC·S + CB·S

Substituting for concentrations of the adsorbed species, CB·S and CC·S yields

(17)
The initial rate of reaction is

(18)
Figure 16 shows the initial rate of reaction as a function of the initial partial pressure of cumene
for the case of surface reaction limiting.

Fig. 16. Surface-reaction-limited.

At low partial pressures of cumene

and we observe that the initial rate will increase linearly with the initial partial pressure of
cumene:

At high partial pressures

and Equation (10-45) becomes

and the initial rate is independent of the initial partial pressure of cumene.
Is the Desorption of Benzene Rate-Limiting?

The rate expression for the desorption of benzene is

(3)

From the rate expression for surface reaction, Equation (2), we set

to obtain

(19)
Similarly, for the adsorption step, Equation (1), we set

to obtain

then substitute for CC·S in Equation (19) to obtain

(20)
Combining Equations (3) and (20) gives us

where KC is the cumene adsorption constant, KS is the surface reaction equilibrium constant,
and KP is the thermodynamic gas-phase equilibrium constant [Equation (10-38)] for the
reaction. The expression for Cu is obtained from a site balance:

Site balance: Ct = Cv + CC·S + CB·S

After substituting for the respective surface concentrations, we solve the site balance for
Cv:

(21)
Replacing Cv in Equation (20) by Equation (21) and multiplying the numerator and
denominator by PP, we obtain the rate expression for desorption control:

(22)
To determine the dependence of the initial rate of reaction on the initial partial pressure of cumene,
we again set PP = PB = 0; and the rate law reduces to

with the corresponding plot of – r’CO = 0 shown in Figure 17. If desorption were rate limiting,
we would see that the initial rate of reaction would be independent of the initial partial
pressure of cumene.

Fig. 17. Desorption-limited reaction

Summary of the Cumene Decomposition

The experimental observations of -rCO as a function of PCO shows that the rate law derived by
assuming that the surface reaction is rate-limiting agrees with the experimental data.